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Exact Diagonal-Quadratic Gradient Steps and Strict Safe-Step Traces
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batch-20261015-advance-20261119-r03-gradient-descent-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
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1. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-1, -2], and point x=[-2, -3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([1, 2], [-1, -2], [-2, -3], [-3, -8], 19, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 2], [-1, -2], [-2, -3], [-3, -8], 19, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
2. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[0, -1], and point x=[-1, -2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([2, 3], [0, -1], [-1, -2], [-2, -7], 9, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 3], [0, -1], [-1, -2], [-2, -7], 9, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
3. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[1, 0], and point x=[0, -1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([3, 4], [1, 0], [0, -1], [1, -4], 2, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 4], [1, 0], [0, -1], [1, -4], 2, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
4. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[2, 1], and point x=[1, 0] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([1, 5], [2, 1], [1, 0], [3, 1], 5/2, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 5], [2, 1], [1, 0], [3, 1], 5/2, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
5. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[-2, 2], and point x=[2, 1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([2, 2], [-2, 2], [2, 1], [2, 4], 3, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 2], [-2, 2], [2, 1], [2, 4], 3, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
6. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[-1, 3], and point x=[-2, 2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([3, 3], [-1, 3], [-2, 2], [-7, 9], 20, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 3], [-1, 3], [-2, 2], [-7, 9], 20, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
7. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[0, -3], and point x=[-1, 3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([1, 4], [0, -3], [-1, 3], [-1, 9], 19/2, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 4], [0, -3], [-1, 3], [-1, 9], 19/2, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
8. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[1, -2], and point x=[0, -3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([2, 5], [1, -2], [0, -3], [1, -17], 57/2, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 5], [1, -2], [0, -3], [1, -17], 57/2, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
9. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[2, -1], and point x=[1, -2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([3, 2], [2, -1], [1, -2], [5, -5], 19/2, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 2], [2, -1], [1, -2], [5, -5], 19/2, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
10. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[-2, 0], and point x=[2, -1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([1, 3], [-2, 0], [2, -1], [0, -3], -1/2, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 3], [-2, 0], [2, -1], [0, -3], -1/2, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
11. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[-1, 1], and point x=[-2, 0] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([2, 4], [-1, 1], [-2, 0], [-5, 1], 6, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 4], [-1, 1], [-2, 0], [-5, 1], 6, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
12. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[0, 2], and point x=[-1, 1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([3, 5], [0, 2], [-1, 1], [-3, 7], 6, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 5], [0, 2], [-1, 1], [-3, 7], 6, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
13. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[1, 3], and point x=[0, 2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([1, 2], [1, 3], [0, 2], [1, 7], 10, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 2], [1, 3], [0, 2], [1, 7], 10, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
14. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[2, -3], and point x=[1, 3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([2, 3], [2, -3], [1, 3], [4, 6], 15/2, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 3], [2, -3], [1, 3], [4, 6], 15/2, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
15. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[-2, -2], and point x=[2, -3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([3, 4], [-2, -2], [2, -3], [4, -14], 26, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 4], [-2, -2], [2, -3], [4, -14], 26, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
16. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[-1, -1], and point x=[-2, -2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([1, 5], [-1, -1], [-2, -2], [-3, -11], 16, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 5], [-1, -1], [-2, -2], [-3, -11], 16, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
17. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[0, 0], and point x=[-1, -1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([2, 2], [0, 0], [-1, -1], [-2, -2], 2, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 2], [0, 0], [-1, -1], [-2, -2], 2, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
18. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[1, 1], and point x=[0, 0] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([3, 3], [1, 1], [0, 0], [1, 1], 0, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 3], [1, 1], [0, 0], [1, 1], 0, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
19. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[2, 2], and point x=[1, 1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([1, 4], [2, 2], [1, 1], [3, 6], 13/2, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 4], [2, 2], [1, 1], [3, 6], 13/2, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
20. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[-2, 3], and point x=[2, 2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([2, 5], [-2, 3], [2, 2], [2, 13], 16, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 5], [-2, 3], [2, 2], [2, 13], 16, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
21. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[-1, -3], and point x=[-2, 3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([3, 2], [-1, -3], [-2, 3], [-7, 3], 8, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 2], [-1, -3], [-2, 3], [-7, 3], 8, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
22. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[0, -2], and point x=[-1, -3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([1, 3], [0, -2], [-1, -3], [-1, -11], 20, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 3], [0, -2], [-1, -3], [-1, -11], 20, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
23. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[1, -1], and point x=[0, -2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([2, 4], [1, -1], [0, -2], [1, -9], 10, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 4], [1, -1], [0, -2], [1, -9], 10, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
24. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[2, 0], and point x=[1, -1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([3, 5], [2, 0], [1, -1], [5, -5], 6, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 5], [2, 0], [1, -1], [5, -5], 6, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
25. A Positive Diagonal Quadratic Has an Exact Ordered Gradient
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-2, 1], and point x=[2, 0] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).
View answer and explanation
([1, 2], [-2, 1], [2, 0], [0, 1], -2, fixed_standard_Euclidean_coordinate_order)
Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 2], [-2, 1], [2, 0], [0, 1], -2, fixed_standard_Euclidean_coordinate_order).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-quadratic-gradient · candidate
26. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-1, -2], and point x=[-2, -3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([-2, -3], [-3, -8], 1/4, [-5/4, -1], 19, 161/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-2, -3], [-3, -8], 1/4, [-5/4, -1], 19, 161/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
27. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[0, -1], and point x=[-1, -2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([-1, -2], [-2, -7], 1/4, [-1/2, -1/4], 9, 19/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, -2], [-2, -7], 1/4, [-1/2, -1/4], 9, 19/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
28. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[1, 0], and point x=[0, -1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([0, -1], [1, -4], 1/4, [-1/4, 0], 2, -5/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([0, -1], [1, -4], 1/4, [-1/4, 0], 2, -5/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
29. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[2, 1], and point x=[1, 0] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([1, 0], [3, 1], 1/4, [1/4, -1/4], 5/2, 7/16, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([1, 0], [3, 1], 1/4, [1/4, -1/4], 5/2, 7/16, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
30. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[-2, 2], and point x=[2, 1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([2, 1], [2, 4], 1/4, [3/2, 0], 3, -3/4, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([2, 1], [2, 4], 1/4, [3/2, 0], 3, -3/4, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
31. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[-1, 3], and point x=[-2, 2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([-2, 2], [-7, 9], 1/4, [-1/4, -1/4], 20, -5/16, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-2, 2], [-7, 9], 1/4, [-1/4, -1/4], 20, -5/16, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
32. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[0, -3], and point x=[-1, 3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
33. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[1, -2], and point x=[0, -3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([0, -3], [1, -17], 1/4, [-1/4, 5/4], 57/2, 39/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([0, -3], [1, -17], 1/4, [-1/4, 5/4], 57/2, 39/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
34. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[2, -1], and point x=[1, -2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([1, -2], [5, -5], 1/4, [-1/4, -3/4], 19/2, 29/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([1, -2], [5, -5], 1/4, [-1/4, -3/4], 19/2, 29/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
35. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[-2, 0], and point x=[2, -1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([2, -1], [0, -3], 1/4, [2, -1/4], -1/2, -61/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([2, -1], [0, -3], 1/4, [2, -1/4], -1/2, -61/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
36. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[-1, 1], and point x=[-2, 0] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([-2, 0], [-5, 1], 1/4, [-3/4, -1/4], 6, 19/16, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-2, 0], [-5, 1], 1/4, [-3/4, -1/4], 6, 19/16, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
37. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[0, 2], and point x=[-1, 1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([-1, 1], [-3, 7], 1/4, [-1/4, -3/4], 6, 0, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, 1], [-3, 7], 1/4, [-1/4, -3/4], 6, 0, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
38. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[1, 3], and point x=[0, 2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([0, 2], [1, 7], 1/4, [-1/4, 1/4], 10, 19/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([0, 2], [1, 7], 1/4, [-1/4, 1/4], 10, 19/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
39. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[2, -3], and point x=[1, 3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([1, 3], [4, 6], 1/4, [0, 3/2], 15/2, -9/8, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([1, 3], [4, 6], 1/4, [0, 3/2], 15/2, -9/8, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
40. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[-2, -2], and point x=[2, -3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([2, -3], [4, -14], 1/4, [1, 1/2], 26, -1, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([2, -3], [4, -14], 1/4, [1, 1/2], 26, -1, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
41. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[-1, -1], and point x=[-2, -2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([-2, -2], [-3, -11], 1/4, [-5/4, 3/4], 16, 43/16, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-2, -2], [-3, -11], 1/4, [-5/4, 3/4], 16, 43/16, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
42. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[0, 0], and point x=[-1, -1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([-1, -1], [-2, -2], 1/4, [-1/2, -1/2], 2, 1/2, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, -1], [-2, -2], 1/4, [-1/2, -1/2], 2, 1/2, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
43. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[1, 1], and point x=[0, 0] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([0, 0], [1, 1], 1/4, [-1/4, -1/4], 0, -5/16, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([0, 0], [1, 1], 1/4, [-1/4, -1/4], 0, -5/16, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
44. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[2, 2], and point x=[1, 1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([1, 1], [3, 6], 1/4, [1/4, -1/2], 13/2, 1/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([1, 1], [3, 6], 1/4, [1/4, -1/2], 13/2, 1/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
45. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[-2, 3], and point x=[2, 2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([2, 2], [2, 13], 1/4, [3/2, -5/4], 16, -19/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([2, 2], [2, 13], 1/4, [3/2, -5/4], 16, -19/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
46. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[-1, -3], and point x=[-2, 3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([-2, 3], [-7, 3], 1/4, [-1/4, 9/4], 8, -43/32, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-2, 3], [-7, 3], 1/4, [-1/4, 9/4], 8, -43/32, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
47. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[0, -2], and point x=[-1, -3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([-1, -3], [-1, -11], 1/4, [-3/4, -1/4], 20, 7/8, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, -3], [-1, -11], 1/4, [-3/4, -1/4], 20, 7/8, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
48. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[1, -1], and point x=[0, -2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([0, -2], [1, -9], 1/4, [-1/4, 1/4], 10, -5/16, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([0, -2], [1, -9], 1/4, [-1/4, 1/4], 10, -5/16, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
49. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[2, 0], and point x=[1, -1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([1, -1], [5, -5], 1/4, [-1/4, 1/4], 6, -1/4, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([1, -1], [5, -5], 1/4, [-1/4, 1/4], 6, -1/4, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
50. Positive Step Size Alone Does Not Guarantee Objective Descent
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-2, 1], and point x=[2, 0] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).
A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee.
View answer and explanation
([2, 0], [0, 1], 1/4, [2, -1/4], -2, -35/16, Yes, positive_step_alone_does_not_guarantee_descent)
Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([2, 0], [0, 1], 1/4, [2, -1/4], -2, -35/16, Yes, positive_step_alone_does_not_guarantee_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-one-step · candidate
51. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-1, -2], and point x=[-2, -3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(2, 1/4, 1, 1, [[0, [-2, -3], 19, [-3, -8]], [1, [-5/4, -1], 161/32, [-9/4, -4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (2, 1/4, 1, 1, [[0, [-2, -3], 19, [-3, -8]], [1, [-5/4, -1], 161/32, [-9/4, -4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
52. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[0, -1], and point x=[-1, -2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(3, 1/4, 2/3, 1, [[0, [-1, -2], 9, [-2, -7]], [1, [-1/2, -1/4], 19/32, [-1, -7/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [-1, -2], 9, [-2, -7]], [1, [-1/2, -1/4], 19/32, [-1, -7/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
53. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[1, 0], and point x=[0, -1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(4, 1/4, 1/2, 1, [[0, [0, -1], 2, [1, -4]], [1, [-1/4, 0], -5/32, [1/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [0, -1], 2, [1, -4]], [1, [-1/4, 0], -5/32, [1/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
54. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[2, 1], and point x=[1, 0] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(5, 1/4, 2/5, 1, [[0, [1, 0], 5/2, [3, 1]], [1, [1/4, -1/4], 7/16, [9/4, -1/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [1, 0], 5/2, [3, 1]], [1, [1/4, -1/4], 7/16, [9/4, -1/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
55. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[-2, 2], and point x=[2, 1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(2, 1/4, 1, 1, [[0, [2, 1], 3, [2, 4]], [1, [3/2, 0], -3/4, [1, 2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (2, 1/4, 1, 1, [[0, [2, 1], 3, [2, 4]], [1, [3/2, 0], -3/4, [1, 2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
56. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[-1, 3], and point x=[-2, 2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(3, 1/4, 2/3, 1, [[0, [-2, 2], 20, [-7, 9]], [1, [-1/4, -1/4], -5/16, [-7/4, 9/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [-2, 2], 20, [-7, 9]], [1, [-1/4, -1/4], -5/16, [-7/4, 9/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
57. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[0, -3], and point x=[-1, 3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(4, 1/4, 1/2, 1, [[0, [-1, 3], 19/2, [-1, 9]], [1, [-3/4, 3/4], -27/32, [-3/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [-1, 3], 19/2, [-1, 9]], [1, [-3/4, 3/4], -27/32, [-3/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
58. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[1, -2], and point x=[0, -3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(5, 1/4, 2/5, 1, [[0, [0, -3], 57/2, [1, -17]], [1, [-1/4, 5/4], 39/32, [1/2, 17/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [0, -3], 57/2, [1, -17]], [1, [-1/4, 5/4], 39/32, [1/2, 17/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
59. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[2, -1], and point x=[1, -2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(3, 1/4, 2/3, 1, [[0, [1, -2], 19/2, [5, -5]], [1, [-1/4, -3/4], 29/32, [5/4, -5/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [1, -2], 19/2, [5, -5]], [1, [-1/4, -3/4], 29/32, [5/4, -5/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
60. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[-2, 0], and point x=[2, -1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(3, 1/4, 2/3, 1, [[0, [2, -1], -1/2, [0, -3]], [1, [2, -1/4], -61/32, [0, -3/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [2, -1], -1/2, [0, -3]], [1, [2, -1/4], -61/32, [0, -3/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
61. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[-1, 1], and point x=[-2, 0] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(4, 1/4, 1/2, 1, [[0, [-2, 0], 6, [-5, 1]], [1, [-3/4, -1/4], 19/16, [-5/2, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [-2, 0], 6, [-5, 1]], [1, [-3/4, -1/4], 19/16, [-5/2, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
62. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[0, 2], and point x=[-1, 1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(5, 1/4, 2/5, 1, [[0, [-1, 1], 6, [-3, 7]], [1, [-1/4, -3/4], 0, [-3/4, -7/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [-1, 1], 6, [-3, 7]], [1, [-1/4, -3/4], 0, [-3/4, -7/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
63. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[1, 3], and point x=[0, 2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(2, 1/4, 1, 1, [[0, [0, 2], 10, [1, 7]], [1, [-1/4, 1/4], 19/32, [3/4, 7/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (2, 1/4, 1, 1, [[0, [0, 2], 10, [1, 7]], [1, [-1/4, 1/4], 19/32, [3/4, 7/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
64. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[2, -3], and point x=[1, 3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(3, 1/4, 2/3, 1, [[0, [1, 3], 15/2, [4, 6]], [1, [0, 3/2], -9/8, [2, 3/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [1, 3], 15/2, [4, 6]], [1, [0, 3/2], -9/8, [2, 3/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
65. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[-2, -2], and point x=[2, -3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(4, 1/4, 1/2, 1, [[0, [2, -3], 26, [4, -14]], [1, [1, 1/2], -1, [1, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [2, -3], 26, [4, -14]], [1, [1, 1/2], -1, [1, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
66. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[-1, -1], and point x=[-2, -2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(5, 1/4, 2/5, 1, [[0, [-2, -2], 16, [-3, -11]], [1, [-5/4, 3/4], 43/16, [-9/4, 11/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [-2, -2], 16, [-3, -11]], [1, [-5/4, 3/4], 43/16, [-9/4, 11/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
67. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[0, 0], and point x=[-1, -1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(2, 1/4, 1, 1, [[0, [-1, -1], 2, [-2, -2]], [1, [-1/2, -1/2], 1/2, [-1, -1]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (2, 1/4, 1, 1, [[0, [-1, -1], 2, [-2, -2]], [1, [-1/2, -1/2], 1/2, [-1, -1]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
68. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[1, 1], and point x=[0, 0] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(3, 1/4, 2/3, 1, [[0, [0, 0], 0, [1, 1]], [1, [-1/4, -1/4], -5/16, [1/4, 1/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [0, 0], 0, [1, 1]], [1, [-1/4, -1/4], -5/16, [1/4, 1/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
69. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[2, 2], and point x=[1, 1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(4, 1/4, 1/2, 1, [[0, [1, 1], 13/2, [3, 6]], [1, [1/4, -1/2], 1/32, [9/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [1, 1], 13/2, [3, 6]], [1, [1/4, -1/2], 1/32, [9/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
70. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[-2, 3], and point x=[2, 2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(5, 1/4, 2/5, 1, [[0, [2, 2], 16, [2, 13]], [1, [3/2, -5/4], -19/32, [1, -13/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [2, 2], 16, [2, 13]], [1, [3/2, -5/4], -19/32, [1, -13/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
71. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[-1, -3], and point x=[-2, 3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(3, 1/4, 2/3, 1, [[0, [-2, 3], 8, [-7, 3]], [1, [-1/4, 9/4], -43/32, [-7/4, 3/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [-2, 3], 8, [-7, 3]], [1, [-1/4, 9/4], -43/32, [-7/4, 3/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
72. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[0, -2], and point x=[-1, -3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(3, 1/4, 2/3, 1, [[0, [-1, -3], 20, [-1, -11]], [1, [-3/4, -1/4], 7/8, [-3/4, -11/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [-1, -3], 20, [-1, -11]], [1, [-3/4, -1/4], 7/8, [-3/4, -11/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
73. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[1, -1], and point x=[0, -2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(4, 1/4, 1/2, 1, [[0, [0, -2], 10, [1, -9]], [1, [-1/4, 1/4], -5/16, [1/2, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [0, -2], 10, [1, -9]], [1, [-1/4, 1/4], -5/16, [1/2, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
74. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[2, 0], and point x=[1, -1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(5, 1/4, 2/5, 1, [[0, [1, -1], 6, [5, -5]], [1, [-1/4, 1/4], -1/4, [5/4, 5/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [1, -1], 6, [5, -5]], [1, [-1/4, 1/4], -1/4, [5/4, 5/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
75. A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace
Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-2, 1], and point x=[2, 0] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).
The safe range is strictly0<alpha<2/L. Records include the initial state and each later state. Finite nonincrease is not an unbounded convergence certificate.
View answer and explanation
(2, 1/4, 1, 1, [[0, [2, 0], -2, [0, 1]], [1, [2, -1/4], -35/16, [0, 1/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)
Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (2, 1/4, 1, 1, [[0, [2, 0], -2, [0, 1]], [1, [2, -1/4], -35/16, [0, 1/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-safe-quadratic-trace · candidate
76. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=oversized_step, complete model={"curvature":1,"point":1,"step_size":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(oversized_step, 1, 1, 3, -2, 1/2, 2, No, positive_but_oversized_step_can_increase_the_objective)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 1, 1, 3, -2, 1/2, 2, No, positive_but_oversized_step_can_increase_the_objective).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
77. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=zero_gradient_nonminimum, complete model={"scale":2}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(zero_gradient_nonminimum, f(x)=-2*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-2*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
78. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=finite_trace_only, complete model={"diagonal":[1,2],"linear":[2,-2],"point":[3,1],"step_size":[1,4],"steps":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(finite_trace_only, 1, [[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
79. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=constrained_confusion, complete model={"point":0,"gradient":4,"step_size":2,"lower":-1,"upper":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(constrained_confusion, -1, 1, 0, -8, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -8, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
80. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=oversized_step, complete model={"curvature":5,"point":5,"step_size":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(oversized_step, 5, 5, 3, -70, 125/2, 12250, No, positive_but_oversized_step_can_increase_the_objective)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 5, 5, 3, -70, 125/2, 12250, No, positive_but_oversized_step_can_increase_the_objective).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
81. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=zero_gradient_nonminimum, complete model={"scale":6}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(zero_gradient_nonminimum, f(x)=-6*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-6*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
82. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=finite_trace_only, complete model={"diagonal":[1,2],"linear":[6,-6],"point":[7,1],"step_size":[1,4],"steps":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(finite_trace_only, 1, [[0, [7, 1], 123/2], [1, [15/4, 2], 689/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [7, 1], 123/2], [1, [15/4, 2], 689/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
83. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=constrained_confusion, complete model={"point":0,"gradient":8,"step_size":2,"lower":-1,"upper":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(constrained_confusion, -1, 1, 0, -16, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -16, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
84. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=oversized_step, complete model={"curvature":4,"point":9,"step_size":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(oversized_step, 4, 9, 3, -99, 162, 19602, No, positive_but_oversized_step_can_increase_the_objective)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 4, 9, 3, -99, 162, 19602, No, positive_but_oversized_step_can_increase_the_objective).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
85. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=zero_gradient_nonminimum, complete model={"scale":10}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(zero_gradient_nonminimum, f(x)=-10*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-10*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
86. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=finite_trace_only, complete model={"diagonal":[1,2],"linear":[10,-10],"point":[11,1],"step_size":[1,4],"steps":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(finite_trace_only, 1, [[0, [11, 1], 323/2], [1, [23/4, 3], 1697/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [11, 1], 323/2], [1, [23/4, 3], 1697/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
87. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=constrained_confusion, complete model={"point":0,"gradient":12,"step_size":2,"lower":-1,"upper":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(constrained_confusion, -1, 1, 0, -24, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -24, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
88. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=oversized_step, complete model={"curvature":3,"point":13,"step_size":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(oversized_step, 3, 13, 3, -104, 507/2, 16224, No, positive_but_oversized_step_can_increase_the_objective)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 3, 13, 3, -104, 507/2, 16224, No, positive_but_oversized_step_can_increase_the_objective).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
89. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=zero_gradient_nonminimum, complete model={"scale":14}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(zero_gradient_nonminimum, f(x)=-14*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-14*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
90. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=finite_trace_only, complete model={"diagonal":[1,2],"linear":[14,-14],"point":[15,1],"step_size":[1,4],"steps":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(finite_trace_only, 1, [[0, [15, 1], 619/2], [1, [31/4, 4], 3153/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [15, 1], 619/2], [1, [31/4, 4], 3153/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
91. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=constrained_confusion, complete model={"point":0,"gradient":16,"step_size":2,"lower":-1,"upper":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(constrained_confusion, -1, 1, 0, -32, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -32, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
92. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=oversized_step, complete model={"curvature":2,"point":17,"step_size":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(oversized_step, 2, 17, 3, -85, 289, 7225, No, positive_but_oversized_step_can_increase_the_objective)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 2, 17, 3, -85, 289, 7225, No, positive_but_oversized_step_can_increase_the_objective).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
93. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=zero_gradient_nonminimum, complete model={"scale":18}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(zero_gradient_nonminimum, f(x)=-18*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-18*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
94. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=finite_trace_only, complete model={"diagonal":[1,2],"linear":[18,-18],"point":[19,1],"step_size":[1,4],"steps":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(finite_trace_only, 1, [[0, [19, 1], 1011/2], [1, [39/4, 5], 5057/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [19, 1], 1011/2], [1, [39/4, 5], 5057/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
95. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=constrained_confusion, complete model={"point":0,"gradient":20,"step_size":2,"lower":-1,"upper":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(constrained_confusion, -1, 1, 0, -40, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -40, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
96. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=oversized_step, complete model={"curvature":1,"point":21,"step_size":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(oversized_step, 1, 21, 3, -42, 441/2, 882, No, positive_but_oversized_step_can_increase_the_objective)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 1, 21, 3, -42, 441/2, 882, No, positive_but_oversized_step_can_increase_the_objective).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
97. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=zero_gradient_nonminimum, complete model={"scale":22}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(zero_gradient_nonminimum, f(x)=-22*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-22*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
98. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=finite_trace_only, complete model={"diagonal":[1,2],"linear":[22,-22],"point":[23,1],"step_size":[1,4],"steps":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(finite_trace_only, 1, [[0, [23, 1], 1499/2], [1, [47/4, 6], 7409/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [23, 1], 1499/2], [1, [47/4, 6], 7409/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
99. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=constrained_confusion, complete model={"point":0,"gradient":24,"step_size":2,"lower":-1,"upper":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(constrained_confusion, -1, 1, 0, -48, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -48, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
100. Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences
Use mode=oversized_step, complete model={"curvature":5,"point":25,"step_size":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).
The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent.
View answer and explanation
(oversized_step, 5, 25, 3, -350, 3125/2, 306250, No, positive_but_oversized_step_can_increase_the_objective)
A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 5, 25, 3, -350, 3125/2, 306250, No, positive_but_oversized_step_can_increase_the_objective).
Reviewed step-by-step derivation
Gradient descent · qf-mod50-gd-boundary · candidate
Canonical question JSONL · English JSONL · Validation report