Question 27875: A Positive Diagonal Quadratic Has an Exact Ordered Gradient

Original question and givens

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).

• G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-gradient-descent

Step-by-step derivation

source roles = [gd-quadratic-gradient]

derived = [[positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention]]

GD ORIGINAL ORIGINAL gradient d*x coordinate0 := mul(1, 2)

GD ORIGINAL ORIGINAL gradient d*x coordinate0 = 2

GD ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(2, -2)

GD ORIGINAL ORIGINAL gradient plus b coordinate0 = 0

GD ORIGINAL ORIGINAL gradient d*x coordinate1 := mul(2, 0)

GD ORIGINAL ORIGINAL gradient d*x coordinate1 = 0

GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(0, 1)

GD ORIGINAL ORIGINAL gradient plus b coordinate1 = 1

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(2, 2)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 4

GD ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(1, 4)

GD ORIGINAL ORIGINAL diagonal times square coordinate0 = 4

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(4, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 2

GD ORIGINAL ORIGINAL linear product coordinate0 := mul(-2, 2)

GD ORIGINAL ORIGINAL linear product coordinate0 = -4

GD ORIGINAL ORIGINAL coordinate objective sum0 := sum(2, -4)

GD ORIGINAL ORIGINAL coordinate objective sum0 = -2

GD ORIGINAL ORIGINAL objective running sum0 := sum(0, -2)

GD ORIGINAL ORIGINAL objective running sum0 = -2

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 := mul(0, 0)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 0

GD ORIGINAL ORIGINAL diagonal times square coordinate1 := mul(2, 0)

GD ORIGINAL ORIGINAL diagonal times square coordinate1 = 0

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(0, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 0

GD ORIGINAL ORIGINAL linear product coordinate1 := mul(1, 0)

GD ORIGINAL ORIGINAL linear product coordinate1 = 0

GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(0, 0)

GD ORIGINAL ORIGINAL coordinate objective sum1 = 0

GD ORIGINAL ORIGINAL objective running sum1 := sum(-2, 0)

GD ORIGINAL ORIGINAL objective running sum1 = -2

source roles = [GD ORIGINAL, [1, 2], [-2, 1], [2, 0]]

derived = [[1, 2], [-2, 1], [2, 0], [0, 1], -2]

Reverse checks

• S018 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S017 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL objective running sum1 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL coordinate objective sum1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL linear product coordinate1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL diagonal times square coordinate1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL objective running sum0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL coordinate objective sum0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL linear product coordinate0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL diagonal times square coordinate0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL gradient plus b coordinate1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL gradient d*x coordinate1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL gradient plus b coordinate0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute GD ORIGINAL ORIGINAL gradient d*x coordinate0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.

Result

([1, 2], [-2, 1], [2, 0], [0, 1], -2, fixed_standard_Euclidean_coordinate_order)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.