Regular Equality Constraints, Multiplier Residuals and Necessity Boundaries

([[1, 0], [0, 1]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 2]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 2]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 3]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 3]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 4]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 4]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 5]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 5]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 6]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 6]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 7]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 7]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 8]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 8]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 9]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 9]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 10]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 10]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 11]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 11]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 12]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 12]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 13]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 13]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 14]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 14]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 15]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 15]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 16]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 16]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 17]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 17]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 18]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 18]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 19]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 19]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 20]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 20]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 21]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 21]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 22]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 22]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 23]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 23]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 24]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 24]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([[1, 0], [0, 25]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)

Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 25]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent).

([1, 2], [[1, 0], [0, 1]], [1, 2], [1, 2], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([1, 2], [[1, 0], [0, 1]], [1, 2], [1, 2], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([2, 4], [[1, 0], [0, 1]], [2, 4], [2, 4], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([2, 4], [[1, 0], [0, 1]], [2, 4], [2, 4], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([3, 6], [[1, 0], [0, 1]], [3, 6], [3, 6], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([3, 6], [[1, 0], [0, 1]], [3, 6], [3, 6], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([4, 8], [[1, 0], [0, 1]], [4, 8], [4, 8], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([4, 8], [[1, 0], [0, 1]], [4, 8], [4, 8], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([5, 10], [[1, 0], [0, 1]], [5, 10], [5, 10], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([5, 10], [[1, 0], [0, 1]], [5, 10], [5, 10], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([6, 12], [[1, 0], [0, 1]], [6, 12], [6, 12], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([6, 12], [[1, 0], [0, 1]], [6, 12], [6, 12], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([7, 14], [[1, 0], [0, 1]], [7, 14], [7, 14], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([7, 14], [[1, 0], [0, 1]], [7, 14], [7, 14], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([8, 16], [[1, 0], [0, 1]], [8, 16], [8, 16], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([8, 16], [[1, 0], [0, 1]], [8, 16], [8, 16], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([9, 18], [[1, 0], [0, 1]], [9, 18], [9, 18], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([9, 18], [[1, 0], [0, 1]], [9, 18], [9, 18], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([10, 20], [[1, 0], [0, 1]], [10, 20], [10, 20], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([10, 20], [[1, 0], [0, 1]], [10, 20], [10, 20], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([11, 22], [[1, 0], [0, 1]], [11, 22], [11, 22], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([11, 22], [[1, 0], [0, 1]], [11, 22], [11, 22], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([12, 24], [[1, 0], [0, 1]], [12, 24], [12, 24], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([12, 24], [[1, 0], [0, 1]], [12, 24], [12, 24], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([13, 26], [[1, 0], [0, 1]], [13, 26], [13, 26], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([13, 26], [[1, 0], [0, 1]], [13, 26], [13, 26], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([14, 28], [[1, 0], [0, 1]], [14, 28], [14, 28], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([14, 28], [[1, 0], [0, 1]], [14, 28], [14, 28], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([15, 30], [[1, 0], [0, 1]], [15, 30], [15, 30], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([15, 30], [[1, 0], [0, 1]], [15, 30], [15, 30], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([16, 32], [[1, 0], [0, 1]], [16, 32], [16, 32], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([16, 32], [[1, 0], [0, 1]], [16, 32], [16, 32], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([17, 34], [[1, 0], [0, 1]], [17, 34], [17, 34], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([17, 34], [[1, 0], [0, 1]], [17, 34], [17, 34], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([18, 36], [[1, 0], [0, 1]], [18, 36], [18, 36], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([18, 36], [[1, 0], [0, 1]], [18, 36], [18, 36], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([19, 38], [[1, 0], [0, 1]], [19, 38], [19, 38], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([19, 38], [[1, 0], [0, 1]], [19, 38], [19, 38], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([20, 40], [[1, 0], [0, 1]], [20, 40], [20, 40], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([20, 40], [[1, 0], [0, 1]], [20, 40], [20, 40], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([21, 42], [[1, 0], [0, 1]], [21, 42], [21, 42], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([21, 42], [[1, 0], [0, 1]], [21, 42], [21, 42], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([22, 44], [[1, 0], [0, 1]], [22, 44], [22, 44], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([22, 44], [[1, 0], [0, 1]], [22, 44], [22, 44], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([23, 46], [[1, 0], [0, 1]], [23, 46], [23, 46], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([23, 46], [[1, 0], [0, 1]], [23, 46], [23, 46], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([24, 48], [[1, 0], [0, 1]], [24, 48], [24, 48], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([24, 48], [[1, 0], [0, 1]], [24, 48], [24, 48], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([25, 50], [[1, 0], [0, 1]], [25, 50], [25, 50], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)

Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([25, 50], [[1, 0], [0, 1]], [25, 50], [25, 50], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda).

([0, 0], [1, 1], 3, 3/2, [3/2, 3/2], 3, [3, 3], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([0, 0], [1, 1], 3, 3/2, [3/2, 3/2], 3, [3, 3], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([1, 1], [1, 2], 4, 1/5, [6/5, 7/5], 2/5, [2/5, 4/5], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([1, 1], [1, 2], 4, 1/5, [6/5, 7/5], 2/5, [2/5, 4/5], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([2, 2], [1, 3], 5, -3/10, [17/10, 11/10], -3/5, [-3/5, -9/5], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([2, 2], [1, 3], 5, -3/10, [17/10, 11/10], -3/5, [-3/5, -9/5], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([3, 3], [1, 4], 6, -9/17, [42/17, 15/17], -18/17, [-18/17, -72/17], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 3], [1, 4], 6, -9/17, [42/17, 15/17], -18/17, [-18/17, -72/17], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([4, 4], [1, 5], 7, -17/26, [87/26, 19/26], -17/13, [-17/13, -85/13], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([4, 4], [1, 5], 7, -17/26, [87/26, 19/26], -17/13, [-17/13, -85/13], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([0, 5], [1, 6], 8, -22/37, [-22/37, 53/37], -44/37, [-44/37, -264/37], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([0, 5], [1, 6], 8, -22/37, [-22/37, 53/37], -44/37, [-44/37, -264/37], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([1, 6], [1, 7], 9, -17/25, [8/25, 31/25], -34/25, [-34/25, -238/25], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([1, 6], [1, 7], 9, -17/25, [8/25, 31/25], -34/25, [-34/25, -238/25], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([2, 0], [1, 8], 10, 8/65, [138/65, 64/65], 16/65, [16/65, 128/65], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([2, 0], [1, 8], 10, 8/65, [138/65, 64/65], 16/65, [16/65, 128/65], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([4, 2], [1, 10], 12, -12/101, [392/101, 82/101], -24/101, [-24/101, -240/101], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([4, 2], [1, 10], 12, -12/101, [392/101, 82/101], -24/101, [-24/101, -240/101], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([0, 3], [1, 11], 13, -10/61, [-10/61, 73/61], -20/61, [-20/61, -220/61], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([0, 3], [1, 11], 13, -10/61, [-10/61, 73/61], -20/61, [-20/61, -220/61], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([1, 4], [1, 12], 14, -7/29, [22/29, 32/29], -14/29, [-14/29, -168/29], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([1, 4], [1, 12], 14, -7/29, [22/29, 32/29], -14/29, [-14/29, -168/29], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([2, 5], [1, 13], 15, -26/85, [144/85, 87/85], -52/85, [-52/85, -676/85], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([2, 5], [1, 13], 15, -26/85, [144/85, 87/85], -52/85, [-52/85, -676/85], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([3, 6], [1, 14], 16, -71/197, [520/197, 188/197], -142/197, [-142/197, -1988/197], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 6], [1, 14], 16, -71/197, [520/197, 188/197], -142/197, [-142/197, -1988/197], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([4, 0], [1, 15], 17, 13/226, [917/226, 195/226], 13/113, [13/113, 195/113], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([4, 0], [1, 15], 17, 13/226, [917/226, 195/226], 13/113, [13/113, 195/113], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([0, 1], [1, 16], 18, 2/257, [2/257, 289/257], 4/257, [4/257, 64/257], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([0, 1], [1, 16], 18, 2/257, [2/257, 289/257], 4/257, [4/257, 64/257], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([1, 2], [1, 17], 19, -8/145, [137/145, 154/145], -16/145, [-16/145, -272/145], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([1, 2], [1, 17], 19, -8/145, [137/145, 154/145], -16/145, [-16/145, -272/145], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([2, 3], [1, 18], 20, -36/325, [614/325, 327/325], -72/325, [-72/325, -1296/325], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([2, 3], [1, 18], 20, -36/325, [614/325, 327/325], -72/325, [-72/325, -1296/325], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([3, 4], [1, 19], 21, -29/181, [514/181, 173/181], -58/181, [-58/181, -1102/181], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 4], [1, 19], 21, -29/181, [514/181, 173/181], -58/181, [-58/181, -1102/181], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([4, 5], [1, 20], 22, -82/401, [1522/401, 365/401], -164/401, [-164/401, -3280/401], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([4, 5], [1, 20], 22, -82/401, [1522/401, 365/401], -164/401, [-164/401, -3280/401], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([0, 6], [1, 21], 23, -103/442, [-103/442, 489/442], -103/221, [-103/221, -2163/221], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([0, 6], [1, 21], 23, -103/442, [-103/442, 489/442], -103/221, [-103/221, -2163/221], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([1, 0], [1, 22], 24, 23/485, [508/485, 506/485], 46/485, [46/485, 1012/485], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([1, 0], [1, 22], 24, 23/485, [508/485, 506/485], 46/485, [46/485, 1012/485], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([2, 1], [1, 23], 25, 0, [2, 1], 0, [0, 0], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([2, 1], [1, 23], 25, 0, [2, 1], 0, [0, 0], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([3, 2], [1, 24], 26, -25/577, [1706/577, 554/577], -50/577, [-50/577, -1200/577], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 2], [1, 24], 26, -25/577, [1706/577, 554/577], -50/577, [-50/577, -1200/577], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

([4, 3], [1, 25], 27, -26/313, [1226/313, 289/313], -52/313, [-52/313, -1300/313], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)

Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([4, 3], [1, 25], 27, -26/313, [1226/313, 289/313], -52/313, [-52/313, -1300/313], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly).

(rank_deficient, [[1, 1], [2, 2]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 1], [2, 2]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(stationary_not_sufficient, f(x,y)=-2*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-2*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum).

(infeasible_point, [1, 3], 2, [3, 0], 3, No, stationarity_never_repairs_missing_feasibility)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 3], 2, [3, 0], 3, No, stationarity_never_repairs_missing_feasibility).

(kkt_confusion, inequality, No, equality_only_Lagrange_condition_does_not_supply_KKT_sign_or_complementarity_conditions)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (kkt_confusion, inequality, No, equality_only_Lagrange_condition_does_not_supply_KKT_sign_or_complementarity_conditions).

(rank_deficient, [[1, 5], [2, 10]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 5], [2, 10]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(stationary_not_sufficient, f(x,y)=-6*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-6*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum).

(infeasible_point, [1, 7], 6, [7, 0], 7, No, stationarity_never_repairs_missing_feasibility)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 7], 6, [7, 0], 7, No, stationarity_never_repairs_missing_feasibility).

(rank_deficient, [[1, 8], [2, 16]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 8], [2, 16]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(rank_deficient, [[1, 9], [2, 18]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 9], [2, 18]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(stationary_not_sufficient, f(x,y)=-10*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-10*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum).

(infeasible_point, [1, 11], 10, [11, 0], 11, No, stationarity_never_repairs_missing_feasibility)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 11], 10, [11, 0], 11, No, stationarity_never_repairs_missing_feasibility).

(rank_deficient, [[1, 12], [2, 24]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 12], [2, 24]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(rank_deficient, [[1, 13], [2, 26]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 13], [2, 26]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(stationary_not_sufficient, f(x,y)=-14*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-14*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum).

(infeasible_point, [1, 15], 14, [15, 0], 15, No, stationarity_never_repairs_missing_feasibility)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 15], 14, [15, 0], 15, No, stationarity_never_repairs_missing_feasibility).

(rank_deficient, [[1, 16], [2, 32]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 16], [2, 32]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(rank_deficient, [[1, 17], [2, 34]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 17], [2, 34]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(stationary_not_sufficient, f(x,y)=-18*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-18*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum).

(infeasible_point, [1, 19], 18, [19, 0], 19, No, stationarity_never_repairs_missing_feasibility)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 19], 18, [19, 0], 19, No, stationarity_never_repairs_missing_feasibility).

(rank_deficient, [[1, 20], [2, 40]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 20], [2, 40]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(rank_deficient, [[1, 21], [2, 42]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 21], [2, 42]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(stationary_not_sufficient, f(x,y)=-22*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-22*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum).

(infeasible_point, [1, 23], 22, [23, 0], 23, No, stationarity_never_repairs_missing_feasibility)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 23], 22, [23, 0], 23, No, stationarity_never_repairs_missing_feasibility).

(rank_deficient, [[1, 24], [2, 48]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 24], [2, 48]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).

(rank_deficient, [[1, 25], [2, 50]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)

Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 25], [2, 50]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum).