Question 27985: Squared Distance with One Regular Linear Constraint Has an Exact Solution

Original question and givens

Minimize squared Euclidean distance to center=[4, 2] subject to normal a=[1, 10], a dot x=12; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).

• G001 — Minimize squared Euclidean distance to center=[4, 2] subject to normal a=[1, 10], a dot x=12; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).

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-lagrange-multiplier-condition

Step-by-step derivation

source roles = [lagrange-single-linear-constraint]

derived = [[center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope]]

Lagrange single ORIGINAL normal squared norm ORIGINAL product0 := mul(1, 1)

Lagrange single ORIGINAL normal squared norm ORIGINAL product0 = 1

Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0 := sum(0, 1)

Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0 = 1

Lagrange single ORIGINAL normal squared norm ORIGINAL product1 := mul(10, 10)

Lagrange single ORIGINAL normal squared norm ORIGINAL product1 = 100

Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1 := sum(1, 100)

Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1 = 101

Lagrange single ORIGINAL normal dot center ORIGINAL product0 := mul(1, 4)

Lagrange single ORIGINAL normal dot center ORIGINAL product0 = 4

Lagrange single ORIGINAL normal dot center ORIGINAL running sum0 := sum(0, 4)

Lagrange single ORIGINAL normal dot center ORIGINAL running sum0 = 4

Lagrange single ORIGINAL normal dot center ORIGINAL product1 := mul(10, 2)

Lagrange single ORIGINAL normal dot center ORIGINAL product1 = 20

Lagrange single ORIGINAL normal dot center ORIGINAL running sum1 := sum(4, 20)

Lagrange single ORIGINAL normal dot center ORIGINAL running sum1 = 24

Lagrange single ORIGINAL rhs minus center value := sub(12, 24)

Lagrange single ORIGINAL rhs minus center value = -12

Lagrange single ORIGINAL normal displacement := div(-12, 101)

Lagrange single ORIGINAL normal displacement = -12/101

Lagrange single ORIGINAL normal shift0 := mul(-12/101, 1)

Lagrange single ORIGINAL normal shift0 = -12/101

Lagrange single ORIGINAL solution coordinate0 := sum(4, -12/101)

Lagrange single ORIGINAL solution coordinate0 = 392/101

Lagrange single ORIGINAL normal shift1 := mul(-12/101, 10)

Lagrange single ORIGINAL normal shift1 = -120/101

Lagrange single ORIGINAL solution coordinate1 := sum(2, -120/101)

Lagrange single ORIGINAL solution coordinate1 = 82/101

Lagrange single ORIGINAL multiplier2tau := mul(2, -12/101)

Lagrange single ORIGINAL multiplier2tau = -24/101

Lagrange single ORIGINAL displacement coordinate0 := sub(392/101, 4)

Lagrange single ORIGINAL displacement coordinate0 = -12/101

Lagrange single ORIGINAL objective gradient0 := mul(2, -12/101)

Lagrange single ORIGINAL objective gradient0 = -24/101

Lagrange single ORIGINAL displacement coordinate1 := sub(82/101, 2)

Lagrange single ORIGINAL displacement coordinate1 = -120/101

Lagrange single ORIGINAL objective gradient1 := mul(2, -120/101)

Lagrange single ORIGINAL objective gradient1 = -240/101

Lagrange single ORIGINAL full feasibility ORIGINAL product0 := mul(1, 392/101)

Lagrange single ORIGINAL full feasibility ORIGINAL product0 = 392/101

Lagrange single ORIGINAL full feasibility ORIGINAL running sum0 := sum(0, 392/101)

Lagrange single ORIGINAL full feasibility ORIGINAL running sum0 = 392/101

Lagrange single ORIGINAL full feasibility ORIGINAL product1 := mul(10, 82/101)

Lagrange single ORIGINAL full feasibility ORIGINAL product1 = 820/101

Lagrange single ORIGINAL full feasibility ORIGINAL running sum1 := sum(392/101, 820/101)

Lagrange single ORIGINAL full feasibility ORIGINAL running sum1 = 12

Lagrange single ORIGINAL lambda times normal0 := mul(-24/101, 1)

Lagrange single ORIGINAL lambda times normal0 = -24/101

Lagrange single ORIGINAL lambda times normal1 := mul(-24/101, 10)

Lagrange single ORIGINAL lambda times normal1 = -240/101

source roles = [[4, 2], [1, 10], 12, 101, 24, -12/101, [392/101, 82/101], -24/101, [-24/101, -240/101]]

derived = [Yes, Yes, ANY feasible y=x+v has a dot v=0;f(y)=f(x)+norm(v)^2]

Reverse checks

• S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S026 · premise_and_result_audit — Recompute Lagrange single ORIGINAL lambda times normal1 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Lagrange single ORIGINAL lambda times normal0 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Lagrange single ORIGINAL full feasibility ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Lagrange single ORIGINAL full feasibility ORIGINAL product1 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Lagrange single ORIGINAL full feasibility ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Lagrange single ORIGINAL full feasibility ORIGINAL product0 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Lagrange single ORIGINAL objective gradient1 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Lagrange single ORIGINAL displacement coordinate1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Lagrange single ORIGINAL objective gradient0 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Lagrange single ORIGINAL displacement coordinate0 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Lagrange single ORIGINAL multiplier2tau with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Lagrange single ORIGINAL solution coordinate1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal shift1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Lagrange single ORIGINAL solution coordinate0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal shift0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal displacement with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Lagrange single ORIGINAL rhs minus center value with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal dot center ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal dot center ORIGINAL product1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal dot center ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal dot center ORIGINAL product0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL product1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL product0 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

([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)

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.