# Question 11976: Separate a Quadratic Objective Certificate from Feasible-Interval Qualification

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0022-11976 · q-20261013-advance-20261025-r03-convex-optimization-problem-0001



## Original question and givens

Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -12, -3], not ascending-power order. The declared interval [lower,upper]=[-2, 3] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.


- G001 — Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -12, -3], not ascending-power order. The declared interval [lower,upper]=[-2, 3] belongs to the function data. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.





## 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-convex-optimization-problem](https://ocme.evemisslab.com/en/mko/mko-convex-optimization-problem/)





## Step-by-step derivation

S001 · impliesBefore

declared interval record=[-2, 3]

After

ordered_nonempty=Yes;nondegenerate=Yes

Conditions: none

This nonempty-interval protocol uses lower≤upper;the nondegenerate certificate additionally needs lower<upper.A reversed record is unqualified,not a declaration that the abstract empty set is nonconvex.A singleton interval is convex.

Step identifiers and dependenciesRule: interval_certificate_qualification_not_empty_set_nonconvexity · Uses: G001





S002 · impliesBefore

univariate quadratic coefficients=[-1, -12, -3];order=[a,b,c]

After

f(t)=(-1)*t^2+(-12)*t+(-3)

Conditions: none

These are quadratic,linear,constant coefficients in descending order,not the constant-first boundary-polynomial order.Retain zero leading coefficients.

Step identifiers and dependenciesRule: quadratic_descending_coefficient_order · Uses: G001, S001





S003 · impliesBefore

f(t)=(-1)*t^2+(-12)*t+(-3);z=lambda*x+(1-lambda)*y

After

rhs-lhs=(-1)*(lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2)

Conditions: none

Expand both Jensen sides.The unit weight sum cancels the linear and constant terms exactly,leaving only the quadratic part.

Step identifiers and dependenciesRule: jensen_linear_and_constant_terms_cancel · Uses: G001, S002





S004 · impliesBefore

lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2

After

lambda*(1-lambda)*(x-y)^2;full gap=(-1)*lambda*(1-lambda)*(x-y)^2

Conditions: none

Expand the square and collect x²,xy,y².The coefficients are lambda(1−lambda),−2lambda(1−lambda),lambda(1−lambda),yielding the(x−y)^2 factor for all real endpoints.

Step identifiers and dependenciesRule: jensen_gap_full_symbolic_factorization · Uses: G001, S003





S005 · impliesBefore

arbitrary x,y in [-2,3];0<=lambda<=1;z=lambda*x+(1-lambda)*y

After

z--2=lambda*(x--2)+(1-lambda)*(y--2)>=0;3-z=lambda*(3-x)+(1-lambda)*(3-y)>=0

Conditions: none

Both slacks are nonnegative sums,hence lower≤z≤upper for every endpoint pair and every permitted lambda.No sampling is used;endpoints and singleton cases are included.

Step identifiers and dependenciesRule: interval_all_segments_slack_identity · Uses: G001, S004





S006 · impliesBefore

global quadratic leading coefficient=-1;interval record=[-2, 3]

After

global objective certificate=No;feasible interval certificate=Yes;qualified=No

Conditions: none

The objective flag is the global quadratic a≥0 criterion:for a<0,endpoints0,1 and half weights give a negative gap.The domain flag separately qualifies a nonempty ordered interval.Joint qualification proves no solution,uniqueness,KKT or solver convergence;failure of the global criterion does not classify every restricted singleton problem.

Step identifiers and dependenciesRule: optimization_two_qualifications_not_solver · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

([-1, -12, -3], [-2, 3], No, Yes, No, qualification_only_no_solution_claim)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 661c4c6beb23e4d0d77783bcb4872298c23851039d1a7f6767bfb5309c920b93
- candidate record SHA-256: fde1e4a3867ed7998fef8c88eeab3e24b74cff6486c74a13749b5bcebdb6de8a
- semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac
- question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b
- Lean: NOT_RUN · human review: NOT_PERFORMED

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