# Question 11967: Compare Only Two Eligible Candidates, Not the Whole Problem

[Return to catalogue page 60](https://ocme.evemisslab.com/en/questions/page/60/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261025-r03-convex-optimization-problem-foundations/#q-20261013-advance-20261025-r03-convex-optimization-problem-0092)

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.

R0022-11967 · q-20261013-advance-20261025-r03-convex-optimization-problem-0092



## Original question and givens

Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 4, -1], not ascending-power order. The declared interval [lower,upper]=[-18, 19] belongs to the function data. left=-1/1, right=-1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. 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]=[0, 4, -1], not ascending-power order. The declared interval [lower,upper]=[-18, 19] belongs to the function data. left=-1/1, right=-1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. 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=[-18, 19]

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=[0, 4, -1];order=[a,b,c]

After

f(t)=(0)*t^2+(4)*t+(-1)

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 · calculationBefore

left candidate lower slack := sub(-1, -18)

After

left candidate lower slack = 17

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

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





S004 · calculationBefore

left candidate upper slack := sub(19, -1)

After

left candidate upper slack = 20

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

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





S005 · impliesBefore

left candidate:interval=[-18, 19];point=-1;slacks=[17, 20]

After

feasible=Yes

Conditions: none

Both inclusive bounds must hold.A reversed record supplies no qualified feasible point.The polynomial may have a value outside the domain,but a value does not establish eligibility.

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





S006 · calculationBefore

right candidate lower slack := sub(-1, -18)

After

right candidate lower slack = 17

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

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





S007 · calculationBefore

right candidate upper slack := sub(19, -1)

After

right candidate upper slack = 20

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

right candidate:interval=[-18, 19];point=-1;slacks=[17, 20]

After

feasible=Yes

Conditions: none

Both inclusive bounds must hold.A reversed record supplies no qualified feasible point.The polynomial may have a value outside the domain,but a value does not establish eligibility.

Step identifiers and dependenciesRule: inclusive_point_feasibility_separate_evaluation · Uses: G001, S007





S009 · calculationBefore

left objective point squared := mul(-1, -1)

After

left objective point squared = 1

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

left objective quadratic term := mul(0, 1)

After

left objective quadratic term = 0

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

left objective linear term := mul(4, -1)

After

left objective linear term = -4

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

left objective polynomial value := sum(0, -4, -1)

After

left objective polynomial value = -5

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

right objective point squared := mul(-1, -1)

After

right objective point squared = 1

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

right objective quadratic term := mul(0, 1)

After

right objective quadratic term = 0

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · calculationBefore

right objective linear term := mul(4, -1)

After

right objective linear term = -4

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

right objective polynomial value := sum(0, -4, -1)

After

right objective polynomial value = -5

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

two candidate value difference := sub(-5, -5)

After

two candidate value difference = 0

Conditions: none

Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · impliesBefore

left feasible=Yes;right feasible=Yes;left value=-5;right value=-5;difference=0

After

eligible pair relation=equal;scope=pairwise_only_no_global_optimum_claim

Conditions: none

Compare values only if both points are feasible;otherwise they are not comparable.Even a lower point only beats this other candidate.No quantification over all feasible points has established a global optimum.

Step identifiers and dependenciesRule: two_eligible_points_not_global_optimum · Uses: G001, S017









## Reverse checks


- S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S017 · premise_and_result_audit — Recompute two candidate value difference with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute right objective polynomial value with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Recompute right objective linear term with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Recompute right objective quadratic term with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute right objective point squared with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute left objective polynomial value with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute left objective linear term with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Recompute left objective quadratic term with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Recompute left objective point squared with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Recompute right candidate upper slack with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute right candidate lower slack with exact rational arithmetic, not rounded decimals.
- 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 — Recompute left candidate upper slack with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute left candidate lower slack with exact rational arithmetic, not rounded decimals.
- 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

([0, 4, -1], [-18, 19], -1, Yes, -5, -1, Yes, -5, equal, pairwise_only_no_global_optimum_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: 69fe6ad2a392b94dd81b937f378e2172d4875f5c14eeeeff3f217bd6591fdc32
- candidate record SHA-256: 3e097be44422db15c7bb0ba011f341830b0f4df7d18d74aa894f87240b59db1c
- semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac
- question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261025-r03-convex-optimization-problem-0092.json)