# Optimization problem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

An optimization problem seeks points minimizing or maximizing an objective on an explicit feasible set; the optimal value and optimizer set are different objects.

mko-optimization-problem · definition · v0.10.0



## Mathematical statement

For a feasible set S and objective f:S→R, minimization seeks x*∈S with f(x*)≤f(x) for every x∈S; maximization reverses the inequality. Several optimal points may attain the same value.





## Representative formula (exact TeX)

f(s)=m





## Assumptions


- Exercises give complete finite candidates, feasibility masks, constraints and objective values.
- Enumeration establishes only the declared finite feasible set, not unlisted candidates or general computational complexity.





## Symbol scope


- f — objective function
- x — feasible candidate
- m — optimal value





## Intuition, definition, and misconceptions

Determine feasibility before comparing objective values. An infeasible point cannot be an optimizer even if its numerical objective is better.

The optimal value is a number; argmin or argmax is the set of every feasible point attaining it, including ties.

Complete enumeration proves optimality within a finite list. It does not settle continuous feasible domains, unbounded problems or efficiency on large instances.

Do not ignore constraints when choosing a small value or report one tied optimizer as uniquely optimal.





## Adopted derivation guide

[Optimization problem: feasibility first, then objective values, over nine integer candidates](https://ocme.evemisslab.com/en/derivations/derivation-mko-optimization-problem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — The feasible domain is a set of candidate points.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — The objective assigns a value to each feasible point.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Order comparisons define minima and maxima.





## Verification boundary

This slice establishes only optimality in a completely declared finite candidate set; it does not solve general linear or nonlinear programming.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Develop linear programming, feasible polyhedra, duality and optimality certificates.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/optimization-problem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-optimization-problem/index.md)