Optimization problem

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Set — The feasible domain is a set of candidate points.
• Function mapping — The objective assigns a value to each feasible point.
• Usual order on the real numbers — 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.