Convex optimization problem

Mathematical statement

A minimization problem qualifies here as convex only when its objective function and feasible domain are convex. This slice checks qualification for a univariate interval and quadratic objective, single-point feasibility, single-point evaluation and a comparison between two feasible candidates.

Representative formula (exact TeX)

C(P)

Assumptions

• The feasible domain is an explicit real interval and the objective is an integer-coefficient quadratic.
• Qualification or one-/two-point checks do not establish a global solution; KKT, general solvers and convergence results are not introduced.

Symbol scope

• C — convex-problem qualification record
• f — objective function

Intuition, definition, and misconceptions

Audit objective convexity and feasible-domain convexity separately.

Checking a feasible point and evaluating its objective are not optimality checks.

Comparing two candidates says only which has the lower value between them, not which is globally optimal.

Do not infer a unique global solution merely from the word convex or call the better of two candidates optimal for the whole problem.

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

• Convex function — Objective convexity is one convex-optimization qualification.
• Optimization problem — General optimization supplies feasible-domain, objective and minimization semantics.
• Real vector space — The feasible domain lies in a real vector space.

Verification boundary

Only problem qualification, feasibility, evaluation and two-point comparison are handled; a solved optimization problem is not claimed.

not_formalized

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

Next obligation

Add optimality certificates, KKT, duality and algorithmic convergence.

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.