---
title: "convex problem"
source: https://www.jemoka.com/posts/kbhconvex_problems/
---

Recall optimization (math). An optimization (math) problem is convex if:
the objective is convex function inequality constrains&rsquo; functions are convex equality constrains are affine Special convex problems Linear Program Optimality Criterion for Differentiable Objective \(x\) is optimal IFF its feasible and
\begin{equation} \nabla f_{0} \qty(x)^{T} \qty(y-x) \geq 0 \end{equation}
for all feasible \(y\).
examples unconstrained problem: \(x\) minimizes \(f_{0}\qty(x)\) IFF \(\nabla f_{0}\qty(x) = 0\) equality constrained problem: \(x\) minimizes \(f_{0}\qty(x)\) subject to \(Ax = b\) IFF there is a \(v\) such that \(Ax = b\), \(\nabla f_{0}\qty(x) + A^{T}v = 0\) Local and Global Optima Any locally optimal point of a convex problem is globally optimal.
