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@jemoka / Jemoka Knowledge Base / wiki/concepts/convex_problems.md
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--- title: "convex problem" type: concept related: [Convex Functions, Linear Constraint Optimization, Optimization Math, Convex Problems] source: https://www.jemoka.com/posts/kbhconvex_problems/ confidence: high status: active --- Recall optimization (math). An optimization (math) problem is convex if: the objective is convex function inequality constrains’ 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.