Suggest edit — convex optimization

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---
title: "convex optimization"
type: concept
related: [Linear Constraint Optimization, Convex Optimization, Optimization Math, Quadratic Programming]
source: https://www.jemoka.com/posts/kbhconvex_optimization/
confidence: high
status: active
---

Tractable optimization problems.
Linear Programming quadratic programming etc. We can solve these problems reliably and efficiently.
constituents where \(x \in \mathbb{R}^{n}\) is a vector of variables \(f_{0}\) is the objective function, &ldquo;soft&rdquo; to be minimized \(f_{1} &hellip; f_{m}\) are the inequality constraints \(g_{1} &hellip; g_{p}\) are the inequality constraints requirements Generally of structure:
\begin{equation} \min f_{0}\qty(x) \end{equation}
subject to:
\begin{align} &amp;f_{i} \qty(x) \leq 0, i = 1 \dots m \\ &amp;Ax = b \end{align}
and have curvature constraints
\(f_{0}, &hellip; f_{m}\) are convex:
\begin{equation} f_{i}\qty(\theta x + \qty(1-\theta)y) \leq \theta f_{i}\qty(x) + \qty(1-\theta)f_{i}\qty(y) \end{equation}
for \(\theta \in [0,1]\). That is, \(f_{i}\) have non-negative curvature.
additional information sign asymmetry FUN IMPORTANT FACT: when an minimization problem maybe convex, sticking a negative sign (casting it to the maximization problem) is not necessarily convex. &ldquo;Negative of things that curve down is things that curve up. This may no longer be convex.&rdquo;
ease &ldquo;convex (nonnegative curvature) is easy&rdquo;, &ldquo;onconvex (negative curvature) is hard&rdquo;.
DSLs for Convex Optimization DSLs for convex optimization allows us to describe problems at a high level + close to the math; can transform the problem into a standard form and then solve it.