Convex Optimization

Abstract

Convex optimization deals with problems of the form

$$\begin{aligned} \begin{array}{lll} P: &{} \text {Min} &{} f\left( x\right) \\ &{} \text {s.t.} &{} x\in F, \end{array} \end{aligned}$$

where $\emptyset \ne F\subset \mathbb {R}^{n}$ is a convex set and $f:F\rightarrow \mathbb {R}$ is a convex function. In this chapter, we analyze four particular cases of problem P in (4.1): (a) Unconstrained convex optimization, where the constraint set F represents a given convex subset of $\mathbb {R}^{n}$ (as $\mathbb {R} _{++}^{n}$). (b) Convex optimization with linear constraints, where

$$\begin{aligned} F=\left\{ x\in \mathbb {R}^{n}:g_{i}\left( x\right) \le 0,i\in I\right\} , \end{aligned}$$

with $I=\left\{ 1,\ldots , m\right\}$, $m\ge 1$, and $g_{i}$ are affine functions for all $i\in I$. In this case, F is a polyhedral convex set (an affine manifold in the particular case where F is the solution set of a system of linear equations, as each equation can be replaced by two inequalities).