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Convex optimization
analysis, algorithms, and engineering applications, pp. xi-xiii
 * Yurii Nesterov (2004), Introductory lectures on convex optimization: a basic course, p. 14
 * Aharon Ben-Tal and Arkadi Nemirovski (2012), Lectures on modern convex optimization, pp. 2-5
 * Aharon Ben-Tal, Arkadiĭ Semenovich Nemirovskiĭ (2001), Lectures on modern convex optimization:
 * Boyd, Vandenberghe, pp. 7-8

Convex optimization problem
A mathematical programming program
 * Aharon Ben-Tal, Arkadiĭ Semenovich Nemirovskiĭ (2001), Lectures on modern convex optimization: analysis, algorithms, and engineering applications, pp. 335-336
 * $$\min_x \{p_0(x):x \in \mathcal{X}\},$$

where $$\mathcal{X} \subset \mathbb{R}^n$$ is the feasible domain and $$p_0(x):\mathbb{R}^n \rightarrow \mathbb{R}$$ is the objective, is called convex if the domain $$\mathcal{X}$$ is a convex set and the objective $$p_0(x)$$ is convex on $$\mathbb{R}^n$$. Convex optimization programs are computationally tractable: there exist methods that efficiently solve every convex optimization program satisfying mild restrictions. In contrast to this, no efficient universal solution methods are known for nonconvex programs, and there are strong reasons to expect that no such methods exist.

The basic convex minimization problem is that of finding some $$\tilde{x} \in \mathcal{C}$$ such that
 * Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal (1996), Convex analysis and minimization algorithms: Fundamentals, p. 291
 * $$f(\tilde{x}) = \inf\{f(x):x \in \mathcal{C}\},$$

where $$\mathcal{C} \subset \mathbb{R}^n$$ is a closed convex set (called the feasible set) and $$f:\mathbb{R}^n \rightarrow \mathbb{R}$$ is a (finite-valued) convex function.

A convex optimization problem is one of the form
 * Boyd, Vandenberghe, pp. 7, 136-137
 * $$\begin{align}

&\operatorname{minimize}& & f_0(x) \\ &\operatorname{subject\;to} & &f_i(x) \leq 0, \quad i = 1,\dots,m, \end{align}$$ where the functions $$f_0 \ldots f_m : \mathbb{R}^n \rightarrow \mathbb{R}$$ are convex.

An optimization problem (also referred to as mathematical programming program or minimization problem) of finding some $$x^\ast \in \mathcal{X}$$ such that
 * $$f_0(x^\ast) = \min \{f_0(x):x \in \mathcal{X}\},$$

where $$\mathcal{X} \subset \mathbb{R}^n$$ is the feasible set and $$f_0(x):\mathbb{R}^n \rightarrow \mathbb{R}$$ is the objective, is called convex if $$\mathcal{X}$$ is a closed convex set and $$f_0(x)$$ is convex on $$\mathbb{R}^n$$. Equivalently, an optimization problem
 * $$\begin{align}

&\operatorname{minimize}& & f_0(x) \\ &\operatorname{subject\;to} & &f_i(x) \leq 0, \quad i = 1,\dots,m \end{align}$$ is called convex if the functions $$f_0 \ldots f_m : \mathbb{R}^n \rightarrow \mathbb{R}$$ are convex.