Convex Optimization - Standard Form

Standard form is the usual and most intuitive form of describing a convex minimization problem. It consists of the following three parts:

  • A convex function to be minimized over the variable
  • Inequality constraints of the form, where the functions are convex
  • Equality constraints of the form, where the functions are affine. In practice, the terms "linear" and "affine" are often used interchangeably. Such constraints can be expressed in the form, where is a column-vector and a real number.

A convex minimization problem is thus written as

\begin{align} &\underset{x}{\operatorname{minimize}}& & f(x) \\ &\operatorname{subject\;to} & &g_i(x) \leq 0, \quad i = 1,\dots,m \\ &&&h_i(x) = 0, \quad i = 1, \dots,p. \end{align}

Note that every equality constraint can be equivalently replaced by a pair of inequality constraints and . Therefore, for theoretical purposes, equality constraints are redundant; however, it can be beneficial to treat them specially in practice.

Following from this fact, it is easy to understand why has to be affine as opposed to merely being convex. If is convex, is convex, but is concave. Therefore, the only way for to be convex is for to be affine.

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