Inequality Constrained Space Optimization Problems

Definition

An optimization problem is $(X, f)$ is an inequality constrained space optimization problem if $X \subset \R ^n$, $f: \R ^n \to \R $, and there exists $g: \R ^n \to \R ^m$ so that

\[ X = \Set{x \in \R ^n}{g(x) \leq 0} \]

For this reason, $(f, g)$ is sometimes called the problem data (abstract problem data) of the problem.

Notation

We often write such problems as: given $f: \R ^n \to \R $ and $g: \R ^n \to \R ^m$, find $x \in \R ^n$ to

\[ \begin{aligned} \text{minimize} & \quad f(x) \\ \text{subject to} & \quad g(x) \leq 0 \\ \end{aligned} \]

Some authors abbreviate inequality constrained space optimization problem as ICP.

Handles equality constraints

Suppose $f: \R ^n \to \R $ and $h: \R ^n \to \R ^m$ are the (abstract) problem data for an equality constrained space optimization problem. Define $g: \R ^n \to \R ^{2m}$ so that

\[ g(x) = (h(x), -h(x)) \quad \text{for all } x \in \R ^n \]

Then the ECP $(f, h)$ and ICP $(f, g)$ have the same feasible set and optimal solutions. In other words, given an equality constrained problem we can always write it as an inequality constrained problem.