Real Optimizers

Why

Often the chain considered in optimization is the real numbers.

Definition

A function from a non-empty set to the real numbers, in the context of optimizers, is called an objective function for the feasible set (its domain). Suppose the feasible set is a subset of real numbers. Often we can specify a function defined for every real number. The objective function is the restriction of this function to those elements which are feasible.

Notation

Let $D$ be a non-empty set, a mnemonic for “domain.” Let $f: D \to \R $. Often we have $g: A \to \R $, where $D \subset A$. So $f$ is the restriction of $g$ to the feasible set $D$. In this case the language feasible set language is especially useful