# Slater condition

Originally for the purely inequality system with g convex, it means there exists $LaTeX: x$ for which $LaTeX: g(x) < 0.$ More generally, for a mathematical program in standard form, it means there exists $LaTeX: x \in X$ for which $LaTeX: \textstyle g(x) < 0 \mbox{ and } h(x) = 0.$

# Mathematical program

Commonly an optimization problem of the form

$LaTeX: \max \{ f(x) : x \in X,\, g(x) \le 0, \, h(x) = 0\},$

where $LaTeX: X$ is a subset of $LaTeX: \mathbb{R}^n$ and is the domain of $LaTeX: f$, $LaTeX: g$ and $LaTeX: h,$ which map into real spaces. The function $LaTeX: f$ is called the objective function, which is typically real-valued. If not, then $LaTeX: f$ maps into $LaTeX: \mathbb{R}^p$ with $LaTeX: p \ge 2,$ and the problem is a multiple objective problem. The feasible region is the collection of $LaTeX: x$ that simultaneously satisfy $LaTeX: x$ in $LaTeX: X,$ $LaTeX: g(x) \le 0,$ and $LaTeX: h(x) = 0,$ which are the program's constraints.