Slater condition

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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.

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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.

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