Method of centers

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This is an interior method defined for a mathematical program without equality constraints with a nonempty strict interior. Considering LaTeX: \max \{ f(x) : x \in X, \, g(x) \le 0 \}, define

LaTeX: G(w, x) = \min \{f(w)-f(x), -g_1(w), \ldots,-g_m(w)\}

Observe that LaTeX: G(w, x) > 0 if, and only if, LaTeX: f(w) > f(x) and LaTeX: g_i(w) < 0 for all LaTeX: i. Then, the algorithm map is LaTeX: \arg\max \{G(w,x): w \in \mathbb{R}^n\}.

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