2023/08/17 by M. I. A. Ghitri, Ghitri, M. I. A., Abderrahim Hantoute +1
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Mathematical Inequalities and Applications #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2308.09111
openalex publication_date 2023/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a subset A× B of a locally convex space X× Y (with A compact) and a function f:A× B→ℝ such that f(⋅,y), y∈ B, are concave and upper semicontinuous, the minimax inequality maxx∈ A infy∈ B f(x,y) ≥ infy∈ B sup_x∈ A0 f(x,y) is shown to hold provided that A0 be the set of x∈ A such that f(x,⋅) is proper, convex and lower semi-contiuous. Moreover, if in addition A× B⊂ f-1(ℝ), then we can take as A0 the set of x∈ A such that f(x,⋅) is convex. The relation to Moreau's biconjugate representation theorem is discussed, and some applications to convex duality are provided. Key words. Minimax theorem, Moreau theorem, conjugate function, convex optimization.