On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory

Abstract In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property for compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. We apply these results to derive necessary and sufficient conditions for strong duality for a general class of optimization problems.

[1]  G. Kassay,et al.  On Borel probability measures and noncooperative game theory , 2002 .

[2]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[3]  I. Joó,et al.  On some convexities , 1989 .

[4]  J. B. G. Frenk,et al.  The level set method of Joó and its use in minimax theory , 2006, Math. Program..

[5]  R. Ash,et al.  Real analysis and probability , 1975 .

[6]  Kellen Petersen August.  Real Analysis , 2009 .

[7]  A. Wald.  Generalization of a Theorem By v. Neumann Concerning Zero Sum Two Person Games , 1945 .

[8]  J. B. G. Frenk,et al.  Fiscal Decentralization in Mexico: The Bailout Problem , 2002 .

[9]  R. Blumenthal.  Review: J. Neveu, Amiel Feinstein, Mathematical Foundations of the Calculus of Probability , 1967 .

[10]  M. Degroot.  Game Theory: Mathematical Models of Conflict , 1980 .

[11]  J. B. G. Frenk,et al.  On Classes of Generalized Convex Functions, Gordan–Farkas Type Theorems, and Lagrangian Duality , 1999 .

[12]  Ulrich Faigle,et al.  Algorithmic principles of mathematical programming , 2002 .

[13]  J. Jahn.  Mathematical vector optimization in partially ordered linear spaces , 1986 .

[14]  J. Neveu,et al.  Mathematical foundations of the calculus of probability , 1965 .

[15]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

[16]  R. A. Silverman,et al.  Introductory Real Analysis , 1972 .

[17]  Stephen Simons,et al.  Minimax Theorems and Their Proofs , 1995 .

[18]  Rossiĭskai︠a︡ akademii︠a︡ nauk,et al.  Functional analysis and its applications , 1967 .

[19]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[20]  J. Neumann.  Zur Theorie der Gesellschaftsspiele , 1928 .

[21]  Yuval Rabani,et al.  Linear Programming , 2007, Handbook of Approximation Algorithms and Metaheuristics.

[22]  A. Friedman.  Foundations of modern analysis , 1970 .

[23]  József Kolumbán,et al.  On a generalized sup-inf problem , 1996 .

[24]  J. Aubin.  Optima and Equilibria: An Introduction to Nonlinear Analysis , 1993 .

[25]  R. Holmes.  Geometric Functional Analysis and Its Applications , 1975 .

[26]  J. Aubin.  Optima and Equilibria , 1993 .