The Saddle Point Problem of Polynomials

Abstract

This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre's hierarchy of semidefinite relaxations. Under some genericity assumptions on defining polynomials, we show that: i) if there exists a saddle point, our algorithm can get one by solving a finite number of Lasserre type semidefinite relaxations; ii) if there is no saddle point, our algorithm can detect its nonexistence.

Nash Equilibrium Problems of Polynomials

“…The following theorem is the convergence for Algorithm 4.4. Its proof follows from [47,Theorem 4.4]. Theorem 4.5.…”

“…For the case that f 1 + • • • + f N = 0, the NEP (1.3) is called a zero-sum game. The two-player zero-sum game is equivalent to a saddle point problem [47]. NEPs have broad applications in Economics modelling [3,38,53,58,61].…”

“…For solving finite games or finding their mixed strategy solutions, we refer to the work [1,12,25,34]. For two-player zero sum games, the NEPs are equivalent to saddle point problems and there are optimization methods for solving them [8,47]. More work for solving NEPs can be found in [20,24,26,27,35,36,51,60].…”