Nash Equilibrium Problems of Polynomials.

This article focuses on solving Nash equilibrium problems of polynomials (NEPPs), where each player’s objective and constraint functions are polynomial. It proposes algorithms that formulate polynomial optimization problems whose solutions correspond to Nash equilibria, using polynomial expressions for Lagrange multipliers under a nonsingularity assumption on constraints. The Moment-sum-of-squares (Moment-SOS) hierarchy of semidefinite relaxations is employed to solve these polynomial optimization problems, enabling computation of one or all Nash equilibria when finitely many exist, or detection of nonexistence otherwise. The paper establishes finite convergence of the algorithms under genericity conditions, supported by proofs that generic NEPPs have finitely many Karush-Kuhn-Tucker (KKT) points. Numerical experiments demonstrate the practical effectiveness of the methods on various examples, including convex and nonconvex NEPs, with applications in economics and electricity markets.