Optimization Algorithms and Equilibrium Analysis for Dynamic Resource Allocation

Abstract

Abstract : This paper develops an optimization-based theory for the existence and uniqueness of equilibria of a non-cooperative game wherein the selfish players' optimization problems are nonconvex and there are side constraints and associated price clearance to be satisfied by the equilibria. A new concept of equilibrium for such a nonconvex game, which we term a quasi-Nash equilibrium (QNE), is introduced as a solution of the variational inequality (VI) obtained by aggregating the first-order optimality conditions of the players' problems while retaining the convex constraints (if any) in the defining set of the VI. Under a second-order sufficiency condition from nonlinear programming, a quasi-Nash equilibrium becomes a local Nash equilibrium of the game. Uniqueness of a QNE is established using a degree-theoretic proof. Under a key boundedness property of the Karush-Kuhn-Tucker multipliers of the nonconvex constraints and the positive definiteness of the Hessians of the players' Lagrangian functions, we establish the single-valuedness of the players' best-response maps, from which the existence of a Nash equilibrium (NE) of the nonconvex game follows. We also present a distributed algorithm for computing a NE of such a game and provide a matrix-theoretic condition for the convergence of the algorithm. An application that pertains to a special multi-leader-follower game wherein the nonconvexity is due to the followers' equilibrium conditions in the leaders' optimization problems is presented. Another application to a cognitive radio paradigm in a signal processing game is described in two separate papers.