The cubic algorithm for global games with application to pursuit-evasion games

Abstract

New algorithms for solution of nonconvex global games defined over a cube are presented. Application to differential games and to pursuit-evasion games is considered, and extension is made for games over general compact and robust sets defined by constraints that may depend on time and on state variables. The methods are deterministic and monotonically set-convergent to deliver, in the limit, the unique exact full globally optimal minimax and maximin solutions for both players, or the entire global saddle set, if it exists. A stopping rule is provided for determining approximate solutions with a given precision in a finite number of iterations. A modification is considered to play on mistakes of the opponent. On-line replacement of cost functionals is possible in the course of iterations, according to changing interests of the players. No separability nor convexity-concavity assumptions are imposed on the cost functional, and variational methods are not used. The ideas are illustrated by examples and application is made to determining the globally optimal closed-loop strategies for the ship-torpedo collision-avoidance differential game with manoeuvrability constraints.