Predefined/Prescribed‐Time Convergence Algorithm of Nonconvex‐Nonconcave Min‐Max Optimization

ABSTRACT In this paper, generalized predefined/prescribed‐time stability theory is first put forward, where i) the powers of the upper bound of the derivative of Lyapunov function are no longer confined to and with ; ii) the upper bound of the settling‐time function is simplified and directly deduced by Euler's reflection formula. In addition, the upper bound of the settling‐time function is still irrelevant to the initial states, and can be arbitrarily tuned in advance. As an application, a predefined/prescribed‐time algorithm is proposed for solving the nonconvex‐nonconcave min‐max optimization issue. Under the two‐sided Polyak‐Lojasiewicz (PL) inequality and the Lipschitz continuous gradient, a new dynamical system is established to ensure that the solution of the dynamic system can reach the saddle point of the min‐max optimization issue in a predefined/prescribed time. Two examples are applied to validate the predefined/prescribed‐time algorithm of nonconvex‐nonconcave min‐max optimization.