Abstract
ABSTRACT In this paper, an aggregate homotopy method is proposed to solve unconstrained minimax problems. The homotopy mapping is constructed by the linear homotopy and the aggregate function for the objective function, and the homotopy parameter is also used as the smoothing parameter of the aggregate function. Under some general assumptions, the existence and global convergence of a smooth homotopy path are proved for almost all starting points in or a ball set, and a stationary point of the unconstrained minimax problem can be obtained. A path-following procedure is introduced to numerically trace the homotopy path. When the smoothing parameter becomes small enough, an alternative strategy of the path-following procedure is given to reduce the ill-conditioning of the aggregate function, it uses the Newton method to solve a nonlinear system, which is derived from the first order optimality conditions for the unconstrained minimax problem. Numerical results show that the proposed method is efficient and robust.
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