A New Method For Minmax Problem

The minmax problem is one of an important non-differentiable optimization problems, it does not only has broader applications in engineering designing , electronic microcircuits programming , game theory and so on, but also has very close realation-ship with nonlinear equations , muti-object programming , nonlinear programmming etc.At present, there are some methods, e.g., line search method, SQP method, trust region method and the active-set method, for solving minmax problems. For example, C. Charalambous and A.R. Conn gave the line search method. W. Murray and L. Overton presented the projected Lagrangian method. A. Vardi presented the trust reigon method with the active-set. These methods have stronger therotical conditions and narrower applications. Recently, the filter method for nonlinear progamming has broader applications and good numerical effect. The motivation from filter method is applied for minmax problem.Generally speaking, the minmax problem is always transfered nonlinear programming with inequality constraints and then it is discussed with a penalty function as a merit function. A new method for solving minmax problem is presented with the motivation of filter method in this paper. At each iteration, the trial step is devided into normal step and tangential step. The normal step will improve the feasibility of the constraints and the tangential will improve the object function. On the other hand, the nonmonotonic frank will be used, which results in the generality of the algorithm and good numerical effect. Under no linear independence of the active constraints, we analyse the global convergence. Finally, we give the primary numerical test.