Aggregate Homotopy Methods For Solving A Kind Of Constrained Sequential Max-Min Problems

Sequential max-min optimization problem is a typical kind of nonconvex, nonsmoothprogramming. They arise in the design of electronic circuits, in the design of path for robotsin the presence of obstacles and VLSI circuits, etc. To our knowledge, the literature dealingwith this problem is relatively small. The main theory result is on its quasi-differential theoryand the main algorithm is that given by E. Polak et al, they changed the problem to a kind ofsmooth programming. The advantage of their method is that it can utilize existing softwarefor solving smooth optimization and, the disadvantage is that it needs to introduce manyauxiliary variables.Guoxin Liu constructed an aggregate homotopy method for USMMP as well as anaggregate homotopy interior point method for CSMMP in his paper "Aggregate HomotopyMethod for Solving Sequential Max-Min Problems, Complimentarily Problem andVariational Inequalities". Under mild assumptions, existence and convergence of the smoothpath to the solution of generalized K-K-T system is proved. However he didn't givenumerical experiment to support the validity of methods.Based on the weak normal cone condition, article [1] proved the existence of K-K-Tpoint of constrain sequential max-min problems. Considering the difficulties of validating thiscondition, this paper discusses the convenient way to validate the weak normal cone condition,and designs arithmetic. Meanwhile, this paper analyzes the method in [2] to compare withaggregate Homotopy Methods. The last numerical experiment shows that aggregatehomotopy method can solve problems which don't satisfy conditions in [2], if numbers ofchose function are large, aggregate homotopy method is faster than smoothing method in [2],this support the validity of aggregate homotopy methods.