A hybrid algorithm for nonlinear minimax problems

Abstract

In this paper, a hybrid algorithm for solving finite minimax problem is presented. In the algorithm, we combine the trust-region methods with the line-search methods and curve-search methods. By means of this hybrid technique, the algorithm, according to the specific situation at each iteration, can adaptively performs the trust-region step, line-search step or curve-search step, so as to avoid possibly solving the trust-region subproblems many times, and make better use of the advantages of different methods. Moreover, we use secondorder correction step to circumvent the difficulties of the Maratos effect occurred in the nonsmooth optimization. Under mild conditions, we prove that the new algorithm is of global convergence and locally superlinear convergence. The preliminary experiments show that the new algorithm performs efficiently.

A Parallel Algorithm for Solving Large Convex Minimax Problems

“…The algorithm was tested (using Matlab) on various small convex optimization problems ( [12], [13]) and results compared with performance of algorithm as presented in [11].…”

Nonmonotone algorithm for minimax optimization problems

“…In view of the importance of minimax problems (1.1), many algorithms have been developed before (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]), which can be grouped into two classes of methods. One is the method that was used to solve the unconstrained nondifferentiable optimization problems, such as subgradient methods, bundle methods and cutting plane methods [1,2,7,9].…”

“…Another trust-region based algorithm for minimax problems was developed in [12], which combining trust-region methods with line or arc search methods. This technique is similar to the technique of combining trust-region methods with line search methods [17], which may avoid solving trust-region subproblems many times in each iteration, thus speeding up the convergence.…”

“…In this paper, according to the special features of minimax problems, motivated by Zhou and Tits [4], we combine the nonmonotone technique with the hybrid scheme proposed in [12], that is, when /ðx k þ d k þd k Þ does not meet a sufficient decrease condition defined in the nonmonotone trust-region scheme, the method performs a nonmonotone line or curve search to find a stepsize t k , whered k is a second order correction step. In this scheme, the algorithm can produce a new point at each iteration, in which the objective function is not forced to decrease at each iteration, the Maratos effect can be prevented.…”

An improved partial bundle method for linearly constrained minimax problems

“…The case where the component functions f i (i ∈ I) are all continuously differentiable is well studied, see e.g., [17,24,25]. In what follows, we particularly concern the case where the component functions are not necessarily differentiable.…”