Iterative Parametric Minimax Method for a Class of Composite Optimization Problems

Abstract

This paper considers a class of composite optimization problems that are often difficult to solve directly due to large dimension, nonlinearity, nonseparability, and/or nonconvexity of the problem. An iterative parametric minimax method is proposed in which the original optimization problem is embedded into a weighted minimax formulation. The resulting auxiliary parametric optimization problems at the lower level often have simple structures that are readily tackled by efficient solution strategies, such as the decomposition scheme in dynamic programming and in the primal-dual method. The analytical expression of the partial derivatives of systems performance indices with respect to the weighting vector in the parametric minimax formulation is derived. The gradient method can be thus adopted at the upper level to adjust the value of the weighting vector. The solution of the weighted minimax formulation converges to the optimal solution of the original problem in a multilevel iteration process. An application of the proposed iterative parametric minimax method is demonstrated in constrained reliability optimization problems.