An active set strategy to address the ill-conditioning of smoothing methods for solving finite linear minimax problems

In this paper, an active set strategy is presented to address the ill-conditioning of smoothing methods for solving finite linear minimax problems. Based on the first order optimality conditions, a concept of the strongly active set composed of a part of active indexes is introduced. In the active set strategy, a strongly active set is obtained by solving a linear system or a linear programming problem, then an optimal solution with its active set and Lagrange multipliers is computed by an iterative process. A hybrid algorithm combining a smoothing algorithm and the active set strategy is proposed for solving finite linear minimax problems, in which an approximate solution is obtained by the smoothing algorithm, then an optimal solution is computed by the active set strategy. The convergences of the active set strategy and the hybrid algorithm are established for general finite linear minimax problems. Preliminary numerical experiments show that the active set strategy and the hybrid algorithm are effective and robust, and the active set strategy can effectively address the ill-conditioning of smoothing methods for solving general finite linear minimax problems.