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This paper is concerned with numerical methods for solving a semi-infinite programming problem. We first reformulate the KKT system derived from the problem into a system of semismooth equations by using the F-B NCP function. Under some conditions, a solution of the system of semismooth equations is a solution of the problem. Then we develop a filter-trust-region method for solving this system of semismooth equations. An advantage of the proposed method is that at each iteration, only a system of linear equations is solved. We prove that under standard assumptions, the iterate sequence generated by this proposed method converges globally and superlinearly. Numerical tests are also reported.

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This paper presents a nonmonotone scaled memoryless BFGS preconditioned conjugate gradient algorithm for solving nonsmooth convex optimization problems, which combines the idea of scaled memoryless BFGS preconditioned conjugate gradient method with the nonmonotone technique and the Moreau-Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. Under mild conditions, the global convergence of the proposed method is established. Preliminary numerical results and related comparisons show that the proposed method can be applied to solve large scale nonsmooth convex optimization problems.