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In this paper, we present a central cutting plane algorithm for solving convex min-max semi-infinite programming problems. Because the objective function here is non-differentiable, we apply a smoothing technique to the considered problem and develop an algorithm based on the entropy function. It is shown that the global convergence of the proposed algorithm can be obtained under weaker conditions. Some numerical results are presented to show the potential of the proposed algorithm.In this paper, we present a central cutting plane algorithm for solving convex min-max semi-infinite programming problems. Because the objective function here is non-differentiable, we apply a smoothing technique to the considered problem and develop an algorithm based on the entropy function. It is shown that the global convergence of the proposed algorithm can be obtained under weaker conditions. Some numerical results are presented to show the potential of the proposed algorithm.

Abstract In this paper, we introduce the bilevel decision problems with parametric generalized semi-infinite optimization for fuzzy mappings as the lower-level problem, and its corresponding MPEC problems. For these problems, we establish two models which are different in the feasible region setting of lower-level problems. Some new existence results are obtained in rather weak conditions.

This paper is concerned with numerical methods for solving a semi-infinite programming problem. We reformulate the equations and nonlinear complementarity conditions of the first order optimality condition of the problem into a system of semismooth equations. By using a perturbed Fischer–Burmeister function, we develop a smoothing Newton 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 the smoothing Newton method converges superlinearly/quadratically.

We study a variational inequality problem VI(X,F) with Xbeing defined by infinitely many inequality constraints and Fbeing a pseudomonotone function. It is shown that such problem can be reduced to a problem of finding a feasible point in a convex set defined by infinitely many constraints. An analytic center based cutting plane algorithm is proposed for solving the reduced problem. Under proper assumptions, the proposed algorithm finds an ∈-optimal solution in O*(n2/ρ2) iterations, where O*(·) represents the leading order, nis the dimension of X, ε is a user-specified tolerance, and ρ is the radius of a ball contained in the ε-solution set of VI(X,F).

In this paper we present some semismooth Newton methods for solving the semi-infinite programming problem. We first reformulate the equations and nonlinear complementarity conditions derived from the problem into a system of semismooth equations by using NCP functions. Under some conditions a solution of the system of semismooth equations is a solution of the problem. Then some semismooth Newton methods are proposed for solving this system of semismooth equations. These methods are globally and superlinearly convergent. Numerical results are also given.

In this paper, we propose some dual formulations of generalized vector equilibrium problems. By using such dual formulations, we prove the existence of a solution to the generalized vector equilibrium problem under generalized pseudomonotonicity conditions. The results of this paper extend and generalize the results of I.V. Konnov and S. Schaible (2000, J. Optim. Theory Appl., 104,395–408).

This paper studies the difference between finite-dimensional linear programming problems and infinite dimensional linear programming problems. We discuss a special class of continuous linear programming problems. We develop the structure of extreme points of feasible region for this problem. Under some conditions we can characterize all extreme points of this problem. We show that under some conditions the optimal value for this problem may be finite but there is no optimal solution for it. Finally, we construct an example for this problem which has an optimal solution but all extreme points for this problem are not optimal solutions.

We introduce a type of complementarity problems posed over a measure space. Some examples of its application are given, and an algorithm for its solution is described and illustrated with an example.

The paper studies the linear semi-infinite programming problems and its dual problems. The main purpose is to develop an applicable algorithm to solve such kinds of problems.