The purpose of this paper is to study the recent development of parametric semi-infinite linear optimization. Especially, we will investigate continuity properties of the feasible set mapping and of the optimal value. Further, we will investigate the structure of certain parameter sets. We begin by stating some definitions and by recalling a necessary and sufficient condition that a set of feasible points is optimal. In Section 2, we derive various conditions for the continuity of the feasible set mapping Z. In particular, we derive a result of Fischer [16] that Z is upper semicontinuous at a parameter σ0 if and only if \(Z_{\sigma _0 }\) is compact. In Section 3, we investigate the continuity of the optimal value. We derive various conditions for the continuity of E, e.g. E is locally Lipschitz at σ0 if \(P_{\sigma _0 }\) is compact and \(Z_{\sigma _0 }\) satisfies the Slater-condition. Finally, we show that certain subsets of the parameter space are convex. These results imply that E is piecewise linear and convex (resp. concave) on certain subsets of the parameter space.

Parametric semi-infinite linear programming I. continuity of the feasible set and of the optimal value

This paper is devoted to the study of continuity properties of Pareto solution maps for parametric semi-infinite vector optimization problems (PSVO). We establish new necessary conditions for lower and upper semicontinuity of Pareto solution maps under functional perturbations of both objective functions and constraint sets. We also show that the necessary condition becomes sufficient for the lower and upper semicontinuous properties in the special case where the constraint set mapping is lower semicontinuous at the reference point. Examples are given to illustrate the obtained results.

Stability of semi-infinite vector optimization problems under functional perturbations

Pseudo-Lipschitz property of linear semi-infinite vector optimization problems

This paper is devoted to the study of the pseudo-Lipschitz property of the efficient (Pareto) solution map for the perturbed convex semi-infinite vector optimization problem (CSVO). We establish sufficient conditions for the pseudo-Lipschitz property of the efficient solution map of (CSVO) under continuous perturbations of the right-hand side of the constraints and functional perturbations of the objective function. Examples are given to illustrate the obtained results.

Sufficient conditions for pseudo-Lipschitz property in convex semi-infinite vector optimization problems

In this paper, we treat optimization problems of the following type: minimize a linear functional p on a closed, convex subset of Euclidean m-space. (Semi- infinite linear optimization problems are of this type.) We investigate upper semicontinuity of the set of optimal solutions and the continuous dependence of the optimal value on p. Thereby we essentially work with the concept of characteristic cones of closed, convex sets.

Continuity properties in semi-infinite parametric linear optimization

This paper treats stability theory for optimization problems of the following type: Minimize (p,x) subject to x ∊ F(φ), where p ∊ IRm and F(φ) is a non-void, closed, convex subset of IRm for φ ∊ Φ which is a topological space. We investigate upper and lower semicontinuous dependence of the set of optimal solutions on (p,φ). The obtained results imply piecewise continuity of time-optimal controls of certain control systems governed by ordinary differential equations.

Parametric Optimization and an Application to Optimal Control Theory

We consider parametric optimization problems of the following type; Minimize p(x,t) subject to x∊C, for all t∊T, where C is a weakly locally compact, closed, and convex subset of a Hilbert space H, T is a topological space, and p is a real-valued function on H×T lower semicontinuous if H is provided with the weak topology, and continuous in the second variable for all x∊H. We investigate how the optimal value, and the set of optimal solutions depend on tET, We apply our results to show that the discrete eigenvalues, and the corresponding normed eigenelements of certain self-adjoint operators in Hilbert spaces depend (the latter strongly) continuous on parameters under reasonable conditions.

Rainer Colgen

Papers

Continuity properties in semi-infinite parametric linear optimization

Parametric Optimization and an Application to Optimal Control Theory

On parametric optimization in hilbert spaces