Georg Still

Semi-infinite programming

A semi-infinite programming problem is an optimization problem in which finitely many variables appear in infinitely many constraints. This model naturally arises in an abundant number of applications in different fields of mathematics, economics and engineering. The paper, which intends to make a compromise between an introduction and a survey, treats the theoretical basis, numerical methods, applications and historical background of the field.

Solving semi-infinite optimization problems with interior point techniques

We introduce a new numerical solution method for semi-infinite optimization problems with convex lower level problems. The method is based on a reformulation of the semi-infinite problem as a Stackelberg game and the use of regularized nonlinear complementarity problem functions. This approach leads to central path conditions for the lower level problems, where for a given path parameter a smooth nonlinear finite optimization problem has to be solved. The solution of the semi-infinite optimization problem then amounts to driving the path parameter to zero.We show convergence properties of the method and give a number of numerical examples from design centering and from robust optimization, where actually so-called generalized semi-infinite optimization problems are solved. The presented method is easy to implement, and in our examples it works well for dimensions of the semi-infinite index set at least up to 150.

Discretization in semi-infinite programming: the rate of convergence

Abstract.The discretization approach for solving semi-infinite optimization problems is considered. We are interested in the convergence rate of the error between the solution of the semi-infinite problem and the solution of the discretized program depending on the discretization mesh-size. It will be shown how this rate depends on whether the minimizer is strict of order one or two and on whether the discretization includes boundary points of the index set in a specific way. This is done for ordinary and for generalized semi-infinite problems.

On optimality conditions for generalized semi-infinite programming problems

Generalized semi-infinite optimization problems (GSIP) are considered. We generalize the well-known optimality conditions for minimizers of order one in standard semi-infinite programming to the GSIP case. We give necessary and sufficient conditions for local minimizers of order one without the assumption of local reduction. The necessary conditions are derived along the same lines as the first-order necessary conditions for GSIP in a recent paper of Jongen, Rückmann, and Stein (Ref. 1) by assuming the so-called extended Mangasarian–Fromovitz constraint qualification. Using the ideas of a recent paper of Rückmann and Shapiro, we give short proofs of necessary and sufficient optimality conditions for minimizers of order one under the additional assumption of the Mangasarian–Fromovitz constraint qualification at all local minimizers of the so-called lower-level problem.

Interior points of the completely positive cone.

A matrix A is called completely positive if it can be decomposed as A = BB^T with an entrywise nonnegative matrix B. The set of all such matrices is a convex cone. We provide a characterization of the interior of this cone as well as of its dual.

Solving bilevel programs with the KKT-approach

Bilevel programs (BL) form a special class of optimization problems. They appear in many models in economics, game theory and mathematical physics. BL programs show a more complicated structure than standard finite problems. We study the so-called KKT-approach for solving bilevel problems, where the lower level minimality condition is replaced by the KKT- or the FJ-condition. This leads to a special structured mathematical program with complementarity constraints. We analyze the KKT-approach from a generic viewpoint and reveal the advantages and possible drawbacks of this approach for solving BL problems numerically.

A new relaxation method for the generalized minimum spanning tree problem

We consider a generalization of the minimum spanning tree problem, called the generalized minimum spanning tree problem, denoted by GMST. It is known that the GMST problem is -hard. We present several mixed integer programming formulations of the problem. Based on a new formulation of the problem we give a new solution procedure that finds the optimal solution of the GMST problem for graphs with nodes up to 240. We discuss the advantages of our approach in comparison with earlier methods.

Equilibrium constrained optimization problems

We consider equilibrium constrained optimization problems, which have a general formulationthat encompasses well-known models such as mathematical programs with equilibrium constraints, bilevel programs, and generalized semi-infinite programming problems. Based on the celebrated K K M lemma, we prove the existence of feasible points for the equilibrium constraints. Moreover, we analyze the topological and analytical structure of the feasible set. Alternative formulations of an equilibrium constrained optimization problem (ECOP) that are suitable for numerical purposes are also given. As an important _rst step for developing ef_cient algorithms, we provide a genericity analysis for the feasible set of a particular ECOP, for which all the functions are assumed to be linear.

Optimality conditions in smooth nonlinear programming

This survey is concerned with necessary and sufficient optimality conditions for smooth nonlinear programming problems with inequality and equality constraints. These conditions deal with strict local minimizers of order one and two and with isolated minimizers. In most results, no constraint qualification is required. The optimality conditions are formulated in such a way that the gaps between the necessary and sufficient conditions are small and even vanish completely under mild constraint qualifications.

Mathematical Programs with Complementarity Constraints: Convergence Properties of a Smoothing Method

In this paper, optimization problems P with complementarity constraints are considered. Characterizations for local minimizers \bar{x} of P of Orders 1 and 2 are presented. We analyze a parametric smoothing approach for solving these programs in which P is replaced by a perturbed problem P_{\tau} depending on a (small) parameter \tau . We are interested in the convergence behavior of the feasible set \cal{F}_{\tau} and the convergence of the solutions \bar{x}_{\tau} of P_{\tau} for \tau\to 0 . In particular, it is shown that, under generic assumptions, the solutions \bar{x}_{\tau} are unique and converge to a solution \bar{x} of P with a rate \cal{O}(\sqrt{\tau}) . Moreover, the convergence for the Hausdorff distance d(\cal{F}_{\tau}, \cal{F}) between the feasible sets of P_{\tau} and P is of order \cal{O}(\sqrt{\tau}) .