Harald Günzel

On the closure of the feasible set in generalized semi-infinite programming

In generalized semi-infinite programming the feasible set is known to be not closed in general. In this paper, under natural and generic assumptions, the closure of the feasible set is described in explicit terms.

On Logarithmic Smoothing of the Maximum Function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation gε of f itself, where ε>0 denotes the approximation parameter. The one-parametric family gε converges – relative to a compact subset – uniformly to the function f as ε tends to zero. Under nondegeneracy assumptions we show that the stationary points of gε and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x(ε) of stationary points for gε, where each x(ε) converges to the corresponding stationary point of f as ε tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve x(ε).

Generalized Semi-Infinite Programming: on generic local minimizers

In this paper a basic structural problem in Generalized Semi-Infinite Programming is solved. In fact, under natural and generic assumptions we show that at any (local) minimizer the “Symmetric Reduction Ansatz” holds.

The universal partition theorem for oriented matroids

We consider realization spaces of a family of oriented matroids of rank three as point configurations in the affine plane. The fundamental problem arises as to which way these realization spaces partition their embedding space. The Universal Partition Theorem roughly states that such a partition can be as complicated as any partition of ℝn into elementary semialgebraic sets induced by an arbitrary finite set of polynomials in ℤ[X]. We present the first proof of the Universal Partition Theorem. In particular, it includes the first complete proof of the so-called Universality Theorem.

Generalized semi-infinite programming: the Symmetric Reduction Ansatz

We introduce the Symmetric Reduction Ansatz at a point from the closure of the feasible set in generalized semi-infinite programming. A corresponding Symmetric Reduction Lemma is shown for the local description of the latter set, and optimality conditions as well as topological properties are derived. We conjecture that the Symmetric Reduction Ansatz holds at all local minimizers of generic generalized semi-infinite programs.

Optimal Structural Design in Nonsmooth Mechanics

Optimal structural design problems concern with the choice of the design parameters (material variables, shapes etc) of a structure such as to make optimal certain response variables; this task is measured by an appropriate cost or gain function. They are mathematical programs with equilibrium constraints. The use of bilevel optimization techniques permit us to separate the optimal design problem and the structural analysis problem in two levels (subproblems) and, accordingly, allow for the use of existing methods and software for the solution of each one subproblem. In particular the utilisation of the powerful finite element analysis software is indispensable, since it permits the effective treatment of large scale problems.

The structured jet transversality theorem

A structured jet transversality theorem is presented, dealing with functions possibly depending differently on any variable involved. A method for modelling highly complex situations to fit within the framework of the structured jet transversality theorem is described. Some applications to optimization theory are given. †Dedicated to Bert Jongen on the occasion of his 60th birthday.

On the Topology of the Karush--Kuhn--Tucker Set Under Mangasarian--Fromovitz Constraint Qualification

This paper deals with smooth optimization problems

On Reducibility in Bilevel Problems

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Strong stability implies Mangasarian–Fromovitz Constraint Qualification

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