Diethard Pallaschke

Nondifferentiable Optimization: Motivations and Applications

IIASA has been involved in research on nondifferentiable optimization since 1976. The Institutes research in this field has been very productive, leading to many important theoretical, algorithmic and applied results. Nondifferentiable optimization has now become a recognized and rapidly developing branch of mathematical programming. To continue this tradition and to review developments in this field IIASA held this Workshop in Sopron (Hungary) in September 1984. This volume contains selected papers presented at the Workshop. It is divided into four sections dealing with the following topics: (I) Concepts in Nonsmooth Analysis; (II) Multicriteria Optimization and Control Theory; (III) Algorithms and Optimization Methods; (IV) Stochastic Programming and Applications.

On linearization and continuous selections of functions

Let f n i = 1,…p, be a finite number of differentiate functions on the Euclidean n-space. We study continuous selections of these functions. The concept of a (non-degenerate) critical point is introduced. In a neighborhood of a noncritical point, the continuous selection turns out to be topologicaily equivalent with a linear function. This result remains true for locally Lipschitz functions. Around a nondegenerate critical point, the continuous selection f is shown to be topologically equivalent with CS(L) +Q, where CS(L) is a specific continuous selection of (k+ 1) linear functions on the Euclidean k-space, and where Q is the sum of (positive and negative) squares of the remaining (n-k) coordinate functions. Finally, the case p≤3 is treated in detail, and we make a link with quasi-differentiate functions.

Reduction of quasidifferentials and minimal representations

Some criterias for the non-minimality of pairs of compact convex sets of a real locally convex topological vector space are proved, based on a reduction technique via cutting planes and excision of compact convex subsets. Following an example of J. Grzybowski, we construct a class of equivalent minimal pairs of compact convex sets which are not connected by translations.

On Khachian's algorithm and minimal ellipsoids

SummaryKhachians algorithm for solving a system of linear inequalities is accelerated by choosing smaller ellipsoids than in the original version. Furthermore, certain inequalities can be successively eliminated, from the constraints, yielding a different stopping rule.

On Morse Theory for Piecewise Smooth Functions

Lower level sets of continuous selections of C2-functions defined on a smooth manifold in the vicinity of a nondegenerate critical point in the sense of [11] are studied. It is shown that the lower level set is homotopy equivalent to the join of the lower level sets of the smooth and the nonsmooth part, respectively, of the corresponding normal form. Some generalized Morse inequalities are deduced from this result.

Some criteria for the minimality of pairs of compact convex sets

Some criteria for minimality for a pair of compact convex subsets of a real locally convex vector space are proved. Moreover several examples are given.

On locally-Lipschitz quasi-differentiate functions in Banach-spaces

In this paper quasi-differentiable functions in the sense of V. F. Demyanov and A. N. Rubinov [1] are studied. For locally Lipschitz functions defined on finite dimensional Banach-spaces, a characterization of the quasi-differentiability is given, which is closely related to the Fe-differentiability.

Computing the Minkowski sum of prisms

AbstractWithin this paper we study the Minkowski sum of prisms (“Cephoids”) in a finite dimensional vector space. For a vector

On the Separation and Order Law of Cancellation for Bounded Sets

a \in \mathbb{R}^n