Diethard Klatte

Asymptotic constraint qualifications and global error bounds for convex inequalities

In this paper we study various asymptotic constraint qualifications for the existence of global error bounds for approximate solutions of convex inequalities. Many known conditions that ensure the existence of such a global error bound are shown to be equivalent to one of the following three conditions: (i) the bounded excess condition, (ii) Slater condition together with the asymptotic constraint qualification defined by Auslender and Crouzeix [1], and (iii) positivity of normal directional derivatives of the maximum of the constraint functions introduced by Lewis and Pang [12].

A Frank–Wolfe Type Theorem for Convex Polynomial Programs

In 1956, Frank and Wolfe extended the fundamental existence theorem of linear programming by proving that an arbitrary quadratic function f attains its minimum over a nonempty convex polyhedral set X provided f is bounded from below over X. We show that a similar statement holds if f is a convex polynomial and X is the solution set of a system of convex polynomial inequalities. In fact, this result was published by the first author already in a 1977 book, but seems to have been unnoticed until now. Further, we discuss the behavior of convex polynomial sets under linear transformations and derive some consequences of the Frank–Wolfe type theorem for perturbed problems.

Metric Regularity in Convex Semi-Infinite Optimization under Canonical Perturbations

This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right-hand side of the constraints and linear perturbations of the objective function. In this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the Slater constraint qualification, together with a certain additional requirement in the Karush-Kuhn-Tucker conditions. For linear problems this sufficient condition turns out to be also necessary for the metric regularity, and it is equivalent to some well-known stability concepts.

Implicit functions and sensitivity of stationary points

AbstractWe consider the spaceL(D) consisting of Lipschitz continuous mappings fromD to the Euclideann-space ℝn,D being an open bounded subset of ℝn. LetF belong toL(D) and suppose that

Regularity and Stability in Nonlinear Semi-Infinite Optimization

\bar x

Stable local minimizers in semi-infinite optimization: regularity and second-order conditions

solves the equationF(x) = 0. In case that the generalized Jacobian ofF at

Stability Properties of Infima and Optimal Solutions of Parametric Optimization Problems

\bar x