A NOTE ON LOGARITHMIC SMOOTHING IN SEMI-INFINITE OPTIMIZATION UNDER REDUCTION APPROACH*

This note deals with a semi-infinite optimization problem which is defined by infinitely many inequality constraints. By applying a logarithmic barrier function, a family of interior point approximations of the feasible set is obtained where locally the original feasible set and its approximations are homeomorphic. Under generic assumptions on the structure of the original feasible set, strongly stable stationary points of the original problem are considered and it is shown that there is a one-to-one correspondence between the stationary points (and their stationary indices) of the original problem and those of its approximations. Corresponding convergence results, global aspects and a relationship to a standard interior-point approach are discussed

A modified standard embedding with jumps in nonlinear optimization

The paper deals with a combination of pathfollowing methods (embedding approach) and feasible descent direction methods (so-called jumps) for solving a non-linear optimization problem with equality and inequality constraints. Since the method that we propose here uses jumps from one connected component to another one, more than one connected component of the solution set of the corresponding one-parametric problem can be followed numerically. It is assumed that the problem under consideration belongs to a generic subset which was introduced by Jongen, Jonker and Twilt. There already exist methods of this type for which each starting point of a jump has to be an endpoint of a branch of local minimizers. In this paper the authors propose a new method by allowing a larger set of starting points for the jumps which can be constructed at bifurcation and turning points of the solution set. The topological properties of those cases where the method is not successful are analyzed and the role of constraint qualifications in this context is discussed. Furthermore,this new method is applied to a so-called modified standard embedding which is a particular construction without equality constraints. Finally, an algorithmic version of this new method as well as computational results are presented

Augmented Lagrangians in semi-infinite programming

We consider the class of semi-infinite programming problems which became in recent years a powerful tool for the mathematical modeling of many real-life problems. In this paper, we study an augmented Lagrangian approach to semi-infinite problems and present necessary and sufficient conditions for the existence of corresponding augmented Lagrange multipliers. Furthermore, we discuss two particular cases for the augmenting function: the proximal Lagrangian and the sharp Lagrangian

A note on strict complementarity for the doubly non-negative cone

In this paper, we consider a closed convex cone given by the intersection of two cones and . We study faces and complementary faces of in terms of and . Based on complementary faces, the tangent spaces of can be characterized as well. Moreover, many numerical methods assume regularity conditions such as strict complementarity. We provide necessary and sufficient conditions for strict complementarity for the cone . All these results can be applied to the doubly non-negative cone. Finally, a numerically efficient procedure for checking strict complementarity of for the doubly non-negative cone is provided when X has exactly one zero eigenvalue