Hubertus Th. Jongen

Semi-infinite optimization structure and stability of the feasible set

The problem of the minimization of a functionf: ℝn→ℝ under finitely many equality constraints and perhaps infinitely many inequality constraints gives rise to a structural analysis of the feasible setM[H, G]={x∈ℝn¦H(x)=0,G(x, y)≥0,y∈Y} with compactY⊂ℝr. An extension of the well-known Mangasarian-Fromovitz constraint qualification (EMFCQ) is introduced. The main result for compactM[H, G] is the equivalence of the topological stability of the feasible setM[H, G] and the validity of EMFCQ. As a byproduct, we obtain under EMFCQ that the feasible set admits local linearizations and also thatM[H, G] depends continuously on the pair (H, G). Moreover, EMFCQ is shown to be satisfied generically.

On parametric nonlinear programming

In this tutorial survey we study finite dimensional optimization problems which depend on parameters. It is our aim to work out several basic connections with different mathematical areas. In particular, attention will be paid to unfolding and singularity theory, structural analysis of families of constraint sets, constrained optimization problems and semi-infinite optimization.

Disjunctive optimization: critical point theory

In this paper, we introduce the concepts of (nondegenerate) stationary points and stationary index for disjunctive optimization problems. Two basic theorems from Morse theory, which imply the validity of the (standard) Morse relations, are proved. The first one is a deformation theorem which applies outside the stationary point set. The second one is a cell-attachment theorem which applies at nondegenerate stationary points. The dimension of the cell to be attached equals the stationary index. Here, the stationary index depends on both the restricted Hessian of the Lagrangian and the set of active inequality constraints. In standard optimization problems, the latter contribution vanishes.

One-Parametric Semi-Infinite Optimization: On the Stability of the Feasible Set

This paper studies a global stability property of the (noncompact) feasible set

On Stability and Deformation in Semi-Infinite Optimization

M( H,G,t )

ON STRUCTURE AND COMPUTATION OF GENERALIZED NASH EQUILIBRIA

of a semi-infinite optimization problem defined by finitely many equations

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

H( x,t ) = 0

Generalized Semi-Infinite Programming: on generic local minimizers

and, perhaps, infinitely many inequalities

Bilevel Optimization: on the Structure of the Feasible Set

G( x,t,y ) \leq 0

Nonconvex Optimization and Its Structural Frontiers

that depend on a real parameter t that varies in a compact parameter interval T.Global stability refers to the homeomorphy of