Vladimir Shikhman

ON STRUCTURE AND COMPUTATION OF GENERALIZED NASH EQUILIBRIA

We consider generalized Nash equilibrium problems (GNEP) from a structural and computational point of view. In GNEP the players’ feasible sets may depend on the other players’ strategies. Moreover, the players may share common constraints. In particular, the latter leads to the stable appearance of Nash equilibria which are Fritz-John (FJ) points, but not Karush-Kuhn-Tucker (KKT) points. Basic in our approach is the representation of FJ points as zeros of an appropriate underdetermined system of nonsmooth equations. Here, additional nonsmooth variables are taken into account. We prove that the set of FJ points (together with corresponding active Lagrange multipliers) - generically - constitutes a Lipschitz manifold. Its dimension is (N-1)J_0, where N is the number of players and J_0 is the number of active common constraints. In a structural analysis of Nash equilibria the number (N-1)J_0 plays a crucial role. In fact, the latter number encodes both the possible degeneracies for the players’ parametric subproblems and the dimension of the set of Nash equilibria. In particular, in the nondegenerate case, the dimension of the set of Nash equilibria equals locally (N-1)J_0. For the computation of FJ points we propose a nonsmooth projection method (NPM) which aims at nding solutions of an underdetermined system of nonsmooth equations. NPM is shown to be well-dened for GNEP. Local convergence of NPM is conjectured for GNEP under generic assumptions and its proof is challenging. However, we indicate special cases (known from the literature) in which convergence holds.

Quasi-monotone Subgradient Methods for Nonsmooth Convex Minimization

In this paper, we develop new subgradient methods for solving nonsmooth convex optimization problems. These methods guarantee the best possible rate of convergence for the whole sequence of test points. Our methods are applicable as efficient real-time stabilization tools for potential systems with infinite horizon. Preliminary numerical experiments confirm a high efficiency of the new schemes.

General Semi-Infinite Programming: Symmetric Mangasarian-Fromovitz Constraint Qualification and the Closure of the Feasible Set

The feasible set

MPCC: Critical Point Theory

M

Mathematical programs with vanishing constraints: critical point theory

in general semi-infinite programming (GSIP) need not be closed. This fact is well known. We introduce a natural constraint qualification, called symmetric Mangasarian-Fromovitz constraint qualification (Sym-MFCQ). The Sym-MFCQ is a nontrivial extension of the well-known (extended) MFCQ for the special case of semi-infinite programming (SIP) and disjunctive programming. Under the Sym-MFCQ the closure

On Stability of the Feasible Set of a Mathematical Problem with Complementarity Problems

\overline{M}

Bilevel Optimization: on the Structure of the Feasible Set

has an easy and also natural description. As a consequence, we get a description of the interior and boundary of

Generalized Semi-Infinite Programming: The Nonsmooth Symmetric Reduction Ansatz

M

Characterization of strong stability for C-stationary points in MPCC

. The Sym-MFCQ is shown to be generic and stable under

On Jet-convex Functions and their Tensor Products

C^1