Jiří V. Outrata

Nonsmooth Approach to Optimization Problems with Equilibrium Constraints

Preface. List of Notations. List of Acronyms. Part I: Theory. 1.Introduction. 2. Auxiliary Results. 3. Algorithms of Nonsmooth Optimization. 4. Generalized Equations. 5. Stability of Solutions to Perturbed Generalized Equations. 6. Derivatives of Solutions to Perturbed Generalized Equations. 7. Optimality Conditions and a Solution Method. Part II: Applications. 8. Introduction. 9. Membrane with Obstacle. 10. Elasticity Problems with Internal Obstacles. 11. Contact Problem with Coulomb Friction. 12. Economic Applications. Appendices: A. Cookbook. B. Basic Facts on Elliptic Boundary Value problems. C. Complementarity Problems. References. Index.

Calmness of constraint systems with applications

The paper is devoted to the analysis of the calmness property for constraint set mappings. After some general characterizations, specific results are obtained for various types of constraints, e.g., one single nonsmooth inequality, differentiable constraints modeled by polyhedral sets, finitely and infinitely many differentiable inequalities. The obtained conditions enable the detection of calmness in a number of situations, where the standard criteria (via polyhedrality or the Aubin property) do not work. Their application in the framework of generalized differential calculus is explained and illustrated by examples associated with optimization and stability issues in connection with nonlinear complementarity problems or continuity of the value-at-risk.

On Optimization Problems with Variational Inequality Constraints

This paper is devoted to a class of optimization problems that contain variational inequality or nonlinear complementarity constraints. Problems of this kind arise, for example, in game theory, bilevel programming, and the design of networks subject to equilibrium conditions. For such problems first-order necessary optimality conditions are derived and a numerical approach is proposed, based on nondifferentiable optimization techniques. This approach is illustrated by simple examples.

On mathematical programs with complementarity constraints

The paper deals with mathematical programs, where a complementarity problem arises among the constraints. The main attention is paid to optimality conditions and the respective constraint qualification. In addition, a simple numerical approach is proposed based on the exact penalization of the complementarity constraint.

A Newton method for a class of quasi-variational inequalities

A variant of the Newton method for nonsmooth equations is applied to solve numerically quasivariational inequalities with monotone operators. For this purpose, we investigate the semismoothness of a certain locally Lipschitz operator coming from the quasi-variational inequality, and analyse the generalized Jacobian of this operator to ensure local convergence of the method. A simplified variant of this approach, applicable to implicit complementarity problems, is also studied. Small test examples have been computed.

On a class of quasi-variational inequalities

In this paper we give an existence result for a class of quasi-variational inequalities. Further, we propose a nonsmooth variant of the Newton method for their numerical solution. Using the tools of sensitivity and stability theory and nonsmooth analysis, criteria are formulated ensuring the local superlinear convergence. The method is applied to the discretized contact problem with the Coulomb friction model

Mathematical Programs with Equilibrium Constraints: Theory and Numerical Methods

The lecture notes deal with optimization problems, where a generalized equation (modeling an equilibrium) arises among the constraints. The main attention is paid to necessary optimality conditions and methods to the numerical solution of such problems. The applications come from continuum mechanics.

Optimization problems with equilibrium constraints and their numerical solution

Abstract.We consider a class of optimization problems with a generalized equation among the constraints. This class covers several problem types like MPEC (Mathematical Programs with Equilibrium Constraints) and MPCC (Mathematical Programs with Complementarity Constraints). We briefly review techniques used for numerical solution of these problems: penalty methods, nonlinear programming (NLP) techniques and Implicit Programming approach (ImP). We further present a new theoretical framework for the ImP technique that is particularly useful in case of difficult equilibria. Finally, three numerical examples are presented: an MPEC that can be solved by ImP but can hardly be formulated as a nonlinear program, an MPCC that cannot be solved by ImP and finally an MPEC solvable by both, ImP and NLP techniques. In the last example we compare the efficiency of the two approaches.

Shape Optimization in Contact Problems with Coulomb Friction

The paper deals with a discretized problem of the shape optimization of elastic bodies in unilateral contact. The aim is to extend existing results to the case of contact problems following the Coulomb friction law. Mathematical modelling of the Coulomb friction problem leads to a quasi-variational inequality. It is shown that for small coefficients of friction the discretized problem with Coulomb friction has a unique solution and that this solution is Lipschitzian as a function of a control variable describing the shape of the elastic body. The shape optimization problem belongs to a class of so-called mathematical programs with equilibrium constraints (MPECs). The uniqueness of the equilibria for fixed controls enables us to apply the so-called implicit programming approach. Its main idea consists of minimizing a nonsmooth composite function generated by the objective and the (single-valued) control-state mapping. In this paper, the control-state mapping is much more complicated than in most MPECs solved in the literature so far, and the generalization of the relevant results is by no means straightforward. Numerical examples illustrate the efficiency and reliability of the suggested approach.

ON THE AUBIN PROPERTY OF CRITICAL POINTS TO PERTURBED SECOND-ORDER CONE PROGRAMS *

Two gaps were found in the proof of the main theorems (Theorems 21 and 26) of the paper “On the Aubin property of critical points to perturbed second-order cone programs” [SIAM J. Optim. 21 (2011), 3, pp. 798--823] by J. V. Outrata and H. Ramirez C. In this note both these gaps will be filled. As to the second one, a new technical result will be employed which may possibly be used also in other situations.