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Mathematical Programs with Equilibrium Constraints

Optimal control problems, in which the controlled systems are governed by variational inequalities, arise frequently in optimum design (e.g., shape optimization of contact problems, of elastic-plastic bodies, etc.). The numerical solution of these problems remains to be a difficult task. The classical regularization technique, using smooth penalization of the inequality, leads either to a low accuracy or to ill-conditioned problems. We developed a new approach, in which we handle the variational inequality as a nondifferentiable controlled system by using the tools of nondifferential analysis. In this way one can achieve a substantially higher accuracy compared to the regularization technique. Moreover, the incidental state-space constraints may be treated by exact penalties, which further contributes to the quality of the results.

In our approach we assume that the lower-level problem has for each x a unique solution S(x) (this is a natural assumption in many problems from continuum mechanics) and rewrite the original bi-level problem as a one-level one. The new problem is a nonsmooth optimization problem which can be solved by NSO software, e.g. by the BT code. We gave the necessary theoretical results and applied the approach to the solution of the following problems: shape optimization of a membrane with a rigid obstacle and state-space constraint Kocvara-Outrata (1991) and Kocvara-Outrata (1995a); shape optimization of elastic-plastic bodies obeying Hencky's law Kocvara-Outrata (1994b); shape optimization of masonry structures Kocvara-Outrata (1994c).

The above approach was extended also to the case when, instead of the state variational inequality, we solve the Quasi-Variational Inequality (QVI). In the first stage we concentrated on a particular case when the QVI reduces to an Implicit Complementarity Problem (ICP). If this problem has for each control x a unique solution S(x), we are able to modify the above approach to this situation. Solving of the ICP is already a difficult task. We rewrote it as a nonsmooth equation and solved this equation by a nonsmooth variant of the Newton method. The outer optimization problem was again solved by the BT code. The corresponding theory and results for shape optimization problem of a membrane with a compliant obstacle was presented in Kocvara-Outrata (1993).

An important application of our approach is the solution and control of the contact problems with Coulomb friction. Using the reciprocal variational formulation of he problem, we can write it as a QVI or even as an LCP with nonsymemtric and indefinite system matrix. The state problems can be efficiently solved by a nonsmooth variant of the Newton's method; cf. Kocvara-Outrata (1995b). By the above approach we can solve control problems where the control parameter is the distribution of the coefficient of friction along the contact boundary Kocvara-Outrata (1997a).

All this (and much more) can be found in the monograph

J. Outrata, M. Kocvara and J. Zowe: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Kluwer Academic Publishers, Dordrecht, 1998.
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Michal Kocvara
March 1, 2001