Nonsmooth Approach to Optimization Problems with Equilibrium Constraints

The central subject of this book is an optimization problem with a special side constraint. Its definition and some illustrative examples are displayed in Section 1.1. Section 1.2 briefly describes various optimality conditions and numerical methods. Section 1.3 states two simple existence results.

This chapter collects some nontrivial results which will be needed in the rest of the book. We try to keep the presentation as self-contained as possible.

As we will see later, variational inequalities (and complementarity problems) provide a convenient and elegant tool for characterizing manifold equilibria. The aim of this chapter is to spell out how these models can be brought into the equally useful form of a generalized equation where C[ℝ ^k fℝ ^k ] is a continuous mapping, Q a nonempty, closed, convex subset of ℝ^k and N ^Q (z) its normal cone to Q at z; cf. Definition 2.6. Q is called the feasible set of the GE (4.1). Oftentimes, the rewriting as a “nonsmooth equation” is not only possible but very helpful. While proceeding, we also collect several basic results on existence and uniqueness needed in the later chapters. Our objective is to prepare for subsequent analysis and computations.

In this chapter we apply the preceding theory to establish first-order necessary optimality conditions and to construct an efficient and robust numerical method for the solution of the considered MPECs. This numerical method will extensively be used in the second (“applied”) part of the book.

The idea to use the implicit programming approach in optimum shape design problems is in fact quite old. It was extensively used for solving problems with equilibrium constraints given by variational equations. In this case it is straightforward to plug the unique solution of the equation into the objective function and compute derivatives of such a composite function by means of the implicit-function theorem. If the equilibrium problem is more complicated, e.g., it is a variational inequality, this technique becomes less straightforward. In classic shape optimization, one circumvents the difficulty (switch from equality to inequality) by means of the regularization technique.

We start the application part of the book with a classic example of an elliptic variational inequality—an elastic membrane with an obstacle. First, we describe the state problem and then, in the subsequent sections, three shape optimization problems of graded complexity.

Although the membrane problems discussed in the previous chapter have practical applications, they are usually considered as only model problems. Truly interesting practical examples arise from modelling of two- and three-dimensional elastic bodies. Their behaviour is again described by elliptic variational equations or inequalities.

In the previous two chapters we have introduced problems with obstacles and the problem of linear elasticity. The goal of this chapter is to combine those two and to introduce probably the most important ”nonsmooth“ problem of mechanics: the problem of an elastic body in contact with a rigid obstacle. A simple way how to define this problem is to restrict the normal displacement of certain boundary points in the elasticity problem by means of the unilateral contact constraints known from Chapter 9. This leads to a formulation known as contact problem without friction; in most cases, this formulation does not fully reflect the physical reality. To achieve this, one has to take into account the friction between the body and the obstacle—this friction restricts the tangential component of the displacement on the ”contact boundary“. There are several models of contact problems with friction (cf., e.g., Klarbring, 1986; Lemaitre and Chaboche, 1994). The most realistic one is probably the model of Coulomb friction. In the following sections we will introduce this model, derive its reciprocal (dual) formulation which leads to a quasi-variational inequality and apply the nonsmooth Newton’s method of Chapter 3 to the numerical solution of a suitable discrete approximation. We further show that the discretized problem can be formulated as a linear complementarity problem which, again, can be solved by the nonsmooth Newton’s method. Finally, we formulate a design problem; the control (design) variable is the coefficient of friction (a property of the material) and the goal to maximize the tangential adhesion between the body and the obstacle. This design problem is an MPEC with an LCP as equilibrium constraint.

Central to many aspects of economic analysis (and even political theory) is the concept of equilibrium. Similarly as in mechanics, formal characterizations of this concept typically come as complementarity problems or variational inequalities.