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RESEARCH PRODUCT

Nonsmooth Optimization Methods

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From the previous chapters we know that after the discretization, elliptic and parabolic hemivariational inequalities can be transformed into substationary point type problems for locally Lipschitz superpotentials and as such will be solved. There is a class of mathematical programming methods especially developed for this type of problems. The aim of this chapter is to give an overview of nonsmooth optimization techniques with special emphasis on the first and the second order bundle methods. We present their basic ideas in the convex case and necessary modifications for nonconvex optimization. We shall use them in the next chapter for the numerical realization of several model examples. Let us mention also that bundle type methods can be applied to a large class of locally Lipschitz superpotentials without any special requirement on their properties (such as the difference of two convex functions, e.g.).