Nonsmooth Optimization Algorithms

Abstract

Quasidifferentiable and codifferentiable optimization algorithms are based on gradientlike, descent, iterative techniques whereas gradient information is replaced by the setvalued quasidifferential or the codifferential. Then the steepest descent finding subproblems are appropriately replaced by quadratic programming subproblems with a polyhedral approximation of the aforementioned set—valued quantities. Since supergradients (resp. hyperdifferentials) pose a combinatorial problem in the descent direction finding subproblem, which can effectively be treated after making the polyhedral approximation by repeated solution of a number of similar subproblems or simply by solving one of them (supergradient—like technique), the basic methods used are the ones of hypodifferential optimization. These techniques will be described in the sequel (for more details we refer to [3], [9], [5]). It should be mentioned here that first order quasidifferential and codifferential optimization schemes treat more effectively, in a correct way vertical branches of laws and boundary conditions in mechanical problems, or equivalently, the nonsmoothness of the respective potentials. If at a neighbourhood of the solution the problem is essentially smooth, i.e. the solution lies far away from a point of nondifferentiability, classical methods of nonlinear computational mechanics (e.g. Newton—type methods and its derivates) can be used for the refinement of the accuracy and for speeding up the rate of convergence. Nevertheless if multiple points of nondifferentiability (cusps) have to be passed along a given loading path the general methods presented here must be used (see also Chapter 8).