Nonsmooth Optimization; Proceedings of an IIASA Workshop, March 28 - April 8, 1977

Optimization, a central methodological tool of systems analysis, is used in many of IIASA's research areas, including the Energy Systems and Food and Agriculture Programs. IIASA's activity in the field of optimization is strongly connected with nonsmooth or nondifferentiable extreme problems, which consist of searching for conditional or unconditional minima of functions that, due to their complicated internal structure, have no continuous derivatives. Particularly significant for these kinds of extreme problems in systems analysis is the strong link between nonsmooth or nondifferentiable optimization and the decomposition approach to large-scale programming. This volume contains the report of the IIASA workshop held from March 28 to April 8, 1977, entitled Nondifferentiable Optimization. However, the title was changed to Nonsmooth Optimization for publication of this volume as we are concerned not only with optimization without derivatives, but also with problems having functions for which gradients exist almost everywhere but are not continous, so that the usual gradient-based methods fail. Because of the small number of participants and the unusual length of the workshop, a substantial exchange of information was possible. As a result, details of the main developments in nonsmooth optimization are summarized in this volume, which might also be considered a guide for inexperienced users. Eight papers are presented: three on subgradient optimization, four on descent methods, and one on applicability. The report also includes a set of nonsmooth optimization test problems and a comprehensive bibliography

Proximal bundle method for contact shape optimization problem

From the mathematical point of view, the contact shape optimization is a problem of nonlinear optimization with a specific structure, which can be exploited in its solution. In this paper, we show how to overcome the difficulties related to the nonsmooth cost function by using the proximal bundle methods. We describe all steps of the solution, including linearization, construction of a descent direction, line search, stopping criterion, etc. To illustrate the performance of the presented algorithm, we solve a shape optimization problem associated with the discretized two-dimensional contact problem with Coulomb's friction

NOA1: A Fortran Package of Nondifferentiable Optimization Algorithms Methodological and User's Guide

This paper is one of the series of 11 Working Papers presenting the software for interactive decision support and software tools for developing decision support systems. These products constitute the outcome of the contracted study agreement between the System and Decision Sciences Program at IIASA and several Polish scientific institutions. The theoretical part of these results is presented in the IIASA Working Paper WP-88-071 entitled "Theory, Software and Testing Examples in Decision Support Systems". This volume contains the theoretical and methodological backgrounds of the software systems developed within the project. This paper constitutes a methodological guide and user's manual for NOA1, a package of Fortran subroutines designed to locate the minimum of a locally Lipschitz continuous function subject to locally Lipschitzian inequality and equality constraints, general linear constraints and simple upper and lower bounds. The user must provide a Fortran subroutine for evaluating the (possibly nondifferentiable and nonconvex) functions being minimized and their subgradients. The package implements several descent methods, and is intended for solving small-scale nondifferentiable minimization problems on a professional microcomputer

On epsilon-Differential Mappings and their Applications in Nondifferentiable Optimization

In Section 1 we give some review of the recent developments in nondifferential optimization and discuss the difficulties of the application of subgradient methods. It is shown that the use of epsilon-subgradient methods may bring computational advantages. Section 2 contains the technical results on continuity of epsilon-subdifferentials. The principal result of this section consists in establishing Lipschitz continuity of epsilon-subdifferential mappings. Section 3 gives some results on convergence of weighted sums of multifunctions. These results will be used in the study of the convergence of epsilon-subgradient method with sequential averages given in Section 4. Section 4 gives the convergence theory for several modifications of this method. It is shown that in some cases it is possible to neglect accuracy control for the solution of internal maximum problems in the minmax problems. The results when this accuracy is nonzero and fixed are of great practical importance