By V. F. Demyanov, C. Lemaréchal, J. Zowe (auth.), Prof. Dr. Vladimir F. Demyanov, Prof. Dr. Diethard Pallaschke (eds.)
The foreign Institute for utilized structures research (IIASA) in Laxenburg, Austria, has been all for learn on nondifferentiable optimization in view that 1976. IIASA-based East-West cooperation during this box has been very efficient, resulting in many very important theoretical, algorithmic and utilized effects. Nondifferentiable optimi zation has now develop into a famous and speedily constructing department of mathematical programming. To proceed this custom, and to study fresh advancements during this box, IIASA held a Workshop on Nondifferentiable Optimization in Sopron (Hungary) in September 1964. The goals of the Workshop have been: 1. to debate the state of the art of nondifferentiable optimization (NDO), its origins and motivation; 2. To compare-various algorithms; three. to guage latest mathematical methods, their purposes and power; four. to increase and deepen business and different purposes of NDO. the subsequent subject matters have been thought of in separate classes: basic motivation for learn in NDO: nondifferentiability in utilized difficulties, nondifferentiable mathematical versions. Numerical equipment for fixing nondifferentiable optimization difficulties, numerical experiments, comparisons and software program. Nondifferentiable research: quite a few generalizations of the idea that of subdifferen tials. business and different functions. This quantity comprises chosen papers offered on the Workshop. it's divided into 4 sections, according to the above themes: I. techniques in Nonsmooth research II. Multicriteria Optimization and keep an eye on concept III. Algorithms and Optimization tools IV. Stochastic Programming and purposes we want to thank the foreign Institute for utilized platforms research, rather Prof. V. Kaftanov and Prof. A.B. Kurzhanski, for his or her help in organiz ing this meeting.
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Additional resources for Nondifferentiable Optimization: Motivations and Applications: Proceedings of an IIASA (International Institute for Applied Systems Analysis) Workshop on Nondifferentiable Optimization Held at Sopron, Hungary, September 17–22, 1984
Example text
Clarke [1] [2] should berjudged. Clarke's theory emphasizes Lipschitzian properties and sturdily combines convex analysis and classical smooth analysis in a single framework. At the present stage of development, thanks to the efforts of many individuals, it has already had strong effects on almost every area of optimization, from nonlinear programming to the calculus of variations, and also on mathematical questions beyond the domain of optimization per se. This is not to say, however, that Clarke's derivatives and sub gradients are the only ones that henceforth need to be considered.
In particu- 53 lar, when f is Hadamard-differentiable at a0f(a) In [12] culus a , one has = {f'(a)} = aPf(a) . a more analytical (but simple) approach to subdifferential cal- is presented which in particular shares this enjoyable property which does not hold with the strict subdifferential asf(a) . H. (1983). Optimization and Non smooth Analysis. Wiley, New-York a [ 2] CORNET B. : Contribution la theorie mathematique des mecanismes dynamiques d'allocation des ressources. These Univ. Paris 9 (1981).
G. [6], [4], [8], [3]) then finding a cr-Newton directLon is equivalent to solving a variant of the cutting plane problem, in which one of the linear pieces is imposed to be active. We also show that a cr-Newton direction can be interpreted in terms of the perturbed second order derivative given in [5], [1]. 26 2. PRELIMINARY RESULTS Let x 1 , ••• , ~ be the iterates generated by the algorithm and let g 1 , ••. , gk be the corresponding subgradients. ' ·~ ~ £(~) K ~ -x. K~ >. ~ :s; a} i=l :s; i :s; k-1 ; observe that gk belongs to Ga.