Topological aspects of nonsmooth optimization

• Topologische Aspekte der nichtglatten Optimierung

Shikhman, Vladimir; Jongen, Hubertus Th. (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2011)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2011

Abstract

The main goal of our study is an attempt to understand and classify nonsmooth structures arising within the optimization setting: P(f,F): min f(x) s.t. x in M[F], where f is a smooth real-valued objective function, F is a smooth vector-valued function and M[F] a feasible set defined by F in some structured way. We focus rather on the underlying nonsmooth structures which fit the smooth function F to define the feasible set M[F]. The basis of our study is the topological approach. It encompasses two objects: the feasible set M[F] and the lower level sets M[f, F]^a. These objects are considered according to topological, optimization and stability issues. On the topology and stability level we deal with topological invariants of M[F] and M[f, F]^a. Here the questionings mainly arise from. They lead to establishing of an adequate theory on the optimization level. For M[F] Lipschitz manifold property and so-called topological stability are discussed. They naturally lead to constraint qualifications for P(f,F). Topological changes of M[f, F]^a give rise to define stationary points and develop critical point theory for P(f,F). Each Chapter 2-5 is devoted to optimization problems with particular type of nonsmoothness: mathematical programming programs with complementarity constraints, general semi-infinite optimization problems, mathematical programming programs with vanishing constraints, bilevel optimization. For these problems above topological and stability issues are elaborated and corresponding optimization concepts are introduced. It is worth to point out that the same topological questionings provide different (analytical) optimization concepts while applied to particular problems. The difference between these analytically described optimization concepts is a key point in understanding and comparing different kinds of nonsmoothness. In Chapter 6 we enlighten the impacts of our topological approach on nonsmooth analysis theory. Topologically regular points of a min-type nonsmooth mappings F are introduced. The crucial property is that for topologically regular value y of F the nonempty set F^{-1}(y) is an n-l dimensional Lipschitz manifold. Corresponding nonsmooth versions of Sard's Theorem are given.

Institutions

• Chair of Mathematics of Information Processing [114510]
• Department of Mathematics [110000]