Nonsmooth and multivalued analysis with applications in optimization

Abstract

The object of this thesis is two - fold . In the first part, we develop analogs of convex and nonconvex analysis for vector - valuedoperators, while in the second part we study the theory of Banach valued multifunctions. In the first part, we start with a study of convex operators. We introduce the notion of algebraic and topological subdifferentials and then derive conditions for those two to be equal. Also we develop a complete subdifferential and e-subdifferential calculus. In the sequence, we deal with the duality theory of convex operators. For that purpose, we introduc a notion of lower semicontinuity of operators and we show that this class is identical with the class of operators that are the upper envelope of continuous affine operators . This allows us to study analogs of the major duality schemes for vector optimization problems. Finally, we conclude our study of convex operators with some probabilistic results on Caratheodory convex integrands. Then we pass to nonconvex operators and introduce the class of locally o-Lipschitz operators. For those operators, we define a generalized subdifferential and develop a corresponding calculus that extends Clarke's theory to a vectorial context. Furthermore, we show that this extension is consistent with the convex theory. Applications to vectorial optimization are given. In the third stage of the process, we consider general operators and using geometric notions we introduce a new subdifferential calculus and provide applications in optimization. We close the first part of the thesis with a detailed study of infinite dimensional Pareto optimization problems and obtain existence and stability results for such problems. In the second part of the thesis, we pass to multivalued analysis. We introduce the vector valued Aumann integral and study its properties.Multifunctions depending on parameters are studied and results are obtained determining which properties of the integrand multifunction are preserved by integration. Then, using a notion of set valued conditional expectation, we introduce set valued martingales and obtain several convergence theorems. Also we study properties of the profile of a multifunction and weakly convergent sequences of multifunctions. Then we consider set valued measures, study their properties and define an integral with respect to a set valued measure and determine its properties. Finally, in the last chapter of our thesis, motivated by Ioffe's recent theory of fans, we introduce the notion of a "normal fan " and develop an integral theory for such fans.