Tomas Roubicek

Relaxation in Optimization Theory and Variational Calculus (Trends in Linguistics) Tomas Roubicek

The purpose of this book is to introduce the reader to a relaxation method based on a continuous extension of various optimization problems on so-called convex compactifications, which proves useful when dealing particularly with problems in optimal control theory, calculus of variations, and non-cooperative game theory. In the first two chapters background results are summarized, and the general theory of convex compactifications developed by the author is introduced. Later in this theory is used to obtain convex locally compact envelopes of the Lebesgue and Sobolev spaces involved in concrete problems From a unified point of view, these nontrivial envelopes cover the classical Young measures as well as various generalizations of them, which can record the 'limit behaviour' of fast oscillation and concentration effects. Existence and stability of the generalized solutions are then guaranteed by compactness, while convexity allows the optimality conditions resulting in Pontryagin and Weirstrab maximum principles to be formulated. This is demonstrated by optimal control problems governed by ordinary or partial differential equations and integral equations, as well as by scalar or vectorial multi-dimensional variational problems. For problems in game theory, both compactness and convexity are needed to obtain existence results (in terms of mixed strategies). Approximation theory and numerical results of sample examples are also presented.