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Erschienen in:
2015 | OriginalPaper | Buchkapitel
4. Examples
verfasst von : Juan Peypouquet
Erschienen in: Convex Optimization in Normed Spaces
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Abstract
The tools presented in the previous chapters are useful, on the one hand, to prove that a wide variety of optimization problems have solutions; and, on the other, to provide useful characterizations allowing to determine them. In this chapter, we present a short selection of problems to illustrate some of those tools. We begin by revisiting some results from functional analysis concerning the maximization of bounded linear functionals and the realization of the dual norm. Next, we discuss some problems in optimal control and calculus of variations. Another standard application of these convex analysis techniques lies in the field of elliptic partial differential equations. We shall review the theorems of Stampacchia and Lax-Milgram, along with some variations of Poisson’s equation, including the obstacle problem and the p-Laplacian. We finish by commenting a problem of data compression and restoration.