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Chapter 2 ELEMENTS OF CONVEX ANALYSIS, DUALITY THEORY The a posteriori error estimates presented in this work are derived based on the duality theory of convex analysis. The first research monograph specifi- cally devoted to the topic of convex analysis is [136], emphasizing the finite- dimensional case. Convex analysis and duality theory in general normed spaces, mostly infinite dimensional ones, are thoroughly discussed in the well-known reference [49]. Another comprehensive treatment of the topic is [159]. Du- ality theory has been also extended for nonconvex systems, see, e.g. [59, 601 where the mathematical theory is motived by duality in natural phenomena with particular emphasis on mechanics. In this chapter, we review some basic notions and results on convex sets, convex functions and their properties as well as the duality theory. Detailed discussions and proofs of the stated results can be found in [49] or [159]. In the theory of convex analysis, it is convenient to consider functions that take on values on the extended real line E. Recall that a functional f : V + is said to be proper if f (v) > -cc b'v E V and f (u) < cx for some u E V. 2.1.
Published: Jan 1, 2005
Keywords: Convex Function; Minimization Problem; Normed Space; Dual Problem; Duality Theory
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