Duality in Optimization Problems
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Duality theory plays a fundamental role in the analysis of optimization and variational problems. It not only provides a powerful theoretical tool in the analysis of these problems, but also paves the way to designng new algorithms for solving them. The basic idea behind a duality framework is greatly similar to the general mathematical line of thinking, namely to transform a hard problem into an easy or at least easier one to be analyzed and to be solved. A key player in any duality framework is the Legendre-Fenchel conjugate transform. Often, duality is associated with convex problems, yet it turns out that duality theory also has a fundamental impact even on the analysis of nonconvex problems. This chapter gives the elements of duality theory for optimization problems. Starting with a very general and abstract scheme for duality based on perturbation functionals, we derive the basic conditions leading to strong duality results, and characterization of optimal solutions. This covers well-known Fenchel and Lagrangian duality schemes, as well as minimax theorems for convex-concave functionals within a unified approach that emphasizes the importance of asymptotic functions.