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Generalized Convex Duality and Its Economic Applications, Juan Enrique Martínez-Legaz
This article presents an approach to generalized convex duality theory based on Fenchel-Moreau conjugations; in particular, it discusses quasiconvex conjugation and duality in detail. It also describes the related topic of microeconomics duality and analyzes the monotonicity of demand functions.
A central topic in optimization is convex duality theory. In its modern approach, mainly due to Rockafeller [110], [111], given a (primal) convex optimization problem one embeds it into a family of perturbed optimization problems and then, relative to these perturbations, one associates to it a so-called dual problem. The deep relations existing between the primal and the dual are helpful for analyzing the properties of the original problem and, in particular, for obtaining optimality conditions; they are also used to devise numerical algorithms. In the case of problems arising in applications to other sciencies, particularly in economics, the dual problems usually have a nice interpretation that shed new light into the nature of the associated primal problems and yield a new perspective for analyzing them.
Contents: 1. Introduction, 2. Generalized convex conjugation, 3. Generalized convex duality, 4. Quasiconvex conjugation, 5. Quasiconvex duality, 6. Duality in consumer theory, 7. Consumer theory without utility.