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UBC Theses and Dissertations

Duality in convex programming Muir, David Charles William

Abstract

Problems of minimizing a convex function or maximizing a concave function over a convex set are called convex programming problems. Duality principles relate two problems, one a minimization problem, the other a maximization problem, in such a way that a solution to one implies a solution to the other and that the minimum value of one is equal to the maximum value of the other. When the functions are linear and the constraint sets are polyhedral, the problems are called linear programming problems. Their duality is well-known. Certain duality results of linear programming can be extended to convex programming by means of the theory of conjugate convex functions introduced by Fenchel ([1], [2]). In this thesis the theory of conjugate functions is generalized and applied to convex programming problems. In particular a duality theorem is given for a class of convex programming problems. This theorem is compared with a duality theorem for convex programming problems given by Dorn [3].

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