Min Common/Max Crossing Duality: A Geometric View of Conjugacy in Convex Optimization1

We provide a unifying framework for the visualization and analysis of duality, and other issues in convex optimization. It is based on two simple optimization problems that are dual to each other: the min common point problem and the max crossing point problem. Within the insightful geometry of these problems, several of the core issues in convex analysis become apparent and can be analyzed in a unified way. These issues are related to conditions for strong duality in constrained optimization and zero sum games, existence of dual optimal solutions and saddle points, existence of subgradients, and theorems of the alternative. The generality and power of our framework is due to its close connection with the Legendre/Fenchel conjugacy framework. However, the two frameworks offer complementary starting points for analysis and provide alternative views of the geometric foundation of duality: conjugacy emphasizes functional/algebraic descriptions, while min common/max crossing emphasizes set/epigraph descriptions. The min common/max crossing framework is simpler, and seems better suited for visualizing and investigating questions of strong duality and existence of dual optimal solutions. The conjugacy framework, with its emphasis on functional descriptions, is more suitable when mathematical operations on convex functions are involved, and the calculus of conjugate functions can be brought to bear for analysis or computation. 1 Supported by NSF Grant ECCS-0801549. The MC/MC framework was initially developed in joint research with A. Nedic, and A. Ozdaglar. This research is described in the book by Bertsekas, Nedic, and Ozdaglar [BNO03]. The present account is an improved and more comprehensive development. In particular, it contains some more streamlined proofs and some new results, particularly in connection with minimax problems and separable problems. 2 Dimitri Bertsekas is with the Dept. of Electr. Engineering and Comp. Science, M.I.T., Cambridge, Mass., 02139.