Min Common/Max Crossing Duality: A Simple Geometric Framework for Convex Optimization and Minimax Theory1

We provide a simple unifying framework for the visualization and analysis of convex programming duality and minimax (saddle point) theory. In particular, we introduce two geometrical problems that are dual to each other: the min common point problem and the max crossing point problem. Within the simple geometry of these problems, the fundamental constraint qualifications needed for strong duality are quite apparent, and admit straightforward proofs. We develop the relevant theorems, and we then obtain as special cases the major results of Lagrangian duality theory for constrained optimization and of convex/concave minimax theory.