James Burke

University of Washington

Convex Duality Theory

We begin by reviewing a few elementary constructions in convex analysis before presenting the modern approach to convex duality theory based on the infimal projection of convex perturbation functions. This approach reveals the deep connections to the sensitivity theory for optimal value functions. Familiar examples are reviewed as well as their connections to Lagrange multiplier theory. We then introduce the more recent notion of gauge functions and gauge duality, and show how gauge duality can be derived using a perturbations analysis. The perturbation approach yields for the first time a sensitivity theory for gauge duality. Again, we illustrate the theory with familiar examples. Finally, we introduce perspective functions and a corresponding new notion of duality called \emph{perspective duality}. Applications of each of these approaches to duality to modern problems in numerical convex optimization are discussed, and a few numerical studies are presented. This talk is based on joint work with Sasha Aravkin, Dima Drusvyatskiy, Michael Friedlander, and Kellie MacPhee. Partial funding for this research was provided by the National Science Foundation of the United States.