OPTIMIZATION SEMINAR

Some Self-Dual Operations on Convex Functions

Rafal Goebel,

Mathematics, UW

Convex conjugacy gives a one to one correspondence between proper, lower semicontinuous, convex functions and their (proper, lower semicontinuous, convex) conjugates. The operations of adding two functions and of multiplying a function by a constant are reflected, through convex conjugacy, in operations involving epigraphs: epi-addition (also called inf-convolution) and epi-multiplication.

The talk will present how the operations mentioned above can be combined to:

• approximate any convex function with a differentiable one, so that the conjugate of the approximate is also differentiable;
• find meaningful "averages" of convex functions even if the effective domains of the functions do not overlap.

In particular, a self-dual approximation technique and a self-dual "proximal average" will be described. Self-duality here means that the conjugate of the approximate is the approximate of the conjugate, and that the conjugate of the proximal average is the proximal average of conjugates.

Mathematics Department University of Washington