Continuous Deformation to Finite Convex Integration

During the course of my master's degree I have studied a variety of topic relating to continuous and numerical optimization. In particular, we are concered with how to transform one function into another. The most general way to do this is the kernel average. The proximal average is a specific example of the kernel average. It turns out that we obtain the proximal average when we fix the kernel to be the normed squared function.

As a variation of the kernel average we consider replacing the distance function of the kernel average by the Bregman distance to obtain the Bregman average. In my report, we discuss and analyze a closed form for Bregman average of two affine functions and present several important open questions.

Finding finite convex antiderivatives quite different then finding standard antiderivatives in the continuous setting. The continuous integration problem is well known to be unique up to a constant. For example, the antiderivative of the continuous function f(x) = x is the function F(x) = 1/2 x^2 + K, where K in R. However, the finite integration problem may have many solutions. The finite convex integration problem is described in my thesis p.61

Finite Convex Antiderivatives Linked to Network Flows

The finite convex integreation problem which has been studied extensively in convex analysis is surprisingly related to the shortest path problem in the field of network flows. For a full example please see p.61-74 in my thesis.

Proximal Average

Similar to the arithmetic average, the proximal average is a unique way to average two functions. It turns out that the the intemdiates of the proximal average contain some unusal and interesting properties. The main results are found in my thesis p.22-27.

Bregman Average

The kernel average can be viewed as a generalization of different averages which depend on the selection of its convex kernel. Indeed, it generalizes the arithematic average, epigraphical average, and proximal average. The Bregman average similarly to the kernel average. By simply replacing the kernel by the Bregman distance we obtain several results which can be viewed here.