We study maximal monotone inclusions from the perspective of (convex) gap functions.
We propose a very natural gap function and will demonstrate how this function arises from the Fitzpatrick function — a convex function used effectively to represent maximal monotone operators.
This approach allows us to use the powerful strong Fitzpatrick inequality to analyse solutions of the inclusion.
This is joint work with Joydeep Dutta.
Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components – when can we conclude that the entire function is convex? Our main result provides sufficient conditions for a piecewise-defined function f to be convex. We also provide a sufficient condition for checking the convexity of a piecewise linear-quadratic function, which play an important role in computer-aided convex analysis.
Based on joint work with Heinz H. Bauschke (Mathematics, UBC Okanagan) and Hung M. Phan (Mathematics, University of Massachusetts Lowell).