The piecewise linear-quadratic model for computational convex analysis

Authors:
Yves Lucet;Heinz H. Bauschke;Mike Trienis
Affiliations:
University of British Columbia--Okanagan, Kelowna, Canada BC V1V 1V7;University of British Columbia--Okanagan, Kelowna, Canada BC V1V 1V7;University of British Columbia--Okanagan, Kelowna, Canada BC V1V 1V7
Venue:
Computational Optimization and Applications
Year:
2009

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Abstract

A new computational framework for computer-aided convex analysis is proposed and investigated. Existing computational frameworks are reviewed and their limitations pointed out. The class of piecewise linear-quadratic functions is introduced to improve convergence and stability. A stable convex calculus is achieved using symbolic-numeric algorithms to compute all fundamental transforms of convex analysis. Our main result states the existence of efficient (linear time) algorithms for the class of piecewise linear-quadratic functions. We also recall that such class is closed under convex transforms. We illustrate the results with numerical examples, and validate numerically the resulting computational framework.