Computing the Subdifferential of Convex Piecewise Linear-Cubic Functions Mahnic, Aaron Matthew
Computational Convex Analysis (CCA) studies the computation of convex operators commonly used in convex analysis. CCA allows researchers to visualize non-trivial convex transforms and build an intuition from these examples. Using the convexity and structure of piecewise linear cubic (PLC) functions as a model, the foundation for a new CCA toolbox has been implemented. The CCA toolbox is built in MATLAB and includes extensive test coverage and examples. In addition, it is free and open source. The CCA toolbox provides several plotting methods to visualize PLC functions and their properties. Plotting the domain of a PLC function is a useful tool for both research and visualizing the correctness of computed output. Within a PLC function, any polyhedral set can be unbounded and plotting the domain of such a function would lead to an unbounded plot which cannot be displayed. We introduce a method that manipulates the PLCVC data structure in the CCA toolbox to plot an unbounded domain within a specified window. A convex analysis operation of considerable interest is the Legendre-Fenchel transform, also known as the Fenchel conjugate. The computation of the Fenchel conjugate can be done in linear time. As a first step in implementing a linear time conjugate computation for the CCA toolbox, we introduce a new class (PLCVP) that implements a method for computing the subdifferential at any point of a PLC function.
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