Computational Convex Analysis

Tanmaya Karmarkar () and Yves Lucet ()
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Tanmaya Karmarkar: University of British Columbia Okanagan, Computer Science, I. K. Barber Faculty of Science
Yves Lucet: University of British Columbia Okanagan, Computer Science, I. K. Barber Faculty of Science

A chapter in Handbook of Visual, Experimental and Computational Mathematics, 2026, pp 1113-1138 from Springer

Abstract: Abstract This chapter introduces computational convex analysis, which deals with the efficient computation of the entire graph of fundamental operators from convex and nonsmooth analysis. The main applications are visualization in low dimensions, building intuition, and illustrating convex analysis results, in addition to several fields that require the computation of the entire graph. Key convex analysis operators – like the Legendre-Fenchel conjugate and subdifferential – are defined and visualized to illustrate important results. Efficient algorithms allow the constructions of a convex toolbox that includes the conjugate, addition, and scalar multiplication. The toolbox allows the computation and visualization of more advanced operators like the proximal average. Extension of the toolbox to nonconvex functions only requires adding one more operator, the convex envelope, which is fundamental in global optimization. Computation requires the manipulation of several classes of functions. They are well known in optimization for the convex case: piecewise linear (linear functions defined on a polyhedral subdivision) and piecewise linear–quadratic (quadratic functions defined on a polyhedral subdivision) functions. For the nonconvex case, new classes of functions include: piecewise parabolic–quadratic (quadratic functions defined on a parabolic subdivision) and piecewise linear–rational (rational functions defined on a polyhedral subdivision).

Keywords: Legendre-Fenchel conjugate; Fenchel conjugate; Moreau envelope; Convex envelope; Piecewise linear–quadratic functions; Polyhedral subdivision (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16368-4_81

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