The Legendre-Fenchel Conjugate: Numerical Computation
نویسنده
چکیده
This paper describes a numerical implementation in Maple V R. 5 of an algorithm to compute the Legendre–Fenchel conjugate, namely the Linear-time Legendre Transform algorithm. After a brief motivation on the importance of the Legendre–Fenchel transform, we illustrate the information the conjugate gives (how to test for convexity or compacity, and how to smooth a convex function), with several examples (including solving a Hamilton-Jacobi equation). The last section shows the convergence behavior. The package is available from the Computational Convex Analysis web page at http://www.cecm.sfu.ca/projects/CCA or directly at http://www.cecm.sfu.ca/projects/CCA/LLT. Introduction Symbolic computation of the Legendre–Fenchel conjugate has been studied recently in [1]. However, it can only be limited to special classes of functions. Therefore several authors studied fast algorithms for numerical computation [2, 4, 7], so called fast Legendre transform algorithms which have a log-linear worst-case computation time. Later, a linear algorithm, the Lineartime Legendre Transform algorithm, was studied in [5, 6]. We describe a package implementing it in Maple V.5. The Legendre–Fenchel conjugate Given a function , its LegendreFenchel conjugate (also called conjugate, convex conjugate, or Fenchel conjugate) is defined by:
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