legendre-fenchel transform

Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

  • Yun-Bin Zhao
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , ,

    While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: \emph{When is the product of finitely many positive … Read more

    What Shape is your Conjugate? A Survey of Computational Convex Analysis and its Applications

  • Yves Lucet
    • Categories Applications - OR and Management Sciences, Convex Optimization Tags , , , , , ,

    Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of Computational Convex Analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolic-numeric algorithms. Our objective is … Read more

    Fast Moreau Envelope Computation I: Numerical Algorithms

  • Yves Lucet
    • Categories Convex Optimization, Nonsmooth Optimization, Problem Solving Environments Tags , , , , , ,

    The present article summarizes the state of the art algorithms to compute the discrete Moreau envelope, and presents a new linear-time algorithm, named NEP for NonExpansive Proximal mapping. Numerical comparisons between the NEP and two existing algorithms: The Linear-time Legendre Transform (LLT) and the Parabolic Envelope (PE) algorithms are performed. Worst-case time complexity, convergence results, … Read more