epi-convergence

Understanding the Douglas-Rachford splitting method through the lenses of Moreau-type envelopes

  • Felipe Atenas
    • Categories Convex and Nonsmooth Optimization, Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , ,

    We analyze the Douglas-Rachford splitting method for weakly convex optimization problems, by the token of the Douglas-Rachford envelope, a merit function akin to the Moreau envelope. First, we use epi-convergence techniques to show that this artifact approximates the original objective function via epigraphs. Secondly, we present how global convergence and local linear convergence rates for … Read more

    Variational Theory and Algorithms for a Class of Asymptotically Approachable Nonconvex Problems

  • Hanyang Li
  • Ying Cui
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    We investigate a class of composite nonconvex functions, where the outer function is the sum of univariate extended-real-valued convex functions and the inner function is the limit of difference-of-convex functions. A notable feature of this class is that the inner function can be merely lower semicontinuous instead of continuously differentiable. It covers a range of … Read more

    Epi-convergent Smoothing with Applications to Convex Composite Functions

  • James V. Burke
  • Tim Hoheisel
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , , ,

    Smoothing methods have become part of the standard tool set for the study and solution of nondifferentiable and constrained optimization problems as well as a range of other variational and equilibrium problems. In this note we synthesize and extend recent results due to Beck and Teboulle on infimal convolution smoothing for convex functions with those … Read more