holder continuity

On Optimal Universal First-Order Methods for Minimizing Heterogeneous Sums

  • Benjamin Grimmer
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , ,

    This work considers minimizing a convex sum of functions, each with potentially different structure ranging from nonsmooth to smooth, Lipschitz to non-Lipschitz. Nesterov’s universal fast gradient method provides an optimal black-box first-order method for minimizing a single function that takes advantage of any continuity structure present without requiring prior knowledge. In this paper, we show … Read more

    Openness, Holder metric regularity and Holder continuity properties of semialgebraic set-valued maps

  • Jae Hyoung Lee
  • Tiên-So'n Pham
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization Tags , , , ,

    Given a semialgebraic set-valued map $F \colon \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ with closed graph, we show that the map $F$ is Holder metrically subregular and that the following conditions are equivalent: (i) $F$ is an open map from its domain into its range and the range of $F$ is locally closed; (ii) the map $F$ is … Read more