non-lipschitz

Some Primal-Dual Theory for Subgradient Methods for Strongly Convex Optimization

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

    We consider (stochastic) subgradient methods for strongly convex but potentially nonsmooth non-Lipschitz optimization. We provide new equivalent dual descriptions (in the style of dual averaging) for the classic subgradient method, the proximal subgradient method, and the switching subgradient method. These equivalences enable $O(1/T)$ convergence guarantees in terms of both their classic primal gap and a … Read more

    A Riemannian smoothing steepest descent method for non-Lipschitz optimization on submanifolds

  • Xiaojun Chen
  • Shiqian Ma
  • Chao Zhang
    • Categories Nonlinear Optimization Tags , , , ,

    In this paper, we propose a Riemannian smoothing steepest descent method to minimize a nonconvex and non-Lipschitz function on submanifolds. The generalized subdifferentials on Riemannian manifold and the Riemannian gradient sub-consistency are defined and discussed. We prove that any accumulation point of the sequence generated by the Riemannian smoothing steepest descent method is a stationary … Read more

    Partially separable convexly-constrained optimization with non-Lipschitz singularities and its complexity

  • Xiaojun Chen
  • Philippe L. Toint
  • Hong Wang
    • Categories Nonsmooth Optimization Tags , ,

    An adaptive regularization algorithm using high-order models is proposed for partially separable convexly constrained nonlinear optimization problems whose objective function contains non-Lipschitzian $\ell_q$-norm regularization terms for $q\in (0,1)$. It is shown that the algorithm using an $p$-th order Taylor model for $p$ odd needs in general at most $O(\epsilon^{-(p+1)/p})$ evaluations of the objective function and … Read more