Xiaojun Chen

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

    High-Order Evaluation Complexity for Convexly-Constrained Optimization with Non-Lipschitzian Group Sparsity Terms

  • Xiaojun Chen
  • Philippe L. Toint
    • Categories Constrained Nonlinear Optimization, Data-Mining, Statistics Tags , , , , ,

    This paper studies high-order evaluation complexity for partially separable convexly-constrained optimization involving non-Lipschitzian group sparsity terms in a nonconvex objective function. We propose a partially separable adaptive regularization algorithm using a $p$-th order Taylor model and show that the algorithm can produce an (epsilon,delta)-approximate q-th-order stationary point in at most O(epsilon^{-(p+1)/(p-q+1)}) evaluations of the objective … Read more

    Convergence Analysis of Sample Average Approximation of Two-stage Stochastic Generalized Equations

  • Xiaojun Chen
  • Alexander Shapiro
  • Hailin Sun
    • Categories Stochastic Programming Tags , , , ,

    A solution of two-stage stochastic generalized equations is a pair: a first stage solution which is independent of realization of the random data and a second stage solution which is a function of random variables. This paper studies convergence of the sample average approximation of two-stage stochastic nonlinear generalized equations. In particular an exponential rate … Read more

    A New First-order Algorithmic Framework for Optimization Problems with Orthogonality Constraints

  • Xiaojun Chen
  • Bin Gao
  • Xin Liu
  • Ya-xiang Yuan
    • Categories Basic Sciences Applications, Constrained Nonlinear Optimization Tags , , , , ,

    In this paper, we consider a class of optimization problems with orthogonality constraints, the feasible region of which is called the Stiefel manifold. Our new framework combines a function value reduction step with a correction step. Different from the existing approaches, the function value reduction step of our algorithmic framework searches along the standard Euclidean … Read more

    On the Global Optimality for Linear Constrained Rank Minimization Problem

  • Xiaojun Chen
  • Xin Liu
  • Hong Wang
  • Ya-xiang Yuan
    • Categories Global Optimization, Nonlinear Optimization Tags , , , ,

    The rank minimization with linear equality constraints has two closely related models, the low rank approximation model, that is to find the best rank-k approximation of a matrix satisfying the linear constraints, and its corresponding factorization model. The latter one is an unconstrained nonlinear least squares problem and hence enjoys a few fast first-order methods … Read more