Nonlinear Optimization

Worst-case evaluation complexity of regularization methods for smooth unconstrained optimization using Hölder continuous gradients

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Unconstrained Optimization Tags , ,

    The worst-case behaviour of a general class of regularization algorithms is considered in the case where only objective function values and associated gradient vectors are evaluated. Upper bounds are derived on the number of such evaluations that are needed for the algorithm to produce an approximate first-order critical point whose accuracy is within a user-defined … Read more

    Solving Power-Constrained Gas Transportation Problems using an MIP-based Alternating Direction Method

  • Björn Geißler
  • Antonio Morsi
  • Lars Schewe
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Constrained Nonlinear Optimization Tags , , , ,

    We present a solution algorithm for problems from steady-state gas transport optimization. Due to nonlinear and nonconvex physics and engineering models as well as discrete controllability of active network devices, these problems lead to hard nonconvex mixed-integer nonlinear optimization models. The proposed method is based on mixed-integer linear techniques using piecewise linear relaxations of the … Read more

    Semivectorial Bilevel Optimization on Riemannian Manifolds

  • Henri Bonnel
  • Leonard Todjihounde
  • Constantin Udriste
    • Categories Nonlinear Optimization Tags , ,

    In this paper we deal with the semivectorial bilevel problem in the Riemannian setting. The upper level is a scalar optimization problem to be solved by the leader, and the lower level is a multiobjective optimization problem to be solved by several followers acting in a cooperative way inside the greatest coalition and choosing among … Read more

    Sequential Threshold Control in Descent Splitting Methods for Decomposable Optimization Problems

  • Igor V. Konnov
    • Categories Constrained Nonlinear Optimization, Data-Mining, Nonsmooth Optimization

    We suggest a modification of the descent splitting methods for decomposable composite optimization problems, which maintains the basic convergence properties, but enables one to reduce the computational expenses per iteration and to provide computations in a distributed manner. It consists in making coordinate-wise steps together with a special threshold control. CitationKazan Federal University, Kazan 420008, … Read more

    On the optimal order of worst case complexity of direct search

  • M. Dodangeh
  • Luis Nunes Vicente
  • Z. Zhang
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , ,

    The worst case complexity of direct-search methods has been recently analyzed when they use positive spanning sets and impose a sufficient decrease condition to accept new iterates. Assuming that the objective function is smooth, it is now known that such methods require at most O(n^2 epsilon^{-2}) function evaluations to compute a gradient of norm below … Read more

    A Filter Active-Set Algorithm for Ball/Sphere Constrained Optimization Problem

  • Shen Chungen
  • Zhang Lei-Hong
  • Weihong Yang
    • Categories Constrained Nonlinear Optimization Tags , , , , , ,

    In this paper, we propose a filter active-set algorithm for the minimization problem over a product of multiple ball/sphere constraints. By making effective use of the special structure of the ball/sphere constraints, a new limited memory BFGS (L-BFGS) scheme is presented. The new L-BFGS implementation takes advantage of the sparse structure of the Jacobian of … Read more

    On proximal subgradient splitting method for minimizing the sum of two nonsmooth convex functions

  • Yunier Bello-Cruz
    • Categories Convex and Nonsmooth Optimization, Global Optimization, Nonlinear Optimization

    In this paper we present a variant of the proximal forward-backward splitting method for solving nonsmooth optimization problems in Hilbert spaces, when the objective function is the sum of two nondifferentiable convex functions. The proposed iteration, which will be call the Proximal Subgradient Splitting Method, extends the classical projected subgradient iteration for important classes of … Read more

    A trust-region method for box-constrained nonlinear semidefinite programs

  • Akihiko Komatsu
  • Makoto Yamashita
    • Categories Bound-constrained Optimization, Semi-definite Programming Tags ,

    We propose a trust-region method for nonlinear semidefinite programs with box-constraints. The penalty barrier method can handle this problem, but the size of variable matrices available in practical time is restricted to be less than 500. We develop a trust-region method based on the approach of Coleman and Li (1996) that utilizes the distance to … Read more

    Simple examples for the failure of Newton’s method with line search for strictly convex minimization

  • Florian Jarre
  • Philippe L. Toint
    • Categories Unconstrained Optimization Tags ,

    In this paper two simple examples of a twice continuously differentiable strictly convex function $f$ are presented for which Newton’s method with line search converges to a point where the gradient of $f$ is not zero. The first example uses a line search based on the Wolfe conditions. For the second example, some strictly convex … 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