Convex and Nonsmooth Optimization

Minimal Residual Methods for Complex Symmetric, Skew Symmetric, and Skew Hermitian Systems

  • Sou-Cheng Choi
    • Categories Convex Optimization, Optimization Software and Modeling Systems, Robust Optimization Tags , , , , , , , , , , , ,

    While there is no lack of efficient Krylov subspace solvers for Hermitian systems, there are few for complex symmetric, skew symmetric, or skew Hermitian systems, which are increasingly important in modern applications including quantum dynamics, electromagnetics, and power systems. For a large consistent complex symmetric system, one may apply a non-Hermitian Krylov subspace method disregarding … Read more

    An interior proximal point method with phi-divergence for Equilibrium Problems

  • Arnaldo Brito
  • Paulo Roberto Oliveira
  • Paulo Santos
  • Afonso Silva
    • Categories Applications - OR and Management Sciences, Convex Optimization

    In this paper, we consider the problem of general equilibrium in a  finite-dimensional space on a closed convex set. For solving this problem, we developed an interior proximal point algorithm with phi-divergence. Under reasonable assumptions, we prove that the sequence generated by the algorithm converges to a solution of the Equilibrium Problem, when the regularization … Read more

    The proximal-proximal gradient algorithm

  • Ting Kei Pong
    • Categories Applications - Science and Engineering, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , ,

    We consider the problem of minimizing a convex objective which is the sum of a smooth part, with Lipschitz continuous gradient, and a nonsmooth part. Inspired by various applications, we focus on the case when the nonsmooth part is a composition of a proper closed convex function P and a nonzero affine map, with the … Read more

    Optimal Primal-Dual Methods for a Class of Saddle Point Problems

  • Yunmei Chen
  • Guanghui Lan
  • Yuyuan Ouyang
    • Categories Convex Optimization, Stochastic Programming

    We present a novel accelerated primal-dual (APD) method for solving a class of deterministic and stochastic saddle point problems (SPP). The basic idea of this algorithm is to incorporate a multi-step acceleration scheme into the primal-dual method without smoothing the objective function. For deterministic SPP, the APD method achieves the same optimal rate of convergence … Read more

    About uniform regularity of collections of sets

  • Alexander Y. Kruger
  • Nguyen H. Thao
    • Categories Convex and Nonsmooth Optimization Tags ,

    We further investigate the uniform regularity property of collections of sets via primal and dual characterizing constants. These constants play an important role in determining convergence rates of projection algorithms for solving feasibility problems. CitationPublished in Serdica Math. J. 39, 287–312 (2013) http://www.math.bas.bg/serdica/2013/2013-287-312.pdfArticleDownload View PDF

    Second-order growth, tilt stability, and metric regularity of the subdifferential

  • Dmitriy Drusvyatskiy
  • Boris S. Mordukhovich
  • Nghia Tran
    • Categories Nonsmooth Optimization Tags , , , , ,

    This paper sheds new light on several interrelated topics of second-order variational analysis, both in finite and infinite-dimensional settings. We establish new relationships between second-order growth conditions on functions, the basic properties of metric regularity and subregularity of the limiting subdifferential, tilt-stability of local minimizers, and positive definiteness/semidefiniteness properties of the second-order subdifferential (or generalized … Read more

    A continuous gradient-like dynamical approach to Pareto-optimization in Hilbert spaces

  • Hédy Attouch
  • Xavier Goudou
    • Categories Applications - OR and Management Sciences, Convex and Nonsmooth Optimization, Multi-Criteria Optimization

    In a Hilbert space setting, we consider new continuous gradient-like dynamical systems for constrained multiobjective optimization. This type of dynamics was first investigated by Cl. Henry, and B. Cornet, as a model of allocation of resources in economics. Based on the Yosida regularization of the discontinuous part of the vector field which governs the system, … Read more

    Worst case complexity of direct search under convexity

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

    In this paper we prove that the broad class of direct-search methods of directional type, based on imposing sufficient decrease to accept new iterates, exhibits the same global rate or worst case complexity bound of the gradient method for the unconstrained minimization of a convex and smooth function. More precisely, it will be shown that … Read more

    A merit function approach for direct search

  • Serge Gratton
  • Luis Nunes Vicente
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , , ,

    In this paper it is proposed to equip direct-search methods with a general procedure to minimize an objective function, possibly non-smooth, without using derivatives and subject to constraints on the variables. One aims at considering constraints, most likely nonlinear or non-smooth, for which the derivatives of the corresponding functions are also unavailable. The novelty of … Read more

    Faster, but Weaker, Relaxations for Quadratically Constrained Quadratic Programs

  • Samuel Burer
  • Sunyoung Kim
  • Masakazu Kojima
    • Categories Convex Optimization, Global Optimization, Quadratic Programming Tags , , ,

    We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than … Read more