Convex and Nonsmooth Optimization

On a generalization of Pólya’s and Putinar-Vasilescu’s Positivstellensätze

  • Peter J.C. Dickinson
  • Janez Povh
    • Categories Convex and Nonsmooth Optimization, Global Optimization, Linear, Cone and Semidefinite Programming

    In this paper we provide a generalization of two well-known positivstellensätze, namely the positivstellensatz from Pólya [Georg Pólya. Über positive darstellung von polynomen vierteljschr. In Naturforsch. Ges. Zürich, 73: 141-145, 1928] and the positivestellensatz from Putinar and Vasilescu [Mihai Putinar and Florian-Horia Vasilescu. Positive polynomials on semialgebraic sets. Comptes Rendus de l’Académie des Sciences – … Read more

    Lagrangian Transformation and Interior Ellipsoid Methods in Convex Optimization

  • Roman Polyak
    • Categories Convex Optimization

    The rediscovery of the affine scaling method in the late 80s was one of the turning points which led to a new chapter in Modern Optimization – the interior point methods (IPMs). Simultaneously and independently the theory of exterior point methods for convex optimization arose. The two seemingly unconnected fields turned out to be intrinsically … Read more

    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

    GENERALIZATIONS OF THE DENNIS-MOR\’E THEOREM II

  • Asen L. Dontchev
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    This paper is a continuation of our previous paper were we presented generalizations of the Dennis-Mor\’e theorem to characterize q-superliner convergences of quasi-Newton methods for solving equations and variational inequalities in Banach spaces. Here we prove Dennis-Mor\’e type theorems for inexact quasi-Newton methods applied to variational inequalities in finite dimensions. We first consider variational inequalities … 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