Convex and Nonsmooth Optimization

A Relaxed-Projection Splitting Algorithm for Variational Inequalities in Hilbert Spaces

  • Yunier Bello-Cruz
  • R. Diaz Millan
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each … Read more

    The inexact projected gradient method for quasiconvex vector optimization problems

  • Yunier Bello-Cruz
  • Glaydston Bento
  • Gemayqzel Bouza
  • R.F.B. Costa
    • Categories Generalized Convexity/Monoticity, Multi-Criteria Optimization, Nonlinear Optimization

    Vector optimization problems are a generalization of multiobjective optimization in which the preference order is related to an arbitrary closed and convex cone, rather than the nonnegative octant. Due to its real life applications, it is important to have practical solution approaches for computing. In this work, we consider the inexact projected gradient-like method for … Read more

    Active Set Methods with Reoptimization for Convex Quadratic Integer Programming

  • Christoph Buchheim
  • Long Trieu
    • Categories (Mixed) Integer Nonlinear Programming, Convex Optimization, Quadratic Programming Tags , ,

    We present a fast branch-and-bound algorithm for solving convex quadratic integer programs with few linear constraints. In each node, we solve the dual problem of the continuous relaxation using an infeasible active set method proposed by Kunisch and Rendl to get a lower bound; this active set algorithm is well suited for reoptimization. Our algorithm … Read more

    An inexact block-decomposition method for extra large-scale conic semidefinite programming

  • Renato D.C. Monteiro
  • Camilo Ortiz
  • Benar Fux Svaiter
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming, Optimization Software and Modeling Systems Tags , , , , , , ,

    In this paper, we present an inexact block-decomposition (BD) first-order method for solving standard form conic semidefinite programming (SDP) which avoids computations of exact projections onto the manifold defined by the affine constraints and, as a result, is able to handle extra large SDP instances. The method is based on a two-block reformulation of the … Read more

    On the Convergence of Alternating Minimization for Convex Programming with Applications to Iteratively Reweighted Least Squares and Decomposition Schemes

  • Amir Beck
    • Categories Convex Optimization

    This paper is concerned with the alternating minimization (AM) for solving convex minimization problems where the decision variables vector is split into two blocks. The objective function is a sum of a differentiable convex function and a separable (possible) nonsmooth extended real-valued convex function, and consequently constraints can be incorporated. We analyze the convergence rate … Read more

    A strongly convergent proximal bundle method for convex minimization in Hilbert spaces

  • Yunier Bello-Cruz
  • Welington de Oliveira
  • Wim van Ackooij
    • Categories Convex and Nonsmooth Optimization

    A key procedure in proximal bundle methods for convex minimization problems is the definition of stability centers, which are points generated by the iterative process that successfully decrease the objective function. In this paper we study a different stability-center classification rule for proximal bundle methods. We show that the proposed bundle variant has three particularly … Read more

    On the Proximal Jacobian Decomposition of ALM for Multiple-block Separable Convex Minimization Problems and its Relationship to ADMM

  • Bingsheng He
  • Xiao-Ming Yuan
  • Hong-Kun Xu
    • Categories Convex Optimization Tags , , , , , ,

    The augmented Lagrangian method (ALM) is a benchmark for solving convex minimization problems with linear constraints. When the objective function of the model under consideration is representable as the sum of some functions without coupled variables, a Jacobian or Gauss-Seidel decomposition is often implemented to decompose the ALM subproblems so that the functions’ properties could … Read more

    Practical Inexact Proximal Quasi-Newton Method with Global Complexity Analysis

  • Katya Scheinberg
  • Xiaocheng Tang
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , ,

    Recently several methods were proposed for sparse optimization which make careful use of second-order information [11, 30, 17, 3] to improve local convergence rates. These methods construct a composite quadratic approximation using Hessian information, optimize this approximation using a first-order method, such as coordinate descent and employ a line search to ensure sufficient descent. Here … Read more

    Differentiability properties of metric projections onto convex sets

  • Alexander Shapiro
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    It is known that directional differentiability of metric projection onto a closed convex set in a finite dimensional space is not guaranteed. In this paper we discuss sufficient conditions ensuring directional differentiability of such metric projections. The approach is based on a general theory of sensitivity analysis of parameterized optimization problems. Article Download View Differentiability … Read more

    Variational analysis in psychological modeling

  • Boris S. Mordukhovich
  • Antoine Soubeyran
  • Bao Q. Truong
    • Categories Applications - OR and Management Sciences, Multi-Criteria Optimization, Nonsmooth Optimization Tags , , , , , ,

    This paper develops some mathematical models arising in psychology and some other areas of behavioral sciences that are formalized via general preferences with variable ordering structures. Our considerations are based on the recent “variational rationality approach” that unifies numerous theories in different branches of behavioral sciences by using, in particular, worthwhile change and stay dynamics … Read more