Convex Optimization

On the Optimal Proximal Parameter of an ADMM-like Splitting Method for Separable Convex Programming

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

    An ADMM-based splitting method is proposed in [11] for solving convex minimization problems with linear constraints and multi-block separable objective functions; while a relatively large proximal parameter is required for theoretically ensuring the convergence. In this paper, we further study this method and find its optimal (smallest) proximal parameter. For succinctness, we focus on the … Read more

    Optimal Linearized Alternating Direction Method of Multipliers for Convex Programming

  • Bingsheng He
  • Feng Ma
  • Xiao-Ming Yuan
    • Categories Convex Optimization Tags , , , , ,

    The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention from many areas because of its efficiency and easy implementation. To theoretically guarantee … Read more

    Linearized version of the generalized alternating direction method of multipliers for three-block separable convex minimization problem

  • Peng Jian-Wen
  • Xue-Qing Zhang
    • Categories Convex Optimization

    Recently, the generalized alternating direction method of multipliers (GADMM) proposed by Eckstein and Bertsekas has received wide attention, especially with respect to numerous applications. In this paper, we develop a new linearized version of generalized alternating direction method of multipliers (L-GADMM) for the linearly constrained separable convex programming whose objective functions are the sum of … Read more

    Constraints reduction programming by subset selection: a study from numerical aspect

  • Yuan Shen
    • Categories 0-1 Programming, Applications - OR and Management Sciences, Convex Optimization Tags , , , , , ,

    We consider a novel method entitled constraints reduction programming which aims to reduce the constraints in an optimization model. This method is derived from various applications of management or decision making, and has potential ability to handle a wider range of applications. Due to the high combinatorial complexity of underlying model, it is difficult to … Read more

    An incremental mirror descent subgradient algorithm with random sweeping and proximal step

  • Axel Böhm
  • Radu Ioan Bot
    • Categories Convex Optimization, Stochastic Programming

    We investigate the convergence properties of incremental mirror descent type subgradient algorithms for minimizing the sum of convex functions. In each step we only evaluate the subgradient of a single component function and mirror it back to the feasible domain, which makes iterations very cheap to compute. The analysis is made for a randomized selection … Read more

    Convergence Analysis of Processes with Valiant Projection Operators in Hilbert Space

  • Yair Censor
  • Rafiq Mansour
    • Categories Constrained Nonlinear Optimization, Convex Optimization

    Convex feasibility problems require to find a point in the intersection of a finite family of convex sets. We propose to solve such problems by performing set-enlargements and applying a new kind of projection operators called valiant projectors. A valiant projector onto a convex set implements a special relaxation strategy, proposed by Goffin in 1971, … Read more

    Proximal-Proximal-Gradient Method

  • Ernest K. Ryu
  • Wotao Yin
    • Categories Convex Optimization

    In this paper, we present the proximal-proximal-gradient method (PPG), a novel optimization method that is simple to implement and simple to parallelize. PPG generalizes the proximal-gradient method and ADMM and is applicable to minimization problems written as a sum of many differentiable and many non-differentiable convex functions. The non-differentiable functions can be coupled. We furthermore … Read more

    Finding a best approximation pair of points for two polyhedra

  • Ron Aharoni
  • Yair Censor
  • Zilin Jiang
    • Categories Convex Optimization, Polyhedra Tags , , ,

    Given two disjoint convex polyhedra, we look for a best approximation pair relative to them, i.e., a pair of points, one in each polyhedron, attaining the minimum distance between the sets. Cheney and Goldstein showed that alternating projections onto the two sets, starting from an arbitrary point, generate a sequence whose two interlaced subsequences converge … Read more

    Implementing the ADMM to Big Datasets: A Case Study of LASSO

  • Xiangfeng Wang
  • Qingzhi Yang
  • Xiao-Ming Yuan
  • Hangrui Yue
    • Categories Convex Optimization

    The alternating direction method of multipliers (ADMM) has been popularly used for a wide range of applications in the literature. When big datasets with high-dimensional variables are considered, subproblems arising from the ADMM must be solved inexactly even though theoretically they may have closed-form solutions. Such a scenario immediately poses mathematical ambiguities such as how … Read more

    Randomized Similar Triangles Method: A Unifying Framework for Accelerated Randomized Optimization Methods (Coordinate Descent, Directional Search, Derivative-Free Method)

  • Pavel Dvurechensky
  • Alexander Gasnikov
  • Alexander Tiurin
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , , , ,

    In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework, which allows to construct different types of accelerated randomized methods for such problems and to prove convergence rate theorems for them. … Read more