Xiao-Ming Yuan

The direct extension of ADMM for three-block separable convex minimization models is convergent when one function is strongly convex

  • Xingju Cai
  • Deren Han
  • Xiao-Ming Yuan
    • Categories Convex Optimization Tags , , ,

    The alternating direction method of multipliers (ADMM) is a benchmark for solving a two-block linearly constrained convex minimization model whose objective function is the sum of two functions without coupled variables. Meanwhile, it is known that the convergence is not guaranteed if the ADMM is directly extended to a multiple-block convex minimization model whose objective … Read more

    Convergence Analysis of Primal-Dual Based Methods for Total Variation Minimization with Finite Element Approximation

  • Wenyi Tian
  • Xiao-Ming Yuan
    • Categories Applications - Science and Engineering Tags , , , ,

    We consider the total variation minimization model with consistent finite element discretization. It has been shown in the literature that this model can be reformulated as a saddle-point problem and be efficiently solved by the primal-dual method. The convergence for this application of the primal-dual method has also been analyzed. In this paper, we focus … Read more

    Block-wise Alternating Direction Method of Multipliers with Gaussian Back Substitution for Multiple-block Convex Programming

  • Xiaoling Fu
  • Bingsheng He
  • Xiangfeng Wang
  • Xiao-Ming Yuan
    • Categories Convex Optimization Tags , , , , , ,

    We consider the linearly constrained convex minimization model with a separable objective function which is the sum of m functions without coupled variables, and discuss how to design an efficient algorithm based on the fundamental technique of splitting the augmented Lagrangian method (ALM). Our focus is the specific big-data scenario where m is huge. A … Read more

    Block-wise Alternating Direction Method of Multipliers for Multiple-block Convex Programming and Beyond

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

    The alternating direction method of multipliers (ADMM) is a benchmark for solving a linearly constrained convex minimization model with a two-block separable objective function; and it has been shown that its direct extension to a multiple-block case where the objective function is the sum of more than two functions is not necessarily convergent. For the … Read more

    Application of the Strictly Contractive Peaceman-Rachford Splitting Method to Multi-block Separable Convex Programming

  • Bingsheng He
  • Xiao-Ming Yuan
  • Han Liu
  • Junwei Lu
    • Categories Convex Optimization Tags , , , , ,

    Recently, a strictly contractive Peaceman- Rachford splitting method (SC-PRSM) was proposed to solve a convex minimization model with linear constraints and a separable objective function which is the sum of two functions without coupled variables. We show by an example that the SC-PRSM cannot be directly extended to the case where the objective function is … Read more

    On the Direct Extension of ADMM for Multi-block Separable Convex Programming and Beyond: From Variational Inequality Perspective

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

    When the alternating direction method of multipliers (ADMM) is extended directly to a multi-block separable convex minimization model whose objective function is in form of more than two functions without coupled variables, it was recently shown that the convergence is not guaranteed. This fact urges to develop efficient algorithms that can preserve completely the numerical … 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

    The Direct Extension of ADMM for Multi-block Convex Minimization Problems is Not Necessarily Convergent

  • Caihua Chen
  • Bingsheng He
  • Yinyu Ye
  • Xiao-Ming Yuan
    • Categories Convex Optimization, Statistics Tags , , , , , , ,

    The alternating direction method of multipliers (ADMM) is now widely used in many fields, and its convergence was proved when two blocks of variables are alternatively updated. It is strongly desirable and practically valuable to extend ADMM directly to the case of a multi-block convex minimization problem where its objective function is the sum of … Read more

    A Generalized Proximal Point Algorithm and its Convergence Rate

  • Etienne Corman
  • Xiao-Ming Yuan
    • Categories Convex Optimization Tags , , ,

    We propose a generalized proximal point algorithm (PPA), in the generic setting of finding a zero point of a maximal monotone operator. In addition to the classical PPA, a number of benchmark operator splitting methods in PDE and optimization literatures such as the Douglas-Rachford splitting method, Peaceman-Rachford splitting method, alternating direction method of multipliers, generalized … Read more

    On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming

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

    The augmented Lagrangian method (ALM) is a benchmark for solving the convex minimization problem with linear constraints. We consider the special case where the objective is in form of the sum of m functions without coupled variables. For solving this separable convex programming model, it is usually required to decompose the ALM subproblem at each … Read more