Convex and Nonsmooth Optimization

E. Lieb convexity inequalities and noncommutative Bernstein inequality in Jordan-algebraic setting

  • Leonid Faybusovich
    • Categories Generalized Convexity/Monoticity, Other Topics Tags , ,

    We describe a Jordan-algebraic version of E. Lieb convexity inequalities. A joint convexity of Jordan-algebraic version of quantum entropy is proven. SA spectral theory on semi-simple complex Jordan algebras is used as atool to prove the convexity results. Possible applications to optimization and statistics are indicated CitationPreprint, University of Notre Dame, August 2014ArticleDownload View PDF

    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

    Self Equivalence of the Alternating Direction Method of Multipliers

  • Ming Yan
  • Wotao Yin
    • Categories Convex Optimization Tags , , , ,

    The alternating direction method of multipliers (ADM or ADMM) breaks a complex optimization problem into much simpler subproblems. The ADM algorithms are typically short and easy to implement yet exhibit (nearly) state-of-the-art performance for large-scale optimization problems. To apply ADM, we first formulate a given problem into the “ADM-ready” form, so the final algorithm depends … Read more

    Block stochastic gradient iteration for convex and nonconvex optimization

  • Yangyang Xu
  • Wotao Yin
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Stochastic Approaches Tags , ,

    The stochastic gradient (SG) method can minimize an objective function composed of a large number of differentiable functions, or solve a stochastic optimization problem, to a moderate accuracy. The block coordinate descent/update (BCD) method, on the other hand, handles problems with multiple blocks of variables by updating them one at a time; when the blocks … Read more

    An improved algorithm for L2-Lp minimization problem

  • Dongdong Ge
  • He Rongchuan
  • He Simai
    • Categories Global Optimization Theory, Nonsmooth Optimization, Unconstrained Optimization

    In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the L2−Lp minimization problem. We propose an iterative algorithm that decides the next iteration based on the local convexity/concavity/sparsity of its current position. We show that our algorithm finds an epsilon-KKT point within O(log(1/epsilon)) iterations. The same result is also … Read more

    A general inertial proximal point method for mixed variational inequality problem

  • Caihua Chen
  • Shiqian Ma
  • Junfeng Yang
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    In this paper, we first propose a general inertial proximal point method for the mixed variational inequality (VI) problem. Based on our knowledge, without stronger assumptions, convergence rate result is not known in the literature for inertial type proximal point methods. Under certain conditions, we are able to establish the global convergence and a $o(1/k)$ … Read more

    Robust Block Coordinate Descent

  • Kimon Fountoulakis
  • Rachael Tappenden
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Unconstrained Optimization Tags , , , ,

    In this paper we present a novel randomized block coordinate descent method for the minimization of a convex composite objective function. The method uses (approximate) partial second-order (curvature) information, so that the algorithm performance is more robust when applied to highly nonseparable or ill conditioned problems. We call the method Robust Coordinate Descent (RCD). At … Read more

    The Principle of Hamilton for Mechanical Systems with Impacts and Unilateral Constraints

  • Kerim Yunt
    • Categories Complementarity and Variational Inequalities, Nonsmooth Optimization Tags , , ,

    An action integral is presented for Hamiltonian mechanics in canonical form with unilateral constraints and/or impacts. The transition conditions on generalized impulses and the energy are presented as variational inequalities, which are obtained by the application of discontinuous transversality conditions. The energetical behavior for elastic, plastic and blocking type impacts are analyzed. A general impact … Read more

    Hypotheses testing on the optimal values of several risk-neutral or risk-averse convex stochastic programs and application to hypotheses testing on several risk measure values

  • Vincent Guigues
    • Categories Convex Optimization, Stochastic Programming

    Given an arbitrary number of risk-averse or risk-neutral convex stochastic programs, we study hypotheses testing problems aiming at comparing the optimal values of these stochastic programs on the basis of samples of the underlying random vectors. We propose non-asymptotic tests based on confidence intervals on the optimal values of the stochastic programs obtained using the … Read more