Month: April 2017

Partially separable convexly-constrained optimization with non-Lipschitz singularities and its complexity

  • Xiaojun Chen
  • Philippe L. Toint
  • Hong Wang
    • Categories Nonsmooth Optimization Tags , ,

    An adaptive regularization algorithm using high-order models is proposed for partially separable convexly constrained nonlinear optimization problems whose objective function contains non-Lipschitzian $\ell_q$-norm regularization terms for $q\in (0,1)$. It is shown that the algorithm using an $p$-th order Taylor model for $p$ odd needs in general at most $O(\epsilon^{-(p+1)/p})$ evaluations of the objective function and … Read more

    Distributionally Robust Mechanism Design

  • Garud Iyengar
  • Çağıl Koçyiğit
  • Daniel Kuhn
  • Wolfram Wiesemann
    • Categories Robust Optimization Tags , , , ,

    We study a mechanism design problem where an indivisible good is auctioned to multiple bidders, for each of whom it has a private value that is unknown to the seller and the other bidders. The agents perceive the ensemble of all bidder values as a random vector governed by an ambiguous probability distribution, which belongs … Read more

    Several variants of the primal-dual hybrid gradient algorithm with applications

  • Jianchao Bai
  • Jicheng Li
  • Zhie Wu
    • Categories Complementarity and Variational Inequalities, Convex Optimization

    By reviewing the primal-dual hybrid algorithm (PDHA) proposed by He, You and Yuan (SIAM J. Imaging Sci. 2014;7(4):2526-2537), in this paper we introduce four improved schemes for solving a class of generalized saddle-point problems. By making use of the variational inequality, weaker conditions are presented to ensure the global convergence of the proposed algorithms, where … Read more

    Bilevel optimization with a multiobjective problem in the lower level

  • Roberto Andreani
  • Sandra Augusta Santos
  • Viviana A. Ramirez
  • Leonardo D. Secchin
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization

    Bilevel problems model instances with a hierarchical structure. Aiming at an efficient solution of a constrained multiobjective problem according with some pre-defined criterion, we reformulate this optimization but non standard problem as a classic bilevel one. This reformulation intents to encompass all the objectives, so that the properly efficient solution set is recovered by means … Read more

    Iteration-Complexity of a Linearized Proximal Multiblock ADMM Class for Linearly Constrained Nonconvex Optimization Problems

  • Jefferson Gonçalves Melo
  • Renato D.C. Monteiro
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    This paper analyzes the iteration-complexity of a class of linearized proximal multiblock alternating direction method of multipliers (ADMM) for solving linearly constrained nonconvex optimization problems. The subproblems of the linearized ADMM are obtained by partially or fully linearizing the augmented Lagrangian with respect to the corresponding minimizing block variable. The derived complexity bounds do not … Read more

    Structural Properties of Affine Sparsity Constraints

  • Miju Ahn
  • Hongbo Dong
  • Jong-Shi Pang
    • Categories Convex and Nonsmooth Optimization, Statistics

    We introduce a new constraint system for sparse variable selection in statistical learning. Such a system arises when there are logical conditions on the sparsity of certain unknown model parameters that need to be incorporated into their selection process. Formally, extending a cardinality constraint, an affine sparsity constraint (ASC) is defined by a linear inequality … Read more

    Small and Strong Formulations for Unions of Convex Sets from the Cayley Embedding

  • Juan Pablo Vielma
    • Categories (Mixed) Integer Nonlinear Programming, 0-1 Programming

    There is often a significant trade-off between formulation strength and size in mixed integer programming (MIP). When modeling convex disjunctive constraints (e.g. unions of convex sets), adding auxiliary continuous variables can sometimes help resolve this trade-off. However, standard formulations that use such auxiliary continuous variables can have a worse-than-expected computational effectiveness, which is often attributed … Read more

    From Data to Decisions: Distributionally Robust Optimization is Optimal

  • Daniel Kuhn
  • Bart Paul Gerard Van Parys
  • Peyman Mohajerin Esfahani
    • Categories Robust Optimization, Second-Order Cone Programming, Stochastic Programming Tags , , , , ,

    We study stochastic programs where the decision-maker cannot observe the distribution of the exogenous uncertainties but has access to a finite set of independent samples from this distribution. In this setting, the goal is to find a procedure that transforms the data to an estimate of the expected cost function under the unknown data-generating distribution, … Read more

    A Bregman alternating direction method of multipliers for sparse probabilistic Boolean network problem

  • Deng Kangkang
  • Peng Zheng
    • Categories Applications - Science and Engineering, Biomedical Applications, Constrained Nonlinear Optimization Tags , , , ,

    The main task of genetic regulatory networks is to construct a sparse probabilistic Boolean network (PBN) based on a given transition-probability matrix and a set of Boolean networks (BNs). In this paper, a Bregman alternating direction method of multipliers (BADMM) is proposed to solve the minimization problem raised in PBN. All the customized subproblem-solvers of … Read more

    Generalized Self-Concordant Functions: A Recipe for Newton-type Methods

  • Tianxiao Sun
  • Quoc Tran-Dinh
    • Categories Convex and Nonsmooth Optimization

    We study the smooth structure of convex functions by generalizing a powerful concept so-called \textit{self-concordance} introduced by Nesterov and Nemirovskii in the early 1990s to a broader class of convex functions, which we call \textit{generalized self-concordant functions}. This notion allows us to develop a unified framework for designing Newton-type methods to solve convex optimization problems. … Read more