Nonsmooth Optimization

A Comparison of Nonsmooth, Nonconvex, Constrained Optimization Solvers for the Design of Time-Delay Compensators

  • Vyacheslav Kungurtsev
  • Tim Mitchell
  • Tomas Vyhlidal
    • Categories Control Applications, Nonsmooth Optimization, Optimization Software Benchmark

    We present a detailed set of performance comparisons of two state-of-the-art solvers for the application of designing time-delay compensators, an important problem in the field of robust control. Formulating such robust control mechanics as constrained optimization problems often involves objective and constraint functions that are both nonconvex and nonsmooth, both of which present significant challenges … Read more

    Basis Pursuit Denoise with Nonsmooth Constraints

  • Aleksandr Y. Aravkin
  • Robert Baraldi
  • Rajiv Kumar
    • Categories Basic Sciences Applications, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    Level-set optimization formulations with data-driven constraints minimize a regularization functional subject to matching observations to a given error level. These formulations are widely used, particularly for matrix completion and sparsity promotion in data interpolation and denoising. The misfit level is typically measured in the l2 norm, or other smooth metrics. In this paper, we present … Read more

    Parallelizing Subgradient Methods for the Lagrangian Dual in Stochastic Mixed-Integer Programming

  • Cong Han Lim
  • Jeff Linderoth
  • James R. Luedtke
  • Stephen Wright
    • Categories Nonsmooth Optimization, Stochastic Programming Tags , ,

    The dual decomposition of stochastic mixed-integer programs can be solved by the projected subgradient algorithm. We show how to make this algorithm more amenable to parallelization in a master-worker model by describing two approaches, which can be combined in a natural way. The first approach partitions the scenarios into batches, and makes separate use of … Read more

    Acceleration of Primal-Dual Methods by Preconditioning and Simple Subproblem Procedures

  • Yanli Liu
  • Yunbei Xu
  • Wotao Yin
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , , ,

    Primal-Dual Hybrid Gradient (PDHG) and Alternating Direction Method of Multipliers (ADMM) are two widely-used first-order optimization methods. They reduce a difficult problem to simple subproblems, so they are easy to implement and have many applications. As first-order methods, however, they are sensitive to problem conditions and can struggle to reach the desired accuracy. To improve … Read more

    A New Sequential Optimality Condition for Constrained Nonsmooth Optimization

  • Elias Salomão Helou
  • Sandra Augusta Santos
  • Lucas E. A. Simões
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization

    We introduce a sequential optimality condition for locally Lipschitz constrained nonsmooth optimization, verifiable just using derivative information, and which holds even in the absence of any constraint qualification. The proposed sequential optimality condition is not only novel for nonsmooth problems, but brings new insights for the smooth case as well. We present a practical algorithm … Read more

    A Unified Framework for Sparse Relaxed Regularized Regression: SR3

  • Aleksandr Y. Aravkin
  • Peng Zheng
  • Travis Askham
  • Steve Brunton
  • Nathan Kutz
    • Categories Nonlinear Systems and Least-Squares, Nonsmooth Optimization, Statistics Tags , , , ,

    Regularized regression problems are ubiquitous in statistical modeling, signal processing, and machine learning. Sparse regression in particular has been instrumental in scientific model discovery, including compressed sensing applications, vari- able selection, and high-dimensional analysis. We propose a broad framework for sparse relaxed regularized regression, called SR3. The key idea is to solve a relaxation of … Read more

    Nonmonotonicity and Quasiconvexity on Equilibrium Problems

  • Lennin Mallma Ramirez
    • Categories Applications - Science and Engineering, Complementarity and Variational Inequalities, Nonsmooth Optimization

    In this note, some results are introduced considering the assumptions of quasiconvexity and nonmonotonicity, finally an application and an idea to solve the quasiconvex equilibrium problem are presented considering these new results. Article Download View Nonmonotonicity and Quasiconvexity on Equilibrium Problems

    Inexact alternating projections on nonconvex sets

  • Dmitriy Drusvyatskiy
  • Adrian Lewis
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex sets is typically intractable, but we show that computationally-cheap inexact projections may suffice instead. In particular, if one set is defined … Read more

    Performance indicators in multiobjective optimization

  • Charles Audet
  • Jean Bigeon
  • Dominique Cartier
  • Sébastien Le Digabel
  • Ludovic Salomon
    • Categories Multi-Criteria Optimization, Nonsmooth Optimization

    In recent years, the development of new algorithms for multiobjective optimization has considerably grown. A large number of performance indicators has been introduced to measure the quality of Pareto front approximations produced by these algorithms. In this work, we propose a review of a total of 63 performance indicators partitioned into four groups according to … Read more

    Discerning the linear convergence of ADMM for structured convex optimization through the lens of variational analysis

  • Xiao-Ming Yuan
  • Shangzhi Zeng
  • Jin Zhang
    • Categories Convex and Nonsmooth Optimization, Nonsmooth Optimization Tags , , , , , ,

    Despite the rich literature, the linear convergence of alternating direction method of multipliers (ADMM) has not been fully understood even for the convex case. For example, the linear convergence of ADMM can be empirically observed in a wide range of applications, while existing theoretical results seem to be too stringent to be satisfied or too … Read more