Convex and Nonsmooth Optimization

Adaptive Importance Sampling Based Surrogation Methods for Bayesian Hierarchical Models, via Logarithmic Integral Optimization

  • Ziyu He
  • Junyi Liu
  • Jong-Shi Pang
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization, Stochastic Programming Tags , ,

    We explore Maximum a Posteriori inference of Bayesian Hierarchical Models (BHMs) with intractable normalizers, which are increasingly prevalent in contemporary applications and pose computational challenges when combined with nonconvexity and nondifferentiability. To address these, we propose the Adaptive Importance Sampling-based Surrogation method, which efficiently handles nonconvexity and nondifferentiability while improving the sampling approximation of the … Read more

    Asynchronous Iterations in Optimization: New Sequence Results and Sharper Algorithmic Guarantees

  • Hamid Reza Feyzmahdavian
  • Mikael Johansson
    • Categories Convex Optimization, Global Optimization, Parallel Algorithms Tags , , , ,

    We introduce novel convergence results for asynchronous iterations that appear in the analysis of parallel and distributed optimization algorithms. The results are simple to apply and give explicit estimates for how the degree of asynchrony impacts the convergence rates of the iterates. Our results shorten, streamline and strengthen existing convergence proofs for several asynchronous optimization … Read more

    Learning in Inverse Optimization: Incenter Cost, Augmented Suboptimality Loss, and Algorithms

  • Pedro Zattoni Scroccaro
  • Bilge Atasoy
  • Peyman Mohajerin Esfahani
    • Categories Convex and Nonsmooth Optimization Tags ,

    In Inverse Optimization (IO), an expert agent solves an optimization problem parametric in an exogenous signal. From a learning perspective, the goal is to learn the expert’s cost function given a dataset of signals and corresponding optimal actions. Motivated by the geometry of the IO set of consistent cost vectors, we introduce the “incenter” concept, … Read more

    The complexity of first-order optimization methods from a metric perspective

  • Adrian Lewis
  • Tonghua Tian
    • Categories Nonsmooth Optimization, Other Topics Tags , , , , ,

    A central tool for understanding first-order optimization algorithms is the Kurdyka-Lojasiewicz inequality. Standard approaches to such methods rely crucially on this inequality to leverage sufficient decrease conditions involving gradients or subgradients. However, the KL property fundamentally concerns not subgradients but rather “slope”, a purely metric notion. By highlighting this view, and avoiding any use of … Read more

    First-Order Methods for Nonsmooth Nonconvex Functional Constrained Optimization with or without Slater Points

  • Zhichao Jia
  • Benjamin Grimmer
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    Constrained optimization problems where both the objective and constraints may be nonsmooth and nonconvex arise across many learning and data science settings. In this paper, we show a simple first-order method finds a feasible, ϵ-stationary point at a convergence rate of O(ϵ−4) without relying on compactness or Constraint Qualification (CQ). When CQ holds, this convergence is measured by … Read more

    A New Inexact Proximal Linear Algorithm with Adaptive Stopping Criteria for Robust Phase Retrieval

  • Zhong Zheng
  • Shiqian Ma
  • Lingzhou Xue
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    This paper considers the robust phase retrieval problem, which can be cast as a nonsmooth and nonconvex optimization problem. We propose a new inexact proximal linear algorithm with the subproblem being solved inexactly. Our contributions are two adaptive stopping criteria for the subproblem. The convergence behavior of the proposed methods is analyzed. Through experiments on … Read more

    A descent method for nonsmooth multiobjective optimization problems on Riemannian manifolds

  • Chunming Tang
  • Hao He
  • Jinbao Jian
  • Miantao Chao
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags

    In this paper, a descent method for nonsmooth multiobjective optimization problems on complete Riemannian manifolds is proposed. The objective functions are only assumed to be locally Lipschitz continuous instead of convexity used in existing methods. A necessary condition for Pareto optimality in Euclidean space is generalized to the Riemannian setting. At every iteration, an acceptable … Read more

    The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets

  • Yair Censor
  • Rafiq Mansour
  • Daniel Reem
    • Categories Approximation Algorithms, Constrained Nonlinear Optimization, Convex Optimization Tags , , , , ,

    Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process … Read more

    Numerical Methods for Convex Multistage Stochastic Optimization

  • Guanghui Lan
  • Alexander Shapiro
    • Categories Convex Optimization, Dynamic Programming, Stochastic Programming Tags , , , , , , ,

    Optimization problems involving sequential decisions in  a  stochastic environment    were studied  in  Stochastic Programming (SP), Stochastic Optimal Control  (SOC) and Markov Decision Processes (MDP). In this paper we mainly concentrate on SP and  SOC modelling   approaches. In these frameworks there are natural situations  when the considered problems are  convex. Classical approach to sequential optimization … Read more

    On the B-differential of the componentwise minimum of two affine vector functions

  • Jean-Pierre Dussault
  • Jean Charles Gilbert
  • Baptiste Plaquevent-Jourdain
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Nonsmooth Optimization Tags , , , , , , , , , , , , , , ,

    This paper focuses on the description and computation of the B-differential of the componentwise minimum of two affine vector functions. This issue arises in the reformulation of the linear complementarity problem with the Min C-function. The question has many equivalent formulations and we identify some of them in linear algebra, convex analysis and discrete geometry. … Read more