Nonlinear Optimization

Algorithms for Difference-of-Convex (DC) Programs Based on Difference-of-Moreau-Envelopes Smoothing

  • Xu Andy Sun
  • Kaizhao Sun
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    In this paper we consider minimization of a difference-of-convex (DC) function with and without linear constraints. We first study a smooth approximation of a generic DC function, termed difference-of-Moreau-envelopes (DME) smoothing, where both components of the DC function are replaced by their respective Moreau envelopes. The resulting smooth approximation is shown to be Lipschitz differentiable, … Read more

    Penetration depth between two convex polyhedra: An efficient global optimization approach

  • Mark A. Abramson
  • Griffin D. Kent
  • Gavin W. Smith
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Global Optimization Applications Tags , , , ,

    During the detailed design phase of an aerospace program, one of the most important consistency checks is to ensure that no two distinct objects occupy the same physical space. Since exact geometrical modeling is usually intractable, geometry models are discretized, which often introduces small interferences not present in the fully detailed model. In this paper, … Read more

    A Proximal Quasi-Newton Trust-Region Method for Nonsmooth Regularized Optimization

  • Aleksandr Y. Aravkin
  • Robert Baraldi
  • Dominique Orban
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , , ,

    We develop a trust-region method for minimizing the sum of a smooth term f and a nonsmooth term h, both of which can be nonconvex. Each iteration of our method minimizes apossibly nonconvex model of f+h in a trust region. The model coincides with f+h in value and subdifferential at the center. We establish global … Read more

    Scalable adaptive cubic regularization methods

  • Jean-Pierre Dussault
  • Dominique Orban
    • Categories Unconstrained Optimization Tags , ,

    Adaptive cubic regularization (ARC) methods for unconstrained optimization compute steps from linear systems involving a shifted Hessian in the spirit of the Levenberg-Marquardt and trust-region methods. The standard approach consists in performing an iterative search for the shift akin to solving the secular equation in trust-region methods. Such search requires computing the Cholesky factorization of … Read more

    Fast cluster detection in networks by first-order optimization

  • Immanuel M. Bomze
  • Francesco Rinaldi
  • Damiano Zeffiro
    • Categories Network Optimization, Nonlinear Optimization Tags , , ,

    Cluster detection plays a fundamental role in the analysis of data. In this paper, we focus on the use of s-defective clique models for network-based cluster detection and propose a nonlinear optimization approach that efficiently handles those models in practice. In particular, we introduce an equivalent continuous formulation for the problem under analysis, and we … Read more

    Complexity of a Projected Newton-CG Method for Optimization with Bounds

  • Yue Xie
  • Stephen Wright
    • Categories Bound-constrained Optimization Tags , , , ,

    This paper describes a method for solving smooth nonconvex minimization problems subject to bound constraints with good worst-case complexity and practical performance. The method contains elements of two existing methods: the classical gradient projection approach for bound-constrained optimization and a recently proposed Newton-conjugate gradient algorithm for unconstrained nonconvex optimization. Using a new definition of approximate … Read more

    An inexact restoration-nonsmooth algorithm with variable accuracy for stochastic nonsmooth convex optimization problems in machine learning and stochastic linear complementarity problems

  • Nataša Krejić
  • Nataša Krklec Jerinkić
  • Tijana Ostojić
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Stochastic Programming Tags , , , ,

    We study unconstrained optimization problems with nonsmooth and convex objective function in the form of a mathematical expectation. The proposed method approximates the expected objective function with a sample average function using Inexact Restoration-based adapted sample sizes. The sample size is chosen in an adaptive manner based on Inexact Restoration. The algorithm uses line search … Read more

    Universal Conditional Gradient Sliding for Convex Optimization

  • Yuyuan Ouyang
  • Trevor Squires
    • Categories Convex Optimization, Nonlinear Optimization Tags , , , ,

    In this paper, we present a first-order projection-free method, namely, the universal conditional gradient sliding (UCGS) method, for solving ε-approximate solutions to convex differentiable optimization problems. For objective functions with Hölder continuous gradients, we show that UCGS is able to terminate with ε-solutions with at most O((1/ε)^(2/(1+3v))) gradient evaluations and O((1/ε)^(4/(1+3v))) linear objective optimizations, where … Read more

    Further developments of methods for traversing regions of non-convexity in optimization problems

  • Michael Bartholomew-Biggs
  • Salah Beddiaf
  • Bruce Christianson
    • Categories Unconstrained Optimization Tags , ,

    This paper continues to address one of its author’s obsession with the well- known problem of dealing with non-convexity during the minimization of a nonlinear function f(x) by Newton-like methods. It builds on some proposals made by the present authors in “A Comparison of methods for traversing regions of non-convexity in optimization problems”. (Numerical Algorithms … Read more

    Robust Price Optimization of Multiple Products under Interval Uncertainties

  • Mahdi Hamzeei
  • Alvin Lim
  • Jiefeng Xu
    • Categories Constrained Nonlinear Optimization, Robust Optimization, Yield Management Tags , ,

    In this paper, we solve the multiple product price optimization problem under interval uncertainties of the price sensitivity parameters in the demand function. The objective of the price optimization problem is to maximize the overall revenue of the firm where the decision variables are the prices of the products supplied by the firm. We propose … Read more