Convex and Nonsmooth Optimization

FrankWolfe.jl: a high-performance and flexible toolbox for Frank-Wolfe algorithms and Conditional Gradients

  • Mathieu Besançon
  • Alejandro Carderera
  • Sebastian Pokutta
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Optimization Software and Modeling Systems Tags , ,

    We present FrankWolfe.jl, an open-source implementation of several popular Frank-Wolfe and Conditional Gradients variants for first-order constrained optimization. The package is designed with flexibility and high-performance in mind, allowing for easy extension and relying on few assumptions regarding the user-provided functions. It supports Julia’s unique multiple dispatch feature, and interfaces smoothly with generic linear optimization … Read more

    On the Weak and Strong Convergence of a Conceptual Algorithm for Solving Three Operator Monotone Inclusions

  • Yunier Bello-Cruz
  • Oday Hazaimah
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Infinite Dimensional Optimization Tags , , , ,

    In this paper, a conceptual algorithm modifying the forward-backward-half-forward (FBHF) splitting method for solving three operator monotone inclusion problems is investigated. The FBHF splitting method adjusts and improves Tseng’s forward-backward-forward (FBF) split- ting method when the inclusion problem has a third-part operator that is cocoercive. The FBHF method recovers the FBF iteration (when this aforementioned … Read more

    Robust Interior Point Method for Quantum Key Distribution Rate Computation

  • Hao Hu
  • Haesol Im
  • Jie Lin
  • Norbert Lutkenhaus
  • Henry Wolkowicz
    • Categories Applications - OR and Management Sciences, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , , ,

    While the security proof method for quantum key distribution, QKD, based on the numerical key rate calculation problem, is powerful in principle, the practicality of the method is limited by computational resources and the efficiency of the underlying algorithm for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model … Read more

    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

    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

    Long-run market equilibria in coupled energy sectors: A study of uniqueness

  • Jonas Egerer
  • Veronika Grimm
  • Julia Grübel
  • Gregor Zöttl
    • Categories Applications - OR and Management Sciences, Convex Optimization, Finance and Economics Tags , , , ,

    We propose an equilibrium model for coupled markets of multiple energy sectors. The agents in our model are operators of sector-specific production and sector-coupling technologies, as well as price-sensitive consumers with varying demand. We analyze long-run investment in production capacity in each sector and investment in coupling capacity between sectors, as well as production decisions … 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

    String-Averaging Methods for Best Approximation to Common Fixed Point Sets of Operators: The Finite and Infinite Cases

  • Yair Censor
  • Ariel Nisenbaum
    • Categories Convex Optimization, Nonlinear Optimization, Parallel Algorithms

    Abstract String-averaging is an algorithmic structure used when handling a family of operators in situations where the algorithm at hand requires to employ the operators in a specific order. Sequential orderings are well-known and a simultaneous order means that all operators are used simultaneously (in parallel). String-averaging allows to use strings of indices, constructed by … Read more

    Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution

  • Bahar Tașkesen
  • Soroosh Shafieezadeh-Abadeh
  • Daniel Kuhn
    • Categories Convex and Nonsmooth Optimization, Robust Optimization, Stochastic Programming Tags , , , , , ,

    Semi-discrete optimal transport problems, which evaluate the Wasserstein distance between a discrete and a generic (possibly non-discrete) probability measure, are believed to be computationally hard. Even though such problems are ubiquitous in statistics, machine learning and computer vision, however, this perception has not yet received a theoretical justification. To fill this gap, we prove that … Read more