Convex and Nonsmooth Optimization

Relationships between constrained and unconstrained multi-objective optimization and application in location theory

  • Christian Günther
  • Christiane Tammer
    • Categories Constrained Nonlinear Optimization, Generalized Convexity/Monoticity, Multi-Criteria Optimization

    This article deals with constrained multi-objective optimization problems. The main purpose of the article is to investigate relationships between constrained and unconstrained multi-objective optimization problems. Under suitable assumptions (e.g., generalized convexity assumptions) we derive a characterization of the set of (strictly, weakly) efficient solutions of a constrained multi-objective optimization problem using characterizations of the sets … Read more

    The rate of convergence of Nesterov’s accelerated forward-backward method is actually (k^{-2})$

  • Hédy Attouch
  • Juan Peypouquet
    • Categories Convex Optimization Tags , ,

    The {\it forward-backward algorithm} is a powerful tool for solving optimization problems with a {\it additively separable} and {\it smooth} + {\it nonsmooth} structure. In the convex setting, a simple but ingenious acceleration scheme developed by Nesterov has been proved useful to improve the theoretical rate of convergence for the function values from the standard … Read more

    Fast convergence of inertial dynamics and algorithms with asymptotic vanishing damping

  • Hédy Attouch
  • Z Chbani
  • Juan Peypouquet
  • P Redont
    • Categories Convex Optimization Tags , , , , , ,

    In a Hilbert space setting $\mathcal H$, we study the fast convergence properties as $t \to + \infty$ of the trajectories of the second-order differential equation \begin{equation*} \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t), \end{equation*} where $\nabla\Phi$ is the gradient of a convex continuously differentiable function $\Phi: \mathcal H \to \mathbb R$, … Read more

    A Stochastic Electricity Market Clearing Formulation with Consistent Pricing Properties

  • Mihai Anitescu
  • John Birge
  • Kibaek Kim
  • Victor M. Zavala
    • Categories Applications - OR and Management Sciences, Convex and Nonsmooth Optimization, Stochastic Programming

    We argue that deterministic market clearing formulations introduce arbitrary distortions between day-ahead and expected real-time prices that bias economic incentives and block diversi cation. We extend and analyze the stochastic clearing formulation proposed by Pritchard et al. (2010) in which the social surplus function induces penalties between day-ahead and real-time quantities. We prove that the formulation … Read more

    From error bounds to the complexity of first-order descent methods for convex functions

  • Jérôme Bolte
  • Trong Phong Nguyen
  • Juan Peypouquet
  • Bruce Suter
    • Categories Convex and Nonsmooth Optimization Tags , , , , , ,

    This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex … Read more

    An optimal first-order primal-dual gap reduction framework for constrained convex optimization

  • Volkan Cevher
  • Quoc Tran-Dinh
    • Categories Convex and Nonsmooth Optimization

    We introduce an analysis framework for constructing optimal first-order primal-dual methods for the prototypical constrained convex optimization tem- plate. While this class of methods offers scalability advantages in obtaining nu- merical solutions, they have the disadvantage of producing sequences that are only approximately feasible to the problem constraints. As a result, it is theoretically challenging … Read more

    Techniques in Iterative Proton CT Image Reconstruction

  • Yair Censor
  • Scott Penfold
    • Categories Biomedical Applications, Convex Optimization Tags , , , , , ,

    This is a review paper on some of the physics, modeling, and iterative algorithms in proton computed tomography (pCT) image reconstruction. The primary challenge in pCT image reconstruction lies in the degraded spatial resolution resulting from multiple Coulomb scattering within the imaged object. Analytical models such as the most likely path (MLP) have been proposed … Read more

    A composition projection method for convex feasibility problems

  • Chen Jiawei
    • Categories Convex and Nonsmooth Optimization

    In this article, we propose a composition projection algorithm for solving feasibility problem in Hilbert space. The convergence of the proposed algorithm are established by using gap vector which does not involve the nonempty intersection assumption. Moreover, we provide the sufficient and necessary condition for the convergence of the proposed method. Article Download View A … Read more

    Manifold Sampling for L1 Nonconvex Optimization

  • Jeffrey Larson
  • Matt Menickelly
  • Stefan M. Wild
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , ,

    We present a new algorithm, called manifold sampling, for the unconstrained minimization of a nonsmooth composite function $h\circ F$ when $h$ has known structure. In particular, by classifying points in the domain of the nonsmooth function $h$ into manifolds, we adapt search directions within a trust-region framework based on knowledge of manifolds intersecting the current … Read more

    On Cournot-Nash-Walras equilibria and their computation

  • Michal Cervinka
  • Michael C. Ferris
  • Jiri V. Outrata
    • Categories Nonsmooth Optimization Tags , , , , ,

    This paper considers a model of Cournot-Nash-Walras (CNW) equilibrium where the Cournot-Nash concept is used to capture equilibrium of an oligopolistic market with non-cooperative players/ rms who share a certain amount of a so-called rare resource needed for their production, and the Walras equilibrium determines the price of that rare resource. We prove the existence of … Read more