Nonlinear Optimization

Optimality conditions and global convergence for nonlinear semidefinite programming

  • Roberto Andreani
  • Gabriel Haeser
  • Daiana O. Santos
    • Categories Nonlinear Optimization, Semi-definite Programming Tags , , ,

    Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization. In this paper, we extend theses concepts for nonlinear semidefinite programming. We define two sequential optimality conditions for nonlinear semidefinite programming. The first is a natural extension of the so-called Approximate-Karush-Kuhn-Tucker … Read more

    MIQP-Based Algorithm for the Global Solution of Economic Dispatch Problems with Valve-Point Effects

  • Pierre-Antoine Absil
  • Benoît Sluysmans
  • Nicolas Stevens
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Global Optimization Tags , , , ,

    Even in a static setting, the economic load dispatch problem (ELDP)—namely the cost-optimal distribution of power among generating units to meet a specific demand subject to system constraints—turns out to be a challenge owing to the consideration of valve-point effects (VPE), which make the cost function nonsmooth and nonconvex. We present a new method, termed … Read more

    Superiorization and perturbation resilience of algorithms: A continuously updated bibliography

  • Yair Censor
    • Categories Biomedical Applications, Constrained Nonlinear Optimization Tags ,

    This document presents a, chronologically ordered, bibliography of scientific publications on the superiorization methodology and perturbation resilience of algorithms which is compiled and continuously updated by us at: http://math.haifa.ac.il/yair/bib-superiorization-censor.html. Since the topic is relatively new it is possible to trace the work that has been published about it since its inception. To the best of … Read more

    A Newton-CG Algorithm with Complexity Guarantees for Smooth Unconstrained Optimization

  • Michael O'Neill
  • Clément W. Royer
  • Stephen Wright
    • Categories Unconstrained Optimization Tags , , ,

    We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton’s method and linear conjugate gradient, with explicit detection and use of negative curvature directions for the Hessian of the objective function. The algorithm tracks Newton-conjugate gradient procedures developed in the 1980s closely, but includes enhancements that allow worst-case complexity … Read more

    A comparison of methods for traversing non-convex regions in optimization problems

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

    This paper considers again the well-known problem of dealing with non-convex regions during the minimization of a nonlinear function F(x) by Newton-like methods. The proposal made here involves a curvilinear search along an approximation to the continuous steepest descent path defined by the solution of the ODE dx/dt = -grad F(x). The algorithm we develop … Read more

    An Improved Method of Total Variation Superiorization Applied to Reconstruction in Proton Computed Tomography

  • Yair Censor
  • Paniz Karbasi
  • Keith E. Schubert
  • Reinhard W. Schulte
  • Blake Schultze
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Convex Optimization Tags , , , , ,

    Previous work showed that total variation superiorization (TVS) improves reconstructed image quality in proton computed tomography (pCT). The structure of the TVS algorithm has evolved since then and this work investigated if this new algorithmic structure provides additional benefits to pCT image quality. Structural and parametric changes introduced to the original TVS algorithm included: (1) … Read more

    On Algebraic Proofs of Stability for Homogeneous Vector Fields

  • Amir Ali Ahmadi
  • Bachir El Khadir
    • Categories Control Applications, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , ,

    We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sum of squares … Read more

    A Distributed Quasi-Newton Algorithm for Empirical Risk Minimization with Nonsmooth Regularization

  • Stephen Wright
  • Cong Han Lim
  • Ching-pei Lee
    • Categories Data-Mining, Nonlinear Optimization, Parallel Algorithms Tags , , , , , , ,

    We propose a communication- and computation-efficient distributed optimization algorithm using second-order information for solving ERM problems with a nonsmooth regularization term. Current second-order and quasi-Newton methods for this problem either do not work well in the distributed setting or work only for specific regularizers. Our algorithm uses successive quadratic approximations, and we describe how to … Read more

    On an Elliptical Trust-Region Procedure for Ill-Posed Nonlinear Least-Squares Problems

  • Stefania Bellavia
  • Elisa Riccietti
    • Categories Nonlinear Systems and Least-Squares Tags , , , ,

    In this paper we address the stable numerical solution of ill-posed nonlinear least-squares problems with small residual. We propose an elliptical trust-region reformulation of a Levenberg-Marquardt procedure. Thanks to an appropriate choice of the trust-region radius, the proposed procedure guarantees an automatic choice of the free regularization parameters that, together with a suitable stopping criterion, … Read more

    Inexact Successive Quadratic Approximation for Regularized Optimization

  • Stephen Wright
  • Ching-pei Lee
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , ,

    Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration complexity focus on the special case of proximal gradient method, or accelerated variants thereof. There have been only a few studies of … Read more