Nonlinear Optimization

Scalable Parallel Nonlinear Optimization with PyNumero and Parapint

  • Jose Rodriguez
  • Robert Parker
  • Carl D. Laird
  • Bethany Nicholson
  • John D. Siirola
  • Michael Lee Bynum
    • Categories Nonlinear Optimization, Parallel Algorithms Tags ,

    We describe PyNumero, an open-source, object-oriented programming framework in Python that supports rapid development of performant parallel algorithms for structured nonlinear programming problems (NLP’s) using the Message Passing Interface (MPI). PyNumero provides three fundamental building blocks for developing NLP algorithms: a fast interface for calculating first and second derivatives with the AMPL Solver Library (ASL), … Read more

    A Local MM Subspace Method for Solving Constrained Variational Problems in Image Recovery

  • Émilie Chouzenoux
  • Ségolène Martin
  • Jean-Christophe Pesquet
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , , ,

    This article introduces a new Penalized Majorization-Minimization Subspace algorithm (P-MMS) for solving smooth, constrained optimization problems. In short, our approach consists of embedding a subspace algorithm in an inexact exterior penalty procedure. The subspace strategy, combined with a Majoration-Minimization step-size search, takes great advantage of the smoothness of the penalized cost function, while the penalty … Read more

    A quasi-Newton method with Wolfe line searches for multiobjective optimization

  • Leandro Fonseca Prudente
  • D. R. Souza
    • Categories Multi-Criteria Optimization, Unconstrained Optimization Tags , , , ,

    We propose a BFGS method with Wolfe line searches for unconstrained multiobjective optimization problems. The algorithm is well defined even for general nonconvex problems. Global convergence and R-linear convergence to a Pareto optimal point are established for strongly convex problems. In the local convergence analysis, if the objective functions are locally strongly convex with Lipschitz … Read more

    An extended delayed weighted gradient algorithm for solving strongly convex optimization problems

  • Roberto Andreani
  • Harry Oviedo
  • Marcos Raydan
  • Leonardo D. Secchin
    • Categories Unconstrained Optimization Tags , , ,

    The recently developed delayed weighted gradient method (DWGM) is competitive with the well-known conjugate gradient (CG) method for the minimization of strictly convex quadratic functions. As well as the CG method, DWGM has some key optimality and orthogonality properties that justify its practical performance. The main difference with the CG method is that, instead of … Read more

    A Cubic Regularization of Newton’s Method with Finite-Difference Hessian Approximations

  • Max L. N. Gonçalves
  • Geovani Nunes Grapiglia
  • Gilson Silva
    • Categories Nonlinear Optimization

    In this paper, we present a version of the Cubic Regularization of Newton’s method for unconstrained nonconvex optimization, in which the Hessian matrices are approximated by forward finite difference Hessians. The regularization parameter of the cubic models and the accuracy of the Hessian approximations are jointly adjusted using a nonmonotone line-search criterion. Assuming that the … Read more

    Nonlinear matrix recovery using optimization on the Grassmann manifold

  • Coralia Cartis
  • Armin Eftekhari
  • Florentin Goyens
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization Tags , , ,

    We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by … Read more

    Two limited-memory optimization methods with minimum violation of the previous quasi-Newton equations

  • Ladislav Lukšan
  • Jan Vlček
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , , , ,

    Limited-memory variable metric methods based on the well-known BFGS update are widely used for large scale optimization. The block version of the BFGS update, derived by Schnabel (1983), Hu and Storey (1991) and Vl·cek and Luk·san (2019), satis¯es the quasi-Newton equations with all used di®erence vectors and for quadratic objective functions gives the best improvement … Read more

    SABRINA: A Stochastic Subspace Majorization-Minimization Algorithm

  • Jean-Baptiste Fest
  • Émilie Chouzenoux
    • Categories Convex Optimization, Nonlinear Optimization Tags , , , , ,

    A wide class of problems involves the minimization of a coercive and differentiable function $F$ on $\mathbb{R}^N$ whose gradient cannot be evaluated in an exact manner. In such context, many existing convergence results from standard gradient-based optimization literature cannot be directly applied and robustness to errors in the gradient is not necessarily guaranteed. This work … Read more

    A Trilevel Model for Segmentation of the Power Transmission Grid Cyber Network

  • Bryan Arguello
  • Jared Gearhart
  • Emma S. Johnson
    • Categories Network Optimization, Nonlinear Optimization, Other Topics Tags , , ,

    Network segmentation of a power grid’s communication system is one way to make the grid more resilient to cyber attacks. We develop a novel trilevel programming model to optimally segment a grid communication system, taking into account the actions of an information technolology (IT) administrator, attacker, and grid operator. The IT administrator is given an … Read more

    Developments in mathematical algorithms and computational tools for proton CT and particle therapy treatment planning

  • Yair Censor
  • Keith E. Schubert
  • Reinhard W. Schulte
    • Categories Basic Sciences Applications, Biomedical Applications, Constrained Nonlinear Optimization Tags , , , , , , , ,

    We summarize recent results and ongoing activities in mathematical algorithms and computer science methods related to proton computed tomography (pCT) and intensitymodulated particle therapy (IMPT) treatment planning. Proton therapy necessitates a high level of delivery accuracy to exploit the selective targeting imparted by the Bragg peak. For this purpose, pCT utilizes the proton beam itself … Read more