Nonlinear Optimization

Towards Simulation Based Mixed-Integer Optimization with Differential Equations

  • Martin Gugat
  • Günter Leugering
  • Martin Schmidt
  • Mathias Sirvent
  • David Wintergerst
  • Alexander Martin
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Systems governed by Differential Equations Optimization Tags , , , ,

    We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with “black-box” nonlinearities, where the latter, e.g., may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refining the mixed-integer linear relaxation of the nonlinearity. We prove that … Read more

    ALGORITHM XXX: SC-SR1: MATLAB SOFTWARE FOR SOLVING SHAPE-CHANGING L-SR1 TRUST-REGION SUBPROBLEMS

  • Johannes Brust
  • Oleg Burdakov
  • Jennifer B. Erway
  • Roummel F. Marcia
  • Ya-xiang Yuan
    • Categories Unconstrained Optimization Tags , , , , ,

    We present a MATLAB implementation of the shape-changing sym- metric rank-one (SC-SR1) method that solves trust-region subproblems when a limited-memory symmetric rank-one (L-SR1) matrix is used in place of the true Hessian matrix. The method takes advantage of two shape-changing norms [4, 3] to decompose the trust-region subproblem into two separate problems. Using one of … Read more

    A derivative-free trust-region augmented Lagrangian algorithm

  • Charles Audet
  • Sébastien Le Digabel
  • Mathilde Peyrega
    • Categories Constrained Nonlinear Optimization

    We present a new derivative-free trust-region (DFTR) algorithm to solve general nonlinear constrained problems with the use of an augmented Lagrangian method. No derivatives are used, neither for the objective function nor for the constraints. An augmented Lagrangian method, known as an effective tool to solve equality and inequality constrained optimization problems with derivatives, is … Read more

    Stochastic Quasi-Newton Methods for Nonconvex Stochastic Optimization

  • Donald Goldfarb
  • Wei Liu
  • Shiqian Ma
  • Xiao Wang
    • Categories Nonlinear Optimization Tags , , , , ,

    In this paper we study stochastic quasi-Newton methods for nonconvex stochastic optimization, where we assume that noisy information about the gradients of the objective function is available via a stochastic first-order oracle ($\SFO$). We propose a general framework for such methods, for which we prove almost sure convergence to stationary points and analyze its worst-case … Read more

    Solving Highly Detailed Gas Transport MINLPs: Block Separability and Penalty Alternating Direction Methods

  • Björn Geißler
  • Antonio Morsi
  • Lars Schewe
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Applications - Science and Engineering, Constrained Nonlinear Optimization Tags , , , ,

    Detailed modeling of gas transport problems leads to nonlinear and nonconvex mixed-integer optimization or feasibility models (MINLPs) because both the incorporation of discrete controls of the network as well as accurate physical and technical modeling is required in order to achieve practical solutions. Hence, ignoring certain parts of the physics model is not valid for … Read more

    Globally Convergent Levenberg-Marquardt Method For Phase Retrieval

  • Xin Liu
  • Zaiwen Wen
  • Chao Ma
    • Categories Nonlinear Optimization Tags , , ,

    In this paper, we consider a nonlinear least squares model for the phase retrieval problem. Since the Hessian matrix may not be positive definite and the Gauss-Newton (GN) matrix is singular at any optimal solution, we propose a modified Levenberg-Marquardt (LM) method, where the Hessian is substituted by a summation of the GN matrix and … Read more

    A Multilevel Proximal Gradient Algorithm for a Class of Composite Optimization Problems

  • Panos Parpas
    • Categories Nonlinear Optimization Tags , ,

    Composite optimization models consist of the minimization of the sum of a smooth (not necessarily convex) function and a non-smooth convex function. Such models arise in many applications where, in addition to the composite nature of the objective function, a hierarchy of models is readily available. It is common to take advantage of this hierarchy … Read more

    Regularized monotonic regression

  • Oleg Burdakov
  • Oleg Sysoev
    • Categories Quadratic Programming, Statistics Tags , , , , ,

    Monotonic (isotonic) Regression (MR) is a powerful tool used for solving a wide range of important applied problems. One of its features, which poses a limitation on its use in some areas, is that it produces a piecewise constant fitted response. For smoothing the fitted response, we introduce a regularization term in the MR formulated … Read more

    A progressive barrier derivative-free trust-region algorithm for constrained optimization

  • Charles Audet
  • Andrew R. Conn
  • Sébastien Le Digabel
  • Mathilde Peyrega
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , ,

    We study derivative-free constrained optimization problems and propose a trust-region method that builds linear or quadratic models around the best feasible and and around the best infeasible solutions found so far. These models are optimized within a trust region, and the progressive barrier methodology handles the constraints by progressively pushing the infeasible solutions toward the … Read more

    A Second-Order Information-Based Gradient and Function Sampling Method for Nonconvex, Nonsmooth Optimization

  • Elias Salomão Helou
  • Sandra Augusta Santos
  • Lucas E. A. Simões
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization

    This paper has the goal to propose a gradient and function sampling method that under special circumstances moves superlinearly to a minimizer of a general class of nonsmooth and nonconvex functions. We present global and local convergence theory with illustrative examples that corroborate and elucidate the theoretical results obtained along the manuscript. ArticleDownload View PDF