Constrained Nonlinear Optimization

A Globally Convergent Stabilized SQP Method: Superlinear Convergence

  • Philip E. Gill
  • Vyacheslav Kungurtsev
  • Daniel P. Robinson
    • Categories Constrained Nonlinear Optimization Tags , , , , , ,

    Regularized and stabilized sequential quadratic programming (SQP) methods are two classes of methods designed to resolve the numerical and theoretical difficulties associated with ill-posed or degenerate nonlinear optimization problems. Recently, a regularized SQP method has been proposed that allows convergence to points satisfying certain second-order KKT conditions (SIAM J. Optim., 23(4):1983–2010, 2013). The method is … Read more

    Local Convergence of an Algorithm for Subspace Identification from Partial Data

  • Laura Balzano
  • Stephen Wright
    • Categories Constrained Nonlinear Optimization, Data-Mining Tags , , ,

    GROUSE (Grassmannian Rank-One Update Subspace Estimation) is an iterative algorithm for identifying a linear subspace of $\R^n$ from data consisting of partial observations of random vectors from that subspace. This paper examines local convergence properties of GROUSE, under assumptions on the randomness of the observed vectors, the randomness of the subset of elements observed at … Read more

    Projection Methods: An Annotated Bibliography of Books and Reviews

  • Andrzej Cegielski
  • Yair Censor
    • Categories Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization Tags , ,

    Projections onto sets are used in a wide variety of methods in optimization theory but not every method that uses projections really belongs to the class of projection methods as we mean it here. Here projection methods are iterative algorithms that use projections onto sets while relying on the general principle that when a family … Read more

    An Efficient Gauss-Newton Algorithm for Symmetric Low-Rank Product Matrix Approximations

  • Xin Liu
  • Zaiwen Wen
  • Yin Zhang
    • Categories Constrained Nonlinear Optimization, Nonlinear Systems and Least-Squares Tags , , ,

    We derive and study a Gauss-Newton method for computing a symmetric low-rank product that is the closest to a given symmetric matrix in Frobenius norm. Our Gauss-Newton method, which has a particularly simple form, shares the same order of iteration-complexity as a gradient method when the size of desired eigenspace is small, but can be … Read more

    Strict Fejér Monotonicity by Superiorization of Feasibility-Seeking Projection Methods

  • Yair Censor
  • Alexander J. Zaslavski
    • Categories Constrained Nonlinear Optimization, Parallel Algorithms Tags , , , , , , ,

    We consider the superiorization methodology, which can be thought of as lying between feasibility-seeking and constrained minimization. It is not quite trying to solve the full fledged constrained minimization problem; rather, the task is to find a feasible point which is superior (with respect to the objective function value) to one returned by a feasibility-seeking … Read more

    Linear equalities in blackbox optimization

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

    The Mesh Adaptive Direct Search (Mads) algorithm is designed for blackbox optimization problems subject to general inequality constraints. Currently, Mads does not support equalities, neither in theory nor in practice. The present work proposes extensions to treat problems with linear equalities whose expression is known. The main idea consists in reformulating the optimization problem into … Read more

    Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods

  • Yair Censor
  • Daniel Reem
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , , , , , ,

    The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, … Read more

    A DC (Difference of Convex functions) approach of the MPECs

  • Rafael Correa
  • Matthieu Marechal
    • Categories Constrained Nonlinear Optimization Tags , ,

    This article deals with a study of the MPEC problem based on a reformulation to a DC problem (Difference of Convex functions). This reformulation is obtained by a partial penalization of the constraints. In this article we prove that a classical optimality condition for a DC program, if a constraint qualification is satisfied for MPEC, … Read more

    An Interior-Point Method for Nonlinear Optimization Problems with Locatable and Separable Nonsmoothness

  • Martin Schmidt
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    A lot of real-world optimization models comprise nonconvex and nonlinear as well as nonsmooth functions leading to very hard classes of optimization models. In this article a new interior-point method for the special but practically relevant class of optimization problems with locatable and separable nonsmooth aspects is presented. After motivating and formalizing the problems under … Read more

    Copositivity-based approximations for mixed-integer fractional quadratic optimization

  • Paula Alexandra Amaral
  • Immanuel M. Bomze
    • Categories Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming Tags , ,

    We propose a copositive reformulation of the mixed-integer fractional quadratic problem (MIFQP) under general linear constraints. This problem class arises naturally in many applications, e.g., for optimizing communication or social networks, or studying game theory problems arising from genetics. It includes several APX-hard subclasses: the maximum cut problem, the $k$-densest subgraph problem and several of … Read more