Nonlinear Optimization

Limited Memory Block Krylov Subspace Optimization for Computing Dominant Singular Value Decompositions

  • Xin Liu
  • Zaiwen Wen
  • Yin Zhang
    • Categories Constrained Nonlinear Optimization, Data-Mining Tags , , ,

    In many data-intensive applications, the use of principal component analysis (PCA) and other related techniques is ubiquitous for dimension reduction, data mining or other transformational purposes. Such transformations often require efficiently, reliably and accurately computing dominant singular value decompositions (SVDs) of large unstructured matrices. In this paper, we propose and study a subspace optimization technique … Read more

    On Differentiability Properties of Player Convex Generalized Nash Equilibrium Problems

  • Nadja Harms
  • Oliver Stein
  • Christian Kanzow
    • Categories Constrained Nonlinear Optimization, Game Theory, Nonsmooth Optimization Tags , , , , ,

    This article studies differentiability properties for a reformulation of a player convex generalized Nash equilibrium problem as a constrained and possibly nonsmooth minimization problem. By using several results from parametric optimization we show that, apart from exceptional cases, all locally minimal points of the reformulation are differentiability points of the objective function. This justifies a … Read more

    Stochastic Optimization Approach to Water Management in Cooling-Constrained Power Plants

  • Emil M. Constantinescu
  • Victor M. Zavala
  • Urmila Diwekar
  • Juan M. Salazar
    • Categories Applications - OR and Management Sciences, Constrained Nonlinear Optimization, Stochastic Programming

    We propose a stochastic optimization framework to perform water management in coolingconstrained power plants. The approach determines optimal set-points to maximize power output in the presence of uncertain weather conditions and water intake constraints. Weather uncertainty is quantified in the form of ensembles using the state-of-the-art numerical weather prediction model WRF. The framework enables us … Read more

    A Low-Memory Approach For Best-State Estimation Of Hidden Markov Models With Model Error

  • Mihai Anitescu
  • Emil M. Constantinescu
  • Xiaoyan Zeng
    • Categories Civil and Environmental Engineering, Statistics, Unconstrained Optimization Tags , , , ,

    We present a low-memory approach for the best-state estimate (data assimilation) of hidden Markov models where model error is considered. In particular, our findings apply for the 4D- Var framework. The novelty of our approach resides in the fact that the storage needed by our estimation framework, while including model error, is dramatically reduced from … Read more

    On optimizing the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere

  • Zhang Lei-Hong
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Global Optimization Tags , , , ,

    Given symmetric matrices $B,D\in R^{n\times n}$ and a symmetric positive definite matrix $W\in R^{n\times n},$ maximizing the sum of the Rayleigh quotient $x^T Dx$ and the generalized Rayleigh quotient $x^T Bx/x^TWx$ on the unit sphere not only is of mathematical interest in its own right, but also finds applications in practice. In this paper, we … Read more

    On the evaluation complexity of cubic regularization methods for potentially rank-deficient nonlinear least-squares problems and its relevance to constrained nonlinear optimization

  • Coralia Cartis
  • Nicholas I. M. Gould
  • Philippe L. Toint
    • Categories Nonlinear Optimization Tags , , , ,

    We propose a new termination criteria suitable for potentially singular, zero or non-zero residual, least-squares problems, with which cubic regularization variants take at most $\mathcal{O}(\epsilon^{-3/2})$ residual- and Jacobian-evaluations to drive either the Euclidean norm of the residual or its gradient below $\epsilon$; this is the best-known bound for potentially singular nonlinear least-squares problems. We then … Read more

    On Relaxing the Mangasarian-Fromovitz Constraint Qualification

  • Alexander Y. Kruger
  • Jiri V. Outrata
  • Leonid Minchenko
    • Categories Constrained Nonlinear Optimization

    For the classical nonlinear program two new relaxations of the Mangasarian-Fromovitz constraint qualification are discussed and their relationship with some standard constraint qualifications is examined. In particular, we establish the equivalence of one of these constraint qualifications with the recently suggested by Andreani et al. Constant rank of the subspace component constraint qualification. As an … Read more

    Global convergence and the Powell singular function

  • Trond Steihaug
  • Sara Suleiman
    • Categories Nonlinear Optimization, Nonlinear Systems and Least-Squares, Unconstrained Optimization

    The Powell singular function was introduced 1962 by M.J.D. Powell as an unconstrained optimization problem. The function is also used as nonlinear least squares problem and system of nonlinear equations. The function is a classic test function included in collections of test problems in optimization as well as an example problem in text books. In … Read more

    Solving trajectory optimization problems via nonlinear programming: the brachistochrone case study

  • Jean-Pierre Dussault
    • Categories Nonlinear Optimization Tags

    This note discusses reformulations the brachistochrone problem suitable for solution via NLP. The availability of solvers and modeling languages such as AMPL makes it tempting to formulate discretized optimization problems and get solutions to the discretized versions of trajectory optimization problems. We use the famous brachistochrone problem to warn that the resulting solutions may be … Read more

    Numerical Optimization of Eigenvalues of Hermitian Matrix Functions

  • Mustafa Kilic
  • E. Alper Yildirim
  • Emre Mengi
    • Categories Global Optimization Applications, Quadratic Programming Tags , , , ,

    The eigenvalues of a Hermitian matrix function that depends on one parameter analytically can be ordered so that each eigenvalue is an analytic function of the parameter. Ordering these analytic eigenvalues from the largest to the smallest yields continuous and piece-wise analytic functions. For multi-variate Hermitian matrix functions that depend on $d$ parameters analytically, the … Read more