Month: March 2012

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

    Reweighted $\ell_1hBcMinimization for Sparse Solutions to Underdetermined Linear Systems

  • Duan Li
  • Yun-Bin Zhao
    • Categories Basic Sciences Applications, Linear Programming, Nonsmooth Optimization Tags , , , , ,

    Numerical experiments have indicated that the reweighted $\ell_1$-minimization performs exceptionally well in locating sparse solutions of underdetermined linear systems of equations. Thus it is important to carry out a further investigation of this class of methods. In this paper, we point out that reweighted $\ell_1$-methods are intrinsically associated with the minimization of the so-called merit … 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

    Irreducible elements of the copositive cone

  • Peter J.C. Dickinson
  • Mirjam Dür
  • Roland Hildebrand
  • Luuk Gijben
    • Categories Linear, Cone and Semidefinite Programming Tags , ,

    An element $A$ of the $n \times n$ copositive cone $\copos{n}$ is called irreducible with respect to the nonnegative cone~$\NNM{n}$ if it cannot be written as a nontrivial sum $A = C+N$ of a copositive matrix $C$ and an elementwise nonnegative matrix $N$. This property was studied by Baumert~\cite{Baumert65} who gave a characterisation of irreducible … 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

    Efficient Cardinality/Mean-Variance Portfolios

  • R. P. Brito
  • Luis Nunes Vicente
    • Categories Finance and Economics, Multi-Criteria Optimization, Nonsmooth Optimization Tags , , , , ,

    A number of variants of the classical Markowitz mean-variance optimization model for portfolio selection have been investigated to render it more realistic. Recently, it has been studied the imposition of a cardinality constraint, setting an upper bound on the number of active positions taken in the portfolio, in an attempt to improve its performance and … Read more