ALGORITHM & DOCUMENTATION: MINRES-QLP for Singular Symmetric and Hermitian Linear Equations and Least-Squares Problems

  • Sou-Cheng Choi
    • Categories Applications - OR and Management Sciences, Applications - Science and Engineering, Optimization Software and Modeling Systems Tags , , , , , , , , ,

    We describe algorithm MINRES-QLP and its FORTRAN 90 implementation for solving symmetric or Hermitian linear systems or least-squares problems. If the system is singular, MINRES-QLP computes the unique minimum-length solution (also known as the pseudoinverse solution), which generally eludes MINRES. In all cases, it overcomes a potential instability in the original MINRES algorithm. A positive-definite … Read more

    S_1/2 Regularization Methods and Fixed Point Algorithms for Affine Rank Minimization Problems

  • Peng Dingtao
  • Yu Jian
  • Naihua Xiu
    • Categories Linear, Cone and Semidefinite Programming, Nonsmooth Optimization

    The affine rank minimization problem is to minimize the rank of a matrix under linear constraints. It has many applications in various areas such as statistics, control, system identification and machine learning. Unlike the literatures which use the nuclear norm or the general Schatten $q~(0<q<1)$ quasi-norm to approximate the rank of a matrix, in this … Read more

    New Fractional Error Bounds for Nonconvex Polynomial Systems with Applications to Holderian Stability in Optimization and Spectral Theory of Tensors

  • Guoyin Li
  • Boris S. Mordukhovich
  • Tiên-So'n Pham
    • Categories Convex and Nonsmooth Optimization Tags , , , , , ,

    In this paper we derive new fractional error bounds for nonconvex polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. The results obtained do not require any regularity assumptions and resolve, in particular, some open questions posed in the literature. The developed techniques are … Read more

    Spectral bounds for the independence ratio and the chromatic number of an operator

  • Christine Bachoc
  • Fernando Mário de Oliveira Filho
  • Frank Vallentin
  • Evan DeCorte
    • Categories Linear, Cone and Semidefinite Programming Tags , , , , ,

    We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L^2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical … Read more

    VSDP: A Matlab toolbox for verified semidefinite-quadratic-linear programming

  • Viktor Härter
  • Christian Jansson
  • Marko Lange
    • Categories Linear, Cone and Semidefinite Programming, Optimization Software and Modeling Systems Tags , , , ,

    VSDP is a software package that is designed for the computation of verified results in conic programming. The current version of VSDP supports the constraint cone consisting of the product of semidefinite cones, second-order cones and the nonnegative orthant. It provides functions for computing rigorous error bounds of the true optimal value, verified enclosures of … Read more

    Transmission Expansion Planning Using an AC Model: Formulations and Possible Relaxations

  • G. Th. Heydt
  • Hans D Mittelmann
  • Vijay Vittal
  • Hui Zhang
    • Categories Facility Planning and Design Tags , , , , ,

    Transmission expansion planning (TEP) is a rather complicated process which requires extensive studies to determine when, where and how many transmission facilities are needed. A well planned power system will not only enhance the system reliability, but also tend to contribute positively to the overall system operating efficiency. Starting with two mixed-integer nonlinear programming (MINLP) … Read more

    Second-Order Variational Analysis in Conic Programming with Applications to Optimality and Stability

  • Boris S. Mordukhovich
  • Jiri V. Outrata
  • Héctor Ramírez
    • Categories Convex and Nonsmooth Optimization Tags , , , , , ,

    This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a second-order generalized di erential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues related … Read more

    A Framework of Constraint Preserving Update Schemes for Optimization on Stiefel Manifold

  • Yu-Hong Dai
  • Bo Jiang
    • Categories Applications - Science and Engineering, Nonlinear Optimization Tags , , , , , , , , , , ,

    This paper considers optimization problems on the Stiefel manifold $X^TX=I_p$, where $X\in \mathbb{R}^{n \times p}$ is the variable and $I_p$ is the $p$-by-$p$ identity matrix. A framework of constraint preserving update schemes is proposed by decomposing each feasible point into the range space of $X$ and the null space of $X^T$. While this general framework … Read more

    Risk-Averse Control of Undiscounted Transient Markov Models

  • Özlem Cavus
  • Andrzej Ruszczyński
    • Categories Dynamic Programming, Stochastic Programming

    We use Markov risk measures to formulate a risk-averse version of the undiscounted total cost problem for a transient controlled Markov process. We derive risk-averse dynamic programming equations and we show that a randomized policy may be strictly better than deterministic policies, when risk measures are employed. We illustrate the results on an optimal stopping … Read more

    A branch-and-bound algorithm for biobjective mixed-integer programs

  • Pietro Belotti
  • Banu Soylu
  • Margaret Wiecek
    • Categories (Mixed) Integer Linear Programming, Multi-Criteria Optimization Tags , , ,

    We propose a branch-and-bound (BB) algorithm for biobjective mixed-integer linear programs (BOMILPs). Our approach makes no assumption on the type of problem and we prove that it returns all Pareto points of a BOMILP. We discuss two techniques upon which the BB is based: fathoming rules to eliminate those subproblems that are guaranteed not to … Read more