Linear Programming

A Polynomial Arc-Search Interior-Point Algorithm for Convex Quadratic Programming

  • Yaguang Yang
    • Categories Linear Programming Tags , ,

    Arc-search is developed for linear programming in \cite{yang09, yang10}. The algorithms search for optimizers along an ellipse that are approximations of the central path. In this paper, the arc-search method is applied to primal-dual path-following interior-point method for convex quadratic programming. A simple algorithm with iteration complexity $O(\sqrt{n}\log(1/\epsilon))$ is devised. Several improvements on the simple … Read more

    Infeasible Constraint-Reduced Interior-Point Methods for Linear Optimization

  • Meiyun Y. He
  • André L. Tits
    • Categories Linear Programming

    Constraint-reduction schemes have been proposed for the solution by means of interior-point methods of linear programs with many more inequality constraints than variables in standard dual form. Such schemes have been shown to be provably convergent and highly efficient in practice. A critical requirement of these schemes is the availability of an initial dual-feasible point. … Read more

    Simultaneous Column-and-Row Generation for Large-Scale Linear Programs with Column-Dependent-Rows

  • Ş. İlker Birbil
  • Kerem Bülbül
  • Ibrahim Muter
  • Guvenc Sahin
    • Categories Linear Programming

    In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many … Read more

    On the Volumetric Path

  • Murat Mut
  • Tamás Terlaky
    • Categories Linear Programming Tags , , , , , ,

    We consider the logarithmic and the volumetric barrier functions used in interior point methods. In the case of the logarithmic barrier function, the analytic center of a level set is the point at which the central path intersects that level set. We prove that this also holds for the volumetric path. For the central path, … Read more

    On the parallel solution of dense saddle-point linear systems arising in stochastic programming

  • Mihai Anitescu
  • Cosmin G. Petra
  • Miles Lubin
    • Categories Linear Programming, Parallel Algorithms, Stochastic Programming Tags , , ,

    We present a novel approach for solving dense saddle-point linear systems in a distributed-memory environment. This work is motivated by an application in stochastic optimization problems with recourse, but the proposed approach can be used for a large family of dense saddle-point systems, in particular those arising in convex programming. Although stochastic optimization problems have … Read more

    Computational and Economic Limitations of Dispatch Operations in the Next-Generation Power Grid

  • Audun Botterud
  • Emil M. Constantinescu
  • Victor M. Zavala
  • Jianhui Wang
    • Categories Dynamic Programming, Linear Programming, Production and Logistics Tags , , , , ,

    We study the interactions between computational and economic performance of dispatch operations under highly dynamic environments. In particular, we discuss the need for extending the forecast horizon of the dispatch formulation in order to anticipate steep variations of renewable power and highly elastic loads. We present computational strategies to solve the increasingly larger optimization problems … Read more

    On the Equivalencey of Linear Programming Problems and Zero-Sum Games

  • Ilan Adler
    • Categories Game Theory, Linear Programming Tags , , , , ,

    In 1951, Dantzig showed the equivalence of linear programming and two-person zero-sum games. However, in the description of his reduction from linear programming to zero-sum games, he noted that there was one case in which his reduction does not work. This also led to incomplete proofs of the relationship between the Minmax Theorem of game … Read more

    Shrink-Wrapping trajectories for Linear Programming

  • Yuriy Zinchenko
    • Categories Linear Programming

    Hyperbolic Programming (HP) –minimizing a linear functional over an affine subspace of a finite-dimensional real vector space intersected with the so-called hyperbolicity cone– is a class of convex optimization problems that contains well-known Linear Programming (LP). In particular, for any LP one can readily provide a sequence of HP relaxations. Based on these hyperbolic relaxations, … Read more

    A Pure L1-norm Principal Component Analysis

  • E.L. Boone
  • J. Paul Brooks
  • JH Dulá
    • Categories Data-Mining, Linear Programming Tags , ,

    The L1 norm has been applied in numerous variations of principal component analysis (PCA). L1-norm PCA is an attractive alternative to traditional L2-based PCA because it can impart robustness in the presence of outliers and is indicated for models where standard Gaussian assumptions about the noise may not apply. Of all the previously-proposed PCA schemes … Read more

    A sufficiently exact inexact Newton step based on reusing matrix information

  • Anders Forsgren
    • Categories Constrained Nonlinear Optimization, Linear Programming, Nonlinear Systems and Least-Squares Tags , ,

    Newton’s method is a classical method for solving a nonlinear equation $F(z)=0$. We derive inexact Newton steps that lead to an inexact Newton method, applicable near a solution. The method is based on solving for a particular $F'(z_{k’})$ during $p$ consecutive iterations $k=k’,k’+1,\dots,k’+p-1$. One such $p$-cycle requires $2^p-1$ solves with the matrix $F'(z_{k’})$. If matrix … Read more