Quadratic Programming

Minimizing convex quadratics with variable precision Krylov methods

  • Serge Gratton
  • Philippe L. Toint
  • Ehouarn Simon
    • Categories Convex Optimization, Optimization Software and Modeling Systems, Quadratic Programming Tags , ,

    Iterative algorithms for the solution of convex quadratic optimization problems are investigated, which exploit inaccurate matrix-vector products. Theoretical bounds on the performance of a Conjugate Gradients and a Full-Orthormalization methods are derived, the necessary quantities occurring in the theoretical bounds estimated and new practical algorithms derived. Numerical experiments suggest that the new methods have significant … Read more

    A hybrid algorithm for the two-trust-region subproblem

  • Saeid Ansary Karbasy
  • Maziar Salahi
    • Categories Global Optimization, Quadratic Programming Tags , , ,

    Two-trust-region subproblem (TTRS), which is the minimization of a general quadratic function over the intersection of two full-dimensional ellipsoids, has been the subject of several recent research. In this paper, to solve TTRS, a hybrid of efficient algorithms for finding global and local-nonglobal minimizers of trust-region subproblem and the alternating direction method of multipliers (ADMM) … Read more

    A conjugate gradient-based algorithm for large-scale quadratic programming problem with one quadratic constraint

  • Maziar Salahi
  • Akram Taati
    • Categories Global Optimization, Quadratic Programming Tags , ,

    In this paper, we consider the nonconvex quadratically constrained quadratic programming (QCQP) with one quadratic constraint. By employing the conjugate gradient method, an efficient algorithm is proposed to solve QCQP that exploits the sparsity of the involved matrices and solves the problem via solving a sequence of positive definite system of linear equations after identifying … Read more

    Quasi-Newton approaches to Interior Point Methods for quadratic problems

  • Jacek Gondzio
  • Francisco N. C. Sobral
    • Categories Nonlinear Systems and Least-Squares, Quadratic Programming Tags , , , ,

    Interior Point Methods (IPM) rely on the Newton method for solving systems of nonlinear equations. Solving the linear systems which arise from this approach is the most computationally expensive task of an interior point iteration. If, due to problem’s inner structure, there are special techniques for efficiently solving linear systems, IPMs enjoy fast convergence and … Read more

    Sensitivity Analysis for Nonlinear Programming in CasADi

  • J A E Andersson
  • J B Rawlings
    • Categories Modeling Languages and Systems, Quadratic Programming Tags , , , ,

    We present an extension of the CasADi numerical optimization framework that allows arbitrary order NLP sensitivities to be calculated automatically and efficiently. The approach, which can be used together with any NLP solver available in CasADi, is based on a sparse QR factorization and an implementation of a primal-dual active set method. The whole toolchain … Read more

    Stable interior point method for convex quadratic programming with strict error bounds

  • Stefania Bellavia
  • Martin Neuenhofen
    • Categories Quadratic Programming Tags , , ,

    We present a short step interior point method for solving a class of nonlinear programming problems with quadratic objective function. Convex quadratic programming problems can be reformulated as problems in this class. The method is shown to have weak polynomial time complexity. A complete proof of the numerical stability of the method is provided. No … Read more

    On a Frank-Wolfe Type Theorem in Cubic Optimization

  • Diethard Klatte
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Quadratic Programming Tags , , , ,

    A classical result due to Frank and Wolfe (1956) says that a quadratic function $f$ attains its supremum on a nonempty polyhedron $M$ if $f$ is bounded from above on $M$. In this note, we present a stringent proof of the extension of this result to cubic optimization (known from Andronov, Belousov and Shironin (1982)). … Read more

    Trust your data or not – StQP remains StQP: Community Detection via Robust Standard Quadratic Optimization

  • Immanuel M. Bomze
  • Michael Kahr
  • Markus Leitner
    • Categories Linear, Cone and Semidefinite Programming, Quadratic Programming, Robust Optimization Tags , , , ,

    We consider the Robust Standard Quadratic Optimization Problem (RStQP), in which an uncertain (possibly indefinite) quadratic form is extremized over the standard simplex. Following most approaches, we model the uncertainty sets by ellipsoids, polyhedra, or spectrahedra, more precisely, by intersections of sub-cones of the copositive matrix cone. We show that the copositive relaxation gap of … Read more

    Solving Quadratic Programs to High Precision using Scaled Iterative Refinement

  • Ambros Gleixner
  • Sebastian Sager
  • Tobias Weber
    • Categories Quadratic Programming Tags , , ,

    Quadratic optimization problems (QPs) are ubiquitous, and solution algorithms have matured to a reliable technology. However, the precision of solutions is usually limited due to the underlying floating-point operations. This may cause inconveniences when solutions are used for rigorous reasoning. We contribute on three levels to overcome this issue. First, we present a novel refinement … Read more

    Exact Semidefinite Formulations for a Class of (Random and Non-Random) Nonconvex Quadratic Programs

  • Samuel Burer
  • Yinyu Ye
    • Categories Global Optimization Theory, Quadratic Programming, Semi-definite Programming Tags , ,

    We study a class of quadratically constrained quadratic programs (QCQPs), called {\em diagonal QCQPs\/}, which contain no off-diagonal terms $x_j x_k$ for $j \ne k$, and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature … Read more