Linear, Cone and Semidefinite Programming

Local superlinear convergence of polynomial-time interior-point methods for hyperbolic cone optimization problems

  • Yurii Nesterov
  • Levent Tunçel
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , , ,

    In this paper, we establish the local superlinear convergence property of some polynomial-time interior-point methods for an important family of conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier function, which must have {\em negative curvature}. We propose a new path-following predictor-corrector scheme, which work only … Read more

    A Facial Reduction Algorithm for Finding Sparse SOS Representations

  • Masakazu Muramatsu
  • Hayato Waki
    • Categories Convex Optimization, Global Optimization Theory, Semi-definite Programming Tags , , ,

    Facial reduction algorithm reduces the size of the positive semidefinite cone in SDP. The elimination method for a sparse SOS polynomial ([3]) removes unnecessary monomials for an SOS representation. In this paper, we establish a relationship between a facial reduction algorithm and the elimination method for a sparse SOS polynomial. CitationTechnical Report CS-09-02, Department of … Read more

    Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs

  • Miguel F. Anjos
  • Bissan Ghaddar
  • Juan C. Vera
    • Categories Combinatorial Optimization, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , ,

    Several types of relaxations for binary quadratic polynomial programs can be obtained using linear, second-order cone, or semidefinite techniques. In this paper, we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming. Using our framework, we re-derive previous relaxation schemes and provide new ones. In particular, we … Read more

    The positive semidefinite Grothendieck problem with rank constraint

  • Jop Briet
  • Fernando Mário de Oliveira Filho
  • Frank Vallentin
    • Categories Combinatorial Optimization, Linear, Cone and Semidefinite Programming Tags , , , , ,

    Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of size m x m, the positive semidefinite Grothendieck problem with rank-n-constraint is (SDP_n) maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, …, x_m \in S^{n-1}. In this paper we design a polynomial time approximation algorithm for SDP_n achieving an approximation … Read more

    On the nonexistence of sum of squares certificates for the BMV conjecture

  • Kristijan Cafuta
  • Igor Klep
  • Janez Povh
    • Categories Basic Sciences Applications, Semi-definite Programming Tags , , , , ,

    The algebraic reformulation of the BMV conjecture is equivalent to a family of dimensionfree tracial inequalities involving positive semidefinite matrices. Sufficient conditions for these to hold in the form of algebraic identities involving polynomials in noncommuting variables have been given by Markus Schweighofer and the second author. Later the existence of these certificates has been … Read more

    Quadratic factorization heuristics for copositive programming

  • Immanuel M. Bomze
  • Florian Jarre
  • Franz Rendl
    • Categories (Mixed) Integer Linear Programming, Global Optimization, Linear, Cone and Semidefinite Programming Tags , ,

    Copositive optimization problems are particular conic programs: extremize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone … Read more

    Matrix-Free Interior Point Method

  • Jacek Gondzio
    • Categories Linear Programming, Quadratic Programming Tags , , , , ,

    In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods … Read more

    Smoothing techniques for solving semidefinite programs with many constraints

  • Michel Baes
  • Michael Bürgisser
    • Categories Nonsmooth Optimization, Semi-definite Programming Tags , ,

    We use smoothing techniques to solve approximately mildly structured semidefinite programs with many constraints. As smoothing techniques require a specific problem format, we introduce an alternative problem formulation that fulfills the structural assumptions. The resulting algorithm has a complexity that depends linearly both on the number of constraints and on the inverse of the accuracy. … Read more

    Solving log-determinant optimization problems by a Newton-CG primal proximal point algorithm

  • Defeng Sun
  • Kim-Chuan Toh
  • Chengjing Wang
    • Categories Convex Optimization, Semi-definite Programming, Statistics Tags , ,

    We propose a Newton-CG primal proximal point algorithm for solving large scale log-determinant optimization problems. Our algorithm employs the essential ideas of the proximal point algorithm, the Newton method and the preconditioned conjugate gradient solver. When applying the Newton method to solve the inner sub-problem, we find that the log-determinant term plays the role of … Read more

    Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry

  • Etienne de Klerk
  • Renata Sotirov
    • Categories Combinatorial Optimization, Linear, Cone and Semidefinite Programming Tags , , ,

    Semidefinite programming (SDP) bounds for the quadratic assignment problem (QAP) were introduced in: [Q. Zhao, S.E. Karisch, F. Rendl, and H. Wolkowicz. Semidefinite Programming Relaxations for the Quadratic Assignment Problem. Journal of Combinatorial Optimization, 2,71–109, 1998.] Empirically, these bounds are often quite good in practice, but computationally demanding, even for relatively small instances. For QAP … Read more