facial reduction

A minimal face constant rank constraint qualification for reducible conic programming

  • Roberto Andreani
  • Gabriel Haeser
  • leomakoto
  • Héctor Ramírez
    • Categories Cone Programming, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , , ,

    \(\) In a previous paper [R. Andreani, G. Haeser, L. M. Mito, H. Ramírez, T. P. Silveira. First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank condition. Mathematical Programming, 2023. DOI: 10.1007/s10107-023-01942-8] we introduced a constant rank constraint qualification for nonlinear semidefinite and second-order cone programming by considering all … Read more

    Closing Duality Gaps of SDPs through Perturbation

  • Takashi Tsuchiya
  • Bruno F. Lourenço
  • Masakazu Muramatsu
  • Takayuki Okuno
    • Categories Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , , , ,

    \(\) Let \(({\bf P},{\bf D})\) be a primal-dual pair of SDPs with a nonzero finite duality gap. Under such circumstances, \({\bf P}\) and \({\bf D}\) are weakly feasible and if we perturb the problem data to recover strong feasibility, the (common) optimal value function \(v\) as a function of the perturbation is not well-defined at … Read more

    Revisiting Degeneracy, Strict Feasibility, Stability, in Linear Programming

  • Haesol Im
  • Henry Wolkowicz
    • Categories Applications - OR and Management Sciences, Linear Programming Tags , , , , , ,

    Currently, the simplex method and the interior point method are indisputably the most popular algorithms for solving linear programs, LPs. Unlike general conic programs, LPs with a finite optimal value do not require strict feasibility in order to establish strong duality. Hence strict feasibility is seldom a concern, even though strict feasibility is equivalent to … Read more

    A Strengthened Barvinok-Pataki Bound on SDP Rank

  • Haesol Im
  • Henry Wolkowicz
    • Categories Semi-definite Programming Tags , , , ,

    The Barvinok-Pataki bound provides an upper bound on the rank of extreme points of a spectrahedron. This bound depends solely on the number of affine constraints of the problem, i.e., on the algebra of the problem. Specifically, the triangular number of the rank r is upper bounded by the number of affine constraints. We revisit … Read more

    SDP-based bounds for the Quadratic Cycle Cover Problem via cutting plane augmented Lagrangian methods and reinforcement learning

  • Frank de Meijer
  • Renata Sotirov
    • Categories Combinatorial Optimization, Semi-definite Programming Tags , , , , ,

    We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the … Read more

    A simplified treatment of Ramana’s exact dual for semidefinite programming

  • Bruno F. Lourenço
  • Gabor Pataki
    • Categories Convex Optimization, Nonlinear Optimization, Semi-definite Programming Tags , , , ,

    In semidefinite programming the dual may fail to attain its optimal value and there could be a duality gap, i.e., the primal and dual optimal values may differ. In a striking paper, Ramana proposed a polynomial size extended dual that does not have these deficiencies and yields a number of fundamental results in complexity theory. … Read more

    A Restricted Dual Peaceman-Rachford Splitting Method for QAP

  • Naomi Graham
  • Hao Hu
  • Xinxin Li
  • Henry Wolkowicz
  • Haesol Im
    • Categories (Mixed) Integer Nonlinear Programming, Linear, Cone and Semidefinite Programming, Optimization Software Benchmark Tags , , , ,

    We revisit and strengthen splitting methods for solving doubly nonnegative, DNN, relaxations of the quadratic assignment problem, QAP. We use a modified restricted contractive splitting method, rPRSM, approach. Our strengthened bounds and new dual multiplier estimates improve on the bounds and convergence results in the literature. Citation Department of Combinatorics & Optimization, University of Waterloo, … Read more

    Facial Reduction for Symmetry Reduced Semidefinite Programs

  • Hao Hu
  • Renata Sotirov
  • Henry Wolkowicz
    • Categories Semi-definite Programming Tags , , , ,

    We consider both facial and symmetry reduction techniques for semidefinite programming, SDP. We show that the two together fit surprisingly well in an alternating direction method of multipliers, ADMM, approach. The combination of facial and symmetry reduction leads to a significant improvement in both numerical stability and running time for both the ADMM and interior … Read more

    A Strictly Contractive Peaceman-Rachford Splitting Method for the Doubly Nonnegative Relaxation of the Minimum Cut Problem

  • Xinxin Li
  • Ting Kei Pong
  • Hao Sun
  • Henry Wolkowicz
    • Categories (Mixed) Integer Nonlinear Programming, Linear, Cone and Semidefinite Programming, Optimization Software Benchmark Tags , , , , , ,

    The minimum cut problem, MC, and the special case of the vertex separator problem, consists in partitioning the set of nodes of a graph G into k subsets of given sizes in order to minimize the number of edges cut after removing the k-th set. Previous work on this topic uses eigenvalue, semidefinite programming, SDP, … Read more

    Error Bounds and Singularity Degree in Semidefinite Programming

  • Stefan Sremac
  • Henry Wolkowicz
  • Hugo J. Woerdeman
    • Categories Convex Optimization, Semi-definite Programming Tags , , ,

    In semidefinite programming a proposed optimal solution may be quite poor in spite of having sufficiently small residual in the optimality conditions. This issue may be framed in terms of the discrepancy between forward error (the unmeasurable `true error’) and backward error (the measurable violation of optimality conditions). In his seminal work, Sturm provided an … Read more