Semi-definite Programming

The Largest Unsolved QAP Instance Tai256c Can Be Converted into A 256-dimensional Simple BQOP with A Single Cardinality Constraint

  • Koichi Fujii
  • Sunyoung Kim
  • Masakazu Kojima
  • Hans D Mittelmann
  • Yuji Shinano
    • Categories 0-1 Programming, Branch and Cut Algorithms, Semi-definite Programming Tags , , , , ,

    Tai256c is the largest unsolved quadratic assignment problem (QAP) instance in QAPLIB; a 1.48\% gap remains between the best known feasible objective value and lower bound of the unknown optimal value. This paper shows that the instance can be converted into a 256 dimensional binary quadratic optimization problem (BQOP) with a single cardinality constraint which … Read more

    Sparse PCA With Multiple Components

  • Ryan Cory-Wright
  • Jean Pauphilet
    • Categories Data Science Theory, Data-Mining, Semi-definite Programming

    Sparse Principal Component Analysis (sPCA) is a cardinal technique for obtaining combinations of features, or principal components (PCs), that explain the variance of high-dimensional datasets in an interpretable manner. This involves solving a sparsity and orthogonality-constrained convex maximization problem, which is extremely computationally challenging. Most existing works address sparse PCA via methods—such as iteratively computing … Read more

    A Strengthened SDP Relaxation for Quadratic Optimization Over the Stiefel Manifold

  • Samuel Burer
  • Kyungchan Park
    • Categories Global Optimization, Quadratic Programming, Semi-definite Programming Tags , ,

    We study semidefinite programming (SDP) relaxations for the NP-hard problem of globally optimizing a quadratic function over the Stiefel manifold. We introduce a strengthened relaxation based on two recent ideas in the literature: (i) a tailored SDP for objectives with a block-diagonal Hessian; (ii) and the use of the Kronecker matrix product to construct SDP relaxations. Using synthetic instances on … Read more

    Certifying Global Optimality of AC-OPF Solutions via sparse polynomial optimization

  • Jie Wang
  • Victor Magron
  • Jean-Bernard Lasserre
    • Categories Global Optimization Applications, Semi-definite Programming Tags , , , , ,

    We report the experimental results on certifying 1% global optimality of solutions of AC-OPF instances from PGLiB via the CS-TSSOS hierarchy — a moment-SOS based hierarchy that exploits both correlative and term sparsity, which can provide tighter SDP relaxations than Shor’s relaxation. Our numerical experiments demonstrate that the CS-TSSOS hierarchy scales well with the problem … Read more

    Decision Rule Approaches for Pessimistic Bilevel Linear Programs under Moment Ambiguity with Facility Location Applications

  • Akshit Goyal
  • Yiling Zhang
  • Chuan He
    • Categories Robust Optimization, Semi-definite Programming Tags , , , ,

    We study a pessimistic stochastic bilevel program in the context of sequential two-player games, where the leader makes a binary here-and-now decision, and the follower responds a continuous wait-and-see decision after observing the leader’s action and revelation of uncertainty. Only the information of the mean, covariance, and support is known. We formulate the problem as … Read more

    Accelerated first-order methods for a class of semidefinite programs

  • Alex L. Wang
  • Fatma Kılınç-Karzan
    • Categories Convex and Nonsmooth Optimization, Quadratic Programming, Semi-definite Programming Tags , , , ,

    This paper introduces a new storage-optimal first-order method (FOM), CertSDP, for solving a special class of semidefinite programs (SDPs) to high accuracy. The class of SDPs that we consider, the exact QMP-like SDPs , is characterized by low-rank solutions, a priori knowledge of the restriction of the SDP solution to a small subspace, and standard … Read more

    Partitioning through projections: strong SDP bounds for large graph partition problems

  • Frank de Meijer
  • Renata Sotirov
  • Angelika Wiegele
  • Shudian Zhao
    • Categories Semi-definite Programming Tags , , , ,

    The graph partition problem (GPP) aims at clustering the vertex set of a graph into a fixed number of disjoint subsets of given sizes such that the sum of weights of edges joining different sets is minimized. This paper investigates the quality of doubly nonnegative (DNN) relaxations, i.e., relaxations having matrix variables that are both … Read more

    Exact SDP relaxations for quadratic programs with bipartite graph structures

  • Godai Azuma
  • Mituhiro Fukuda
  • Sunyoung Kim
  • Makoto Yamashita
    • Categories Semi-definite Programming Tags , , , ,

    For nonconvex quadratically constrained quadratic programs (QCQPs), we first show that, under certain feasibility conditions, the standard semidefinite (SDP) relaxation is exact for QCQPs with bipartite graph structures. The exact optimal solutions are obtained by examining the dual SDP relaxation and the rank of the optimal solution of this dual SDP relaxation under strong duality. … Read more

    An SDP Relaxation for the Sparse Integer Least Square Problem

  • Alberto Del Pia
  • Dekun Zhou
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Semi-definite Programming Tags , , ,

    In this paper, we study the polynomial approximability or solvability of sparse integer least square problem (SILS), which is the NP-hard variant of the least square problem, where we only consider sparse {0, ±1}-vectors. We propose an l1-based SDP relaxation to SILS, and introduce a randomized algorithm for SILS based on the SDP relaxation. In … Read more

    Solving Two-Trust-Region Subproblems using Semidefinite Optimization with Eigenvector Branching

  • Kurt M. Anstreicher
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Semi-definite Programming Tags , , , ,

    Semidefinite programming (SDP) problems typically utilize the constraint that X-xx’ is PSD to obtain a convex relaxation of the condition X=xx’, where x is an n-vector. In this paper we consider a new hyperplane branching method for SDP based on using an eigenvector of X-xx’. This branching technique is related to previous work of Saxeena, … Read more