This paper is concerned with the application of semidefinite programming to the satisfiability problem, and in particular with using semidefinite liftings to efficiently obtain proofs of unsatisfiability. We focus on the Tseitin satisfiability instances which are known to be hard for many proof systems. We present an explicit semidefinite programming problem with dimension linear in … Read more
We introduce a new barrier function to solve a class of Semidefinite Optimization Problems (SOP) with bounded variables. That class is motivated by some (SOP) as the minimization of the sum of the first few eigenvalues of symmetric matrices and graph partitioning problems. We study the primal-dual central path defined by the new barrier and … Read more
We study certain linear and semidefinite programming lifting approximation schemes for computing the stability number of a graph. Our work is based on, and refines De Klerk and Pasechnik’s approach to approximating the stability number via copositive programming (SIAM J. Optim. 12 (2002), 875–892). We provide a closed-form expression for the values computed by the … Read more
The facility layout problem is concerned with the arrangement of a given number of rectangular facilities so as to minimize the total cost associated with the (known or projected) interactions between them. We consider the one-dimensional space allocation problem (ODSAP), also known as the single-row facility layout problem, which consists in finding an optimal linear … Read more
The Semidefinite Program (SDP) is a fundamental problem in mathematical programming. It covers a wide range of applications, such as combinatorial optimization, control theory, polynomial optimization, and quantum chemistry. Solving extremely large-scale SDPs which could not be solved before is a significant work to open up a new vista of future applications of SDPs. Our … Read more
We propose a primal-dual path-following Mehrotra-type predictor-corrector method for solving convex quadratic semidefinite programming (QSDP) problems. For the special case when the quadratic term has the form $\frac{1}{2} X \bul (UXU)$, we compute the search direction at each iteration from the Schur complement equation. We are able to solve the Schur complement equation efficiently via … Read more
SparesPOP is a MATLAB implementation of a sparse semidefinite programming (SDP) relaxation method proposed for polynomial optimization problems (POPs) in the recent paper by Waki et al. The sparse SDP relaxation is based on a hierarchy of LMI relaxations of increasing dimensions by Lasserre, and exploits a sparsity structure of polynomials in POPs. The efficiency … Read more
We consider semidefinite programming problems on which a permutation group is acting. We describe a general technique to reduce the size of such problems, exploiting the symmetry. The technique is based on a low-order matrix *-representation of the commutant (centralizer ring) of the matrix algebra generated by the permutation matrices. We apply it to extending … Read more
When solving large-scale semidefinite programming problems by second-order methods, the storage and factorization of the Newton matrix are the limiting factors. For a particular algorithm based on the modified barrier method, we propose to use iterative solvers instead of the routinely used direct factorization techniques. The preconditioned conjugate gradient method proves to be a viable … Read more
We present a practical approach to Anstreicher and Lee’s masked spectral bound for maximum-entropy sampling, and we describe favorable results that we have obtained with a Branch-&-Bound algorithm based on our approach. By representing masks in factored form, we are able to easily satisfy a semidefiniteness constraint. Moreover, this representation allows us to restrict the … Read more