In this paper, we consider the solution of the standard linear programming (LP). A remarkable result in LP claims that all optimal solutions form an optimal face of the underlying polyhedron. In practice, many real-world problems have infinitely many optimal solutions and pursuing the optimal face, not just an optimal vertex, is quite desirable. The … Read more
We present a parallelization of the revised simplex method for large extensive forms of two-stage stochastic linear programming (LP) problems. These problems have been considered too large to solve with the simplex method; instead, decomposition approaches based on Benders decomposition or, more recently, interior-point methods are generally used. However, these approaches do not provide optimal … Read more
In this paper we discuss two particular layout problems, namely the Single-Row Equidistant Facility Layout Problem (SREFLP) and the Single-Row Facility Layout Problem (SRFLP). Our aim is to consolidate the two respective branches in the layout literature. We show that the SREFLP is not only a special case of the Quadratic Assignment Problem but also … Read more
We discuss the method recently proposed by S. Chubanov for the linear feasibility problem. We present new, concise proofs and interpretations of some of his results. We then show how our proofs can be used to find strongly polynomial time algorithms for special classes of linear feasibility problems. Under certain conditions, these results provide new … Read more
When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented … Read more
We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most $k$. We show that this problem is $\NP$-hard for any fixed integer $k\ge 2$. Equivalently, for $k\ge 2$, it is $\NP$-hard to test membership in the … Read more
Algorithms are presented for evaluating gradients and Hessians of logarithmic barrier functions for two types of convex cones: the cone of positive semidefinite matrices with a given sparsity pattern, and its dual cone, the cone of sparse matrices with the same pattern that have a positive semidefinite completion. Efficient large-scale algorithms for evaluating these barriers … Read more
Implementations of the Simplex method differ only in very specific aspects such as the pivot rule. Similarly, most relaxation methods for mixed-integer programming differ only in the type of cuts and the exploration of the search tree. Implementing instances of those frameworks would therefore be more efficient if linear and mixed-integer programming solvers let users … Read more
Numerical experiments have indicated that the reweighted $\ell_1$-minimization performs exceptionally well in locating sparse solutions of underdetermined linear systems of equations. Thus it is important to carry out a further investigation of this class of methods. In this paper, we point out that reweighted $\ell_1$-methods are intrinsically associated with the minimization of the so-called merit … Read more
An element $A$ of the $n \times n$ copositive cone $\copos{n}$ is called irreducible with respect to the nonnegative cone~$\NNM{n}$ if it cannot be written as a nontrivial sum $A = C+N$ of a copositive matrix $C$ and an elementwise nonnegative matrix $N$. This property was studied by Baumert~\cite{Baumert65} who gave a characterisation of irreducible … Read more