The pooling problem is an important industrial problem in the class of network flow problems for allocating gas flow in pipeline transportation networks. For P-formulation of the pooling problem with time discretization, we propose second order cone programming (SOCP) and linear programming (LP) relaxations and prove that they obtain the same optimal value as the … Read more
We study a class of quadratically constrained quadratic programs (QCQPs), called {\em diagonal QCQPs\/}, which contain no off-diagonal terms $x_j x_k$ for $j \ne k$, and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature … Read more
This paper considers a generalization of the “max-cut-polytope” $\conv\{\ xx^T\mid x\in\real^n, \ \ |x_k| = 1 \ \hbox{for} \ 1\le k\le n\}$ in the space of real symmetric $n\times n$-matrices with all-ones-diagonal to a complex “unit modulus lifting” $\conv\{xx\HH\mid x\in\complex^n, \ \ |x_k| = 1 \ \hbox{for} \ 1\le k\le n\}$ in the space of … Read more
In this paper, we consider a fundamental problem in modern digital communications known as multi-input multi-output (MIMO) detection, which can be formulated as a complex quadratic programming problem subject to unit-modulus and discrete argument constraints. Various semidefinite relaxation (SDR) based algorithms have been proposed to solve the problem in the literature. In this paper, we … Read more
Various conic relaxations of quadratic optimization problems in nonnega- tive variables for combinatorial optimization problems, such as the binary integer quadratic problem, quadratic assignment problem (QAP), and maximum stable set problem have been proposed over the years. The binary and complementarity conditions of the combi- natorial optimization problems can be expressed in several ways, each … Read more
The semidefinite programming (SDP) relaxation has proven to be extremely strong for many hard discrete optimization problems. This is in particular true for the quadratic assignment problem (QAP), arguably one of the hardest NP-hard discrete optimization problems. There are several difficulties that arise in efficiently solving the SDP relaxation, e.g., increased dimension; inefficiency of the … Read more
This paper is concerned with completely positive and semidefinite relaxations of quadratic programs with linear constraints and binary variables as presented by Burer. It observes that all constraints of the relaxation associated with linear constraints of the original problem can be accumulated in a single linear constraint without changing the feasible set of either the … Read more
The Quadratic Convex Reformulation (QCR) method is used to solve quadratic unconstrained binary optimization problems. In this method, the semidefinite relaxation is used to reformulate it to a convex binary quadratic program which is solved using mixed integer quadratic programming solvers. We extend this method to random quadratic unconstrained binary optimization problems. We develop a … Read more
The principal idea of this paper is to exploit Semidefinite Programming (SDP) relaxation within the framework provided by Mixed Integer Nonlinear Programming (MINLP) solvers when tackling Binary Quadratic Problems (BQP). SDP relaxation is well-known to provide strong bounds for BQP in practice. However, the method is not typically implemented in many state-of-the-art MINLP solvers based … Read more
We propose a method to reduce the sizes of SDP relaxation problems for a given polynomial optimization problem (POP). This method is an extension of the elimination method for a sparse SOS polynomial in [Kojima et al., Mathematical Programming] and exploits sparsity of polynomials involved in a given POP. In addition, we show that this … Read more