Exploiting sparsity is essential to improve the efficiency of solving large optimization problems. We present a method for recognizing the underlying sparsity structure of a nonlinear partially separable problem, and show how the sparsity of the Hessian matrices of the problem’s functions can be improved by performing a nonsingular linear transformation in the space corresponding … Read more
Semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the results obtained in the application of semidefinite programming to satisfiability and maximum-satisfiability problems. The … Read more
In this paper we consider the following two types of complex quadratic maximization problems: (i) maximize $z^{\HH} Q z$, subject to $z_k^m=1$, $k=1,…,n$, where $Q$ is a Hermitian matrix with $\tr Q=0$ and $z\in \C^n$ is the decision vector; (ii) maximize $\re y^{\HH}Az$, subject to $y_k^m=1$, $k=1,…,p$, and $z_l^m=1$, $l=1,…,q$, where $A\in \C^{p\times q}$ and … 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
Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) … Read more
A polynomial SDP (semidefinite program) minimizes a polynomial objective function over a feasible region described by a positive semidefinite constraint of a symmetric matrix whose components are multivariate polynomials. Sums of squares relaxations developed for polynomial optimization problems are extended to propose sums of squares relaxations for polynomial SDPs with an additional constraint for the … Read more
Sequences of generalized Lagrangian duals and their SOS (sums of squares of polynomials) relaxations for a POP (polynomial optimization problem) are introduced. Sparsity of polynomials in the POP is used to reduce the sizes of the Lagrangian duals and their SOS relaxations. It is proved that the optimal values of the Lagrangian duals in the … Read more
Semidefinite Programming (SDP) has recently turned out to be a very powerful tool for approximating some NP-hard problems. The nature of the Quadratic Assignment Problem suggests SDP as a way to derive tractable relaxation. We recall some SDP relaxations of QAP and solve them approximately using the Bundle Method. The computational results demonstrate the efficiency … Read more
In this paper, we consider a class of quadratic maximization problems. One important instance in that class is the famous quadratic maximization formulation of the max-cut problem studied by Goemans and Williamson. Since the problem is NP-hard in general, following Goemans and Williamson, we apply the approximation method based on the semidefinite programming (SDP) relaxation. … Read more
Hierarchies of semidefinite relaxations for $0/1$ polytopes have been constructed by Lasserre (2001a) and by Lov\’asz and Schrijver (1991), permitting to find the cut polytope of a graph on $n$ nodes in $n$ steps. We show that $\left\lceil {n\over 2} \right\rceil$ iterations are needed for finding the cut polytope of the complete graph $K_n$. Citation … Read more