Ellipsoidal Approach to Box-Constrained Quadratic Problems

We present a new heuristic for the global solution of box constrained quadratic problems, based on the classical results which hold for the minimization of quadratic problems with ellipsoidal constraints. The approach is tested on several problems randomly generated and on graph instances from the DIMACS challenge, medium size instances of the Maximum Clique Problem. The numerical results seem to suggest some effectiveness of the proposed approach.     

Copositivity detection by difference-of-convex decomposition and ω-subdivision

We present three new copositivity tests based upon difference-of-convex (d.c.) decompositions, and combine them to a branch-and-bound algorithm of ω-subdivision type. The tests employ LP or convex QP techniques, but also can be used heuristically using appropriate test points. We also discuss the selection of efficient d.c. decompositions and propose some preprocessing ideas based on the spectral d.c. decomposition. We report on first numerical experience with this procedure which are very promising.     

New results for molecular formation under pairwise potential minimization

We establish new lower bounds on the distance between two points of a minimum energy configuration of N points in ℝ³ interacting according to a pairwise potential function. For the Lennard-Jones case, this bound is 0.67985 (and 0.7633 in the planar case). A similar argument yields an estimate for the minimal distance in Morse clusters, which improves previously known lower bounds. Moreover, we prove that the optimal configuration cannot be two-dimensional, and establish an upper bound for the distance to the nearest neighbour of every particle, which depends on the position of this particle. On the boundary of the optimal configuration polytope, this is unity while in the interior, this bound depends on the potential function. In the Lennard-Jones case, we get the value . Also, denoting by V _N the global minimum in an N point minimum energy configuration, we prove in Lennard-Jones clusters for all N≥2, while asymptotically holds (as opposed to in the planar case, confirming non-planarity for large N).     

A first-order interior-point method for linearly constrained smooth optimization

We propose a first-order interior-point method for linearly constrained smooth optimization that unifies and extends first-order affine-scaling method and replicator dynamics method for standard quadratic programming. Global convergence and, in the case of quadratic program, (sub)linear convergence rate and iterate convergence results are derived. Numerical experience on simplex constrained problems with 1000 variables is reported.     

On Standard Quadratic Optimization Problems

A standard quadratic optimization problem (QP) consists of finding (global) maximizers of a quadratic form over the standard simplex. Standard QPs arise quite naturally in copositivity-based procedures which enable an escape from local solutions. Furthermore, several important applications yield optimization problems which can be cast into a standard QP in a straightforward way. As an example, a new continuous reformulation of the maximum weight clique problem in undirected graphs is presented which considerably improves previous attacks both as numerical stability and interpretation of the results are concerned. Apparently also for the first time, an equivalence between standard QPs and QPs on the positive orthant is established. Also, a recently presented global optimization procedure (GENF - genetical engineering via negative fitness) is shortly reviewed.     

The combinatorics of pivoting for the maximum weight clique

In this paper we prove the equivalence between a pivoting-based heuristic (PBH) for the maximum weight clique problem and a combinatorial greedy heuristic. It is also proved that PBH always returns a local solution although this is not always guaranteed for Lemke's method, on which PBH is based.     

Perron–Frobenius property of copositive matrices, and a block copositivity criterion

Haynsworth and Hoffman proved in 1969 that the spectral radius of a symmetric copositive matrix is an eigenvalue of this matrix. This note investigates conditions which guarantee that an eigenvector corresponding to this dominant eigenvalue has no negative coordinates, i.e., whether the Perron–Frobenius property holds. Also a block copositivity criterion using the Schur complement is specified which may be helpful to reduce dimension in copositivity checks and which generalizes results proposed by Andersson et al. in 1995, and Johnson and Reams in 2005. Apparently, the latter five researchers were unaware of the more general results by the author precedingly published in 1987 and 1996, respectively.     

Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming

The problem of minimizing a (non-convex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMI's). In particular, we show that our approach leads to a polynomial-time approximation scheme for the standard quadratic optimzation problem. This is an improvement on the previous complexity result by Nesterov who showed that a 2/3-approximation is always possible. Numerical examples from various applications are provided to illustrate our approach.