A robust and efficient method for solving point distance problems by homotopy

Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods are available for the higher degree case. Since high degree objectives are becoming increasingly important in certain application areas, such as computer vision, various techniques have been recently developed to reduce the general case to the quadratic one, at the cost of increasing the number of variables by introducing additional auxiliary variables. In this paper we initiate a systematic study of these quadratization approaches. We provide tight lower and upper bounds on the number of auxiliary variables needed in the worst-case for general objective functions, for bounded-degree functions, and for a restricted class of quadratizations. Our upper bounds are constructive, thus yielding new quadratization procedures. Finally, we completely characterize all “minimal” quadratizations of negative monomials.     

Using an iterative linear solver in an interior-point method for generating support vector machines

This paper concerns the generation of support vector machine classifiers for solving the pattern recognition problem in machine learning. A method is proposed based on interior-point methods for convex quadratic programming. This interior-point method uses a linear preconditioned conjugate gradient method with a novel preconditioner to compute each iteration from the previous. An implementation is developed by adapting the object-oriented package OOQP to the problem structure. Numerical results are provided, and computational experience is discussed.     

A Homological Approach to Two Problems on Finite Sets

We propose a homological approach to two conjectures descended from the Erdös-Ko-Rado Theorem, one due to Chvátal and the other to Frankl and Füredi. We apply the method to reprove, and in one case improve, results of these authors related to their conjectures.     

Using a similarity measure for credible classification

This paper concerns classification by Boolean functions. We investigate the classification accuracy obtained by standard classification techniques on unseen points (elements of the domain, {0,1}ⁿ, for some n) that are similar, in particular senses, to the points that have been observed as training observations. Explicitly, we use a new measure of how similar a point x∈{0,1}ⁿ is to a set of such points to restrict the domain of points on which we offer a classification. For points sufficiently dissimilar, no classification is given. We report on experimental results which indicate that the classification accuracies obtained on the resulting restricted domains are better than those obtained without restriction. These experiments involve a number of standard data-sets and classification techniques. We also compare the classification accuracies with those obtained by restricting the domain on which classification is given by using the Hamming distance.     

Response times in gated M/G/1 queues: The processor-sharing case

A class of single server queues with Poisson arrivals and a gated server is considered. Whenever the server becomes idle the gate separating it from the waiting line opens, admitting all the waiting customers into service, and then closes again. The batch admitted into service may be served according to some arbitrary scheme. The equilibrium waiting time distribution is provided for the subclass of conservative schemes with arbitrary service times and the processor-sharing case is treated in some detail to produce the equilibrium time-in-service and response time distributions, conditional on the length of required service. The LIFO and random order of service schemes and the case of compound Poisson arrivals are treated briefly as examples of the effectiveness of the proposed method of analysis. All distributions are provided in terms of their Laplace transforms except for the case of exponential service times where the L.T. of the waiting time distribution is inverted. The first two moments of the equilibrium waiting and response times are provided for most treated cases and in the exponential service times case the batch size distribution is also presented.     

Dynamics Aggregation in Stochastic Control Problems

For stochastic control problems with mixed state-control constraints, we develop a dynamics aggregation method which replaces the stochastic differential or difference equation with a simpler constraint. Solutions of such simplified problems are used to construct a sequence of approximations to the original problem. Convergence properties of the method for both discrete-time and continuous-time models are analyzed in detail.     

Robust branch-cut-and-price for the Capacitated Minimum Spanning Tree problem over a large extended formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arborescence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms: powerful new cuts expressed over a very large set of variables are added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very significant improvements over previous algorithms. Several open instances could be solved to optimality.     

On the Laplacian Spectrum of (α, ω) -Graphs

We study the Laplacian spectrum of (α, ω)-graphs which play an important role in the theory of perfect graphs. The properties of the spectrum we found allow the establishment of some structural properties of (α, ω)-graphs. We describe, in particular, a class of graphs that are not subgraphs of (α, ω)-graphs.