Geometry and combinatorics of the cutting angle method

Lower approximation of Lipschitz functions plays an important role in deterministic global optimization. This article examines in detail the lower piecewise linear approximation which arises in the cutting angle method. All its local minima can be explicitly enumerated, and a special data structure was designed to process them very efficiently, improving previous results by several orders of magnitude. Further, some geometrical properties of the lower approximation have been studied, and regions on which this function is linear have been identified explicitly. Connection to a special distance function and Voronoi diagrams was established. An application of these results is a black-box multivariate random number generator, based on acceptance-rejection approach.

Polyhedral combinatorics of the cardinality constrained quadratic knapsack problem and the quadratic selective travelling salesman problem

This paper considers the Cardinality Constrained Quadratic Knapsack Problem (QKP) and the Quadratic Selective Travelling Salesman Problem (QSTSP). The QKP is a generalization of the Knapsack Problem and the QSTSP is a generalization of the Travelling Salesman Problem. Thus, both problems are NP hard. The QSTSP and the QKP can be solved using branch-and-cut methods. Good bounds can be obtained if strong constraints are used. Hence it is important to identify strong or even facet-defining constraints. This paper studies the polyhedral combinatorics of the QSTSP and the QKP, i.e. amongst others we identify facet-defining constraints for the QSTSP and the QKP, and provide mathematical proofs that they do indeed define facets.

Facets of the polytope of the asymmetric travelling salesman problem with replenishment arcs

The Asymmetric Travelling Salesman Problem with Replenishment Arcs (RATSP) is a new class of problems arising from work related to aircraft routing. Given a digraph with cost on the arcs, a solution of the RATSP, like that of the Asymmetric Travelling Salesman Problem, induces a directed tour in the graph which minimises total cost. However the tour must satisfy additional constraints: the arc set is partitioned into replenishment arcs and ordinary arcs, each node has a non-negative weight associated with it, and the tour cannot accumulate more than some weight limit before a replenishment arc must be used. To enforce this requirement, constraints are needed. We refer to these as replenishment constraints.

In this paper, we review previous polyhedral results for the RATSP and related problems, then prove that two classes of constraints developed in V. Mak and N. Boland [Polyhedral results and exact algorithms for the asymmetric travelling salesman problem with replenishment arcs, Technical Report TR M05/03, School of Information Technology, Deakin University, 2005] are, under appropriate conditions, facet-defining for the RATS polytope.

New cutting-planes for the time-and/or precedence-constrained ATSP and directed VRP

In this paper, we introduce five classes of new valid cutting planes for the precedence-constrained (PC) and/or time-window-constrained (TW) Asymmetric Travelling Salesman Problems (ATSPs) and directed Vehicle Routing Problems (VRPs). We show that all five classes of new inequalities are facet-defining for the directed VRP-TW, under reasonable conditions and the assumption that vehicles are identical. Similar proofs can be developed for the VRP-PC. As ATSP-TW and PC-ATSP can be formulated as directed identical-vehicle VRP-TW and PC-VRP, respectively, this provides a link to study the polyhedral combinatorics for the ATSP-TW and PC-ATSP. The first four classes of these new cutting planes are cycle-breaking inequalities that are lifted from the well-known D^-_k and D⁺_k inequalities (see Grötschel and Padberg in Polyhedral theory. The traveling salesman problem: a guided tour of combinatorial optimization, Wiley, New York, 1985). The last class of new cutting planes, the TW ₂ inequalities, are infeasible-path elimination inequalities. Separation of these constraints will also be discussed. We also present prelimanry numerical results to demonstrate the strengh of these new cutting planes.

Broadcast anti-jamming systems

In a traditional anti-jamming system a transmitter who wants to send a signal to a single receiver spreads the signal power over a wide frequency spectrum with the aim of stopping a jammer from blocking the transmission. In this paper, we consider the case that there are multiple receivers and the transmitter wants to broadcast a message to all receivers such that colluding groups of receivers cannot jam the reception of any other receiver. We propose efficient coding methods that achieve this goal and link this problem to well-known problems in combinatorics. We also link a generalisation of this problem to the Key Distribution Pattern problem studied in combinatorial cryptography

Learning from ordered sets and applications in collaborative ranking

Ranking over sets arise when users choose between groups of items. For example, a group may be of those movies deemed 5 stars to them, or a customized tour package. It turns out, to model this data type properly, we need to investigate the general combinatorics problem of partitioning a set and ordering the subsets. Here we construct a probabilistic log-linear model over a set of ordered subsets. Inference in this combinatorial space is highly challenging: The space size approaches (N!/2)6.93145N+1 as N approaches infinity. We propose a split-and-merge Metropolis-Hastings procedure that can explore the state-space efficiently. For discovering hidden aspects in the data, we enrich the model with latent binary variables so that the posteriors can be efficiently evaluated. Finally, we evaluate the proposed model on large-scale collaborative filtering tasks and demonstrate that it is competitive against state-of-the-art methods.