# Adam N. LetchfordLancaster University | LU · Department of Management Science

Adam N. Letchford

PhD

## About

142

Publications

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Introduction

For an up-to-date and comprehensive list of my publications, see here:
http://www.lancs.ac.uk/staff/letchfoa/publications.htm

## Publications

Publications (142)

Newsvendor problems are an important and much-studied topic in stochastic inventory control. One strand of the literature on newsvendor problems is concerned with the fact that practitioners often make judgemental adjustments to the theoretically "optimal" order quantities. Although judgemental adjustment is sometimes beneficial, two specific kinds...

Arc Routing Problems (ARPs) are a special kind of Vehicle Routing Problem (VRP), in which the demands are located on edges or arcs, instead of nodes. There is a huge literature on ARPs, and a variety of exact and heuristic algorithms are available. Recently, however, we encountered some real-life ARPs with over ten thousand roads, which is much lar...

Newsvendor problems (NVP) form a classical and important family of stochastic optimisation problems. In this paper, we consider a data-driven method proposed recently by Ban and Rudin. We first examine it from a statistical viewpoint, and establish a connection with quantile regression. We then extend the approach to nonlinear NVP. Finally, we give...

The General Routing Problem (GRP) is a fundamental NP-hard vehicle routing problem, first defined by Orloff in 1974. It contains as special cases the Chinese Postman Problem, the Rural Postman Problem, the Graphical TSP and the Steiner TSP. We examine in detail a known constructive heuristic for the GRP, due to Christofides and others. We show how...

The Quadratic Knapsack Problem (QKP) is a well-known NP-hard combinatorial optimisation problem, with many practical applications. We present a 'cut-and-branch' algorithm for the QKP, in which a cutting-plane phase is followed by a branch-and-bound phase. The cutting-plane phase is more sophisticated than the existing ones in the literature, incorp...

The max-cut problem is a fundamental combinatorial optimisation problem, with many applications. Poljak and Turzik found some facet-defining inequalities for the associated polytope, which we call 2-circulant inequalities. We present a more general family of facet-defining inequalities, an exact separation algorithm that runs in polynomial time, an...

Matchings and T‐joins are fundamental and much‐studied concepts in graph theory and combinatorial optimization. One important application of matchings and T‐joins is in the computation of strong lower bounds for arc routing problems (ARPs). An ARP is a special kind of vehicle routing problem, in which the demands are located along edges or arcs, ra...

When developing an exact algorithm for a combinatorial optimisation problem, it often helps to have a good understanding of certain polyhedra associated with that problem. In the case of quadratic unconstrained Boolean optimisation, the polyhedron in question is called the Boolean quadric polytope. This chapter gives a brief introduction to polyhed...

Perspective functions have long been used to convert fractional programs into convex programs. More recently, they have been used to form tight relaxations of mixed-integer nonlinear programs with so-called indicator variables. Motivated by a practical application (maximising energy efficiency in an OFDMA system), we consider problems that have a f...

In 2013, Buchheim and Wiegele introduced a quadratic optimisation problem, in which the domain of each variable is a closed subset of the reals. This problem includes several other important problems as special cases. We study some convex sets and polyhedra associated with the problem, and derive several families of strong valid inequalities. We al...

The Simple Plant Location Problem (SPLP) is a well-known NP-hard optimisation problem with applications in logistics. Although many families of facet-defining inequalities are known for the associated polyhedron, very little work has been done on separation algorithms. We present the first ever polynomial-time separation algorithm for the SPLP that...

Surrogate and group relaxation have been used to compute bounds for various integer linear programming problems. We prove that (a) when only inequalities are surrogated, the surrogate dual is NP-hard, but solvable in pseudo-polynomial time under certain conditions; (b) when equations are surrogated, the surrogate dual exhibits unusual complexity be...

The stable set polytope is a fundamental object in combinatorial optimisation. Among the many valid inequalities that are known for it, the clique-family inequalities play an important role. Pêcher and Wagler showed that the clique-family inequalities can be strengthened under certain conditions. We show that they can be strengthened even further,...

Suppose that one is given a Vehicle Routing Problem (VRP) on a road network, but does not have access to detailed information about that network. One could obtain a heuristic solution by solving a modified version of the problem, in which true road distances are replaced with planar Euclidean distances. We test this heuristic, on two different type...

We present a new tool for generating cutting planes for NP-hard combinatorial optimisation problems. It is based on the concept of gadgets — small subproblems that are “glued” together to form hard problems — which we borrow from the literature on computational complexity. Using gadgets, we are able to derive huge (exponentially large) new families...

Valid inequalities for the knapsack polytope have proven to be very useful in exact algorithms for mixed-integer linear programming. In this paper, we focus on the knapsack cover inequalities, introduced in 2000 by Carr and co-authors. In general, these inequalities can be rather weak. To strengthen them, we use lifting. Since exact lifting can
be...

Newsvendor problems form a classical and important family of stochastic optimisation problems. The standard solution approach decomposes the problem into two steps: estimation of the demand distribution, then determination of the optimal production quantity (or quantities) for the given distribution. We propose a new, integrated approach, which est...

The stable set problem is a fundamental combinatorial optimisation problem, that is known to be very difficult in both theory and practice. Some of the solution algorithms in the literature are based on 0-1 linear programming formulations. We examine an entire family of such formulations, based on so-called clique and nodal inequalities. As well as...

The matroid parity (MP) problem is a powerful (and NP-hard) extension of the matching problem. Whereas matching polytopes are well understood, little is known about MP polytopes. We prove that, when the matroid is laminar, the MP polytope is affinely congruent to a perfect b-matching polytope. From this we deduce that, even when the matroid is not...

We consider the problem of sequentially choosing observation regions along a line, with an aim of maximising the detection of events of interest. Such a problem may arise when monitoring the movements of endangered or migratory species, detecting crossings of a border, policing activities at sea, and in many other settings. In each case, the key op...

The issue of fairness has received attention from researchers in many fields, including combinatorial optimisation. One way to drive the solution toward fairness is to use a modified objective function that involves so-called $\ell_p$-norms. If done in a naive way, this approach leads to large and symmetric mixed-integer nonlinear programs (MINLPs)...

We consider mixed 0-1 linear programs in which one is given a collection of (not necessarily disjoint) sets of variables and, for each set, a fixed charge is incurred if and only if at least one of the variables in the set takes a positive value. We derive strong valid linear inequalities for these problems, and show that they generalise and domina...

OFDMA is a popular coding scheme for mobile wireless communications. In OFDMA, one must allocate the available resources (bandwidth and power) dynamically, as user requests arrive and depart in a stochastic manner. Several exact and heuristic methods exist to do this, but they all perform poorly in the “over-loaded” case, in which the user demand i...

The most effective software packages for solving mixed 0-1 linear programs use strong valid linear inequalities derived from polyhedral theory. We introduce a new procedure which enables one to take known valid inequalities for the knapsack polytope, and convert them into valid inequalities for the ﬁxed-charge and single-node ﬂow polytopes. The res...

Whilst there are many approaches to detecting changes in mean for a univariate time-series, the problem of detecting multiple changes in slope has comparatively been ignored. Part of the reason for this is that detecting changes in slope is much more challenging. For example, simple binary segmentation procedures do not work for this problem, whils...

Lifted cover inequalities are well-known cutting planes for 0-1 linear programs. We show how one of the earliest lifting procedures, due to Balas, can be significantly improved. The resulting procedure has some unusual properties. For example, (i) it can yield facet-defining inequalities even if the given cover is not minimal, (ii) it can yield fac...

The Capacitated Vehicle Routing Problem (CVRP) is a classic combinatorial optimization problem for which many heuristics, relaxations and exact algorithms have been proposed. Since the CVRP is NP-hard in the strong sense, a natural research topic is relaxations that can be solved in pseudo-polynomial time. We consider several old and new relaxation...

Most OR academics and practitioners are familiar with linear programming (LP) and its applications. Many are however unaware of conic optimisation, which is a powerful generalisation of LP, with a prodigious array of important real-life applications. In this invited paper , we give a gentle introduction to conic optimisation, followed by a survey o...

Maximising the detection of intrusions is a fundamental and often critical aim of perimeter surveillance. Commonly, this requires a decision-maker to optimally allocate multiple searchers to segments of the perimeter. We consider a scenario where the decision-maker may sequentially update the searchers' allocation, learning from the observed data t...

Non-convex quadratic programming with box constraints is a fundamental problem in the global optimization literature, being one of the simplest NP-hard nonlinear programs. We present a new heuristic for this problem, which enables one to obtain solutions of excellent quality in reasonable computing times. The heuristic consists of four phases: bina...

OFDMA is a popular coding scheme for mobile wireless multi-channel multi-user communication systems. In a previous paper, we used mixed-integer nonlinear programming to tackle the problem of maximising energy efficiency, subject to certain quality of service (QoS) constraints. In this paper, we present a heuristic for the same problem. Computationa...

The max-cut problem is a much-studied NP-hard combinatorial optimisation problem. Poljak and Turzik found some facet-defining inequalities for this problem, which we call 2-circulant inequalities. Two polynomial-time separation algorithms have been found for these inequalities, but one is very slow and the other is very complicated. We present a th...

The Simple Plant Location Problem is a well-known (and N P-hard) combinatorial optimisation problem, with applications in logistics. We present a new family of valid inequalities for the associated family of polyhedra, and show that it contains an exponentially large number of new facet-defining members. We also present a new procedure, called faci...

In the k-partition problem (k-PP), one is given an edge-weighted undirected graph, and one must partition the node set into at most k subsets, in order to minimise (or maximise) the total weight of the edges that have their end-nodes in the same cluster. Various hierarchical variants of this problem have been studied in the context of data mining....

In the literature on the quadratic 0-1 knapsack problem, several alternative ways have been given to represent the knapsack constraint in the quadratic space. We extend this work by constructing analogous representations for arbitrary linear inequalities for arbitrary nonconvex mixed-integer quadratic programs with bounded variables.

We consider a challenging resource allocation problem arising in mobile wireless communications. The goal is to allocate the available channels and power in a so-called OFDMA system, in order to maximise the transmission rate, subject to quality of service (QoS) constraints. Standard MINLP software struggled to solve even small instances of this pr...

The k-partition problem is an NP-hard combinatorial optimisation problem with many applications. Chopra and Rao introduced two integer programming formulations of this problem, one having both node and edge variables, and the other having only edge variables. We show that, if we take the polytopes associated with the 'edge-only' formulation, and pr...

The Lovasz theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovasz and the other to Grotschel et al. The former SDP is often thought to be preferable computationally, since it has fewer variab...

The stable set problem is a well-known NP-hard combinatorial optimization problem. As well as being hard to solve (or even approximate) in theory, it is often hard to solve in practice. The main difficulty is that upper bounds based on linear programming (LP) tend to be weak, whereas upper bounds based on semidefinite programming (SDP) take a long...

The sequential ordering problem (SOP) is the generalisation of the asymmetric travelling salesman problem in which there are precedence relations between pairs of nodes. Hern�andez & Salazar introduced a multi-commodity flow (MCF) formulation for a generalisation of the SOP in which the vehicle has a limited capacity. We strengthen this MCF formula...

The Steiner Travelling Salesman Problem (STSP) is a variant of the TSP that is suitable for instances defined on road networks. We consider an extension of the STSP in which the road traversal costs are both stochastic and correlated. This happens, for example, when vehicles are prone to delays due to rush hours, road works or accidents. Following...

The Capacitated Vehicle Routing Problem is a much-studied (and strongly NP-hard) combinatorial optimization problem, for which many integer programming formulations have been proposed. We present two new multi-commodity flow (MCF) formulations, and show that they dominate all of the existing ones, in the sense that their continuous relaxations yiel...

A new exact approach to the stable set problem is presented, which attempts to avoid the pitfalls of existing approaches based on linear and semidefinite programming. The method begins by constructing an ellipsoid that contains the stable set polytope and has the property that the upper bound obtained by optimising over it is equal to the Lov�asz t...

The rapidly growing field of polynomial optimisation (PO) is concerned with optimisation problems in which the objective and constraint functions are all polynomials. There are applications of PO in a surprisingly wide variety of contexts, including, for example, operational research, statistics, applied probability, quantitative finance, theoretic...

The Reformulation-Linearization Technique (RLT), due to Sherali and Adams, can be used to construct hierarchies of linear programming relaxations of mixed 0-1 polynomial programs. As one moves up the hierarchy, the relaxations grow stronger, but the number of variables increases exponentially. We present a procedure that generates cutting planes at...

Several very effective exact algorithms have been developed for vehicle routing problems with time windows. Unfortunately, most of these algorithms cannot be applied to instances that are defined on road networks, because they implicitly assume that the cheapest path between two customers is equal to the quickest path. Garaix and coauthors proposed...

Chvatal-Gomory cutting planes (CG-cuts for short) are a fundamental tool in Integer Programming. Given any single CG-cut, one can derive an entire family of CG-cuts, by "iterating" its multiplier vector modulo one. This leads naturally to two questions: first, which iterates correspond to the strongest cuts, and, second, can we find such strong cut...

A separation algorithm is a procedure for generating cutting planes. Up to now, only a few polynomial-time separation algorithms were known for the Boolean quadric and cut polytopes. These polytopes arise in connection with zero-one quadratic programming and the maxcut problem, respectively. We present a new algorithm, which separates over a class...

Pisinger et al. introduced the concept of ‘aggressive reduction’ for large-scale combinatorial optimization problems. The idea is to spend much time and effort in reducing the size of the instance, in the hope that the reduced instance will then be small enough to be solved by an exact algorithm.
We present an aggressive reduction scheme for the ‘...

Quadratic Convex Reformulation (QCR) is a technique that was originally proposed for quadratic 0–1 programs, and then extended to various other problems. It is used to convert non-convex instances into convex ones, in such a way that the bound obtained by solving the continuous relaxation of the reformulated instance is as strong as possible. In th...

This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family. Some fundamental properties of the...

It is well known that the standard (linear) knapsack problem can be solved exactly by dynamic programming in 𝒪(nc) time, where n is the number of items and c is the capacity of the knapsack. The quadratic knapsack problem, on the other hand, is NP-hard in the strong sense, which makes it unlikely that it can be solved in pseudo-polynomial time. We...

The single row facility layout problem (SRFLP) is the NP-hard problem of arranging facilities on a line, while minimizing a weighted sum of the distances between facility pairs. In this paper, a detailed polyhedral study of the SRFLP is performed, and several huge classes of valid and facet-inducing inequalities are derived. Some separation heurist...

The famous Lovász theta number θ(G)θ(G) is expressed as the optimal solution of a semidefinite program. As such, it can be computed in polynomial time to an arbitrary precision. Nevertheless, computing it in practice yields some difficulties as the size of the graph gets larger and larger, despite recent significant advances of semidefinite program...

The Steiner Traveling Salesman Problem (STSP) is a variant of the Traveling
Salesman Problem (TSP) that is particularly suitable when dealing with sparse
networks, such as road networks. The standard integer programming formulation
of the STSP has an exponential number of constraints, just like the standard
formulation of the TSP. On the other hand...

We examine the issue of how to compute quickly good lower and upper bounds for large instances of the Simple Plant Location Problem. We present three lower-bounding procedures, based on dual ascent, the fastest of which runs inO(mn logm) time, where m and n are the number of locations and clients, respectively. We then present an effective upper-bo...

We prove several complexity results about the gap inequalities for the max-cut problem, including (i) the gap-1 inequalities do not imply the other gap inequalities, unless NP = Co-NP; (ii) there must exist non-redundant gap inequalities with exponentially large coefficients, unless NP = Co-NP; (iii) the associated separation problem can be solved...

Laurent and Poljak introduced a class of valid inequalities for the max-cut problem, called gap inequalities, which include many other known inequalities as special cases. The gap inequalities have received little attention and are poorly understood. This paper presents the first ever computational results. In particular, we describe heuristic sepa...

A wide range of problems arising in practical applications can be formulated as Mixed-Integer Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When nonconvexities are present, however, things become much more difficult, since then...

It is well known that semidenite programming (SDP) can be used to derive useful relaxations for a variety of optimisation problems. Moreover, in the particular case of zero-one quadratic programs, SDP has been used to reformulate problems, rather than merely relax them. The purpose of reformulation is to strengthen the continuous relaxation of the...

We consider the positive semidefinite (psd) matrices with binary entries, along with the corresponding integer polytopes. We begin by establishing some basic properties of these matrices and polytopes. Then, we show that several families of integer polytopes in the literature (the cut, boolean quadric, multicut and clique partitioning polytopes) ar...

In recent years, there has been an increased literature on so-called Generalized Network Design Problems, such as the Generalized Minimum Spanning Tree Problem and the Generalized Traveling Salesman Problem. In such problems, the node set of a graph is partitioned into clusters, and the feasible solutions must contain one node from each cluster. Up...

A new exact approach to the stable set problem is presented, which attempts to avoids the pitfalls of existing approaches based on linear and semidefinite programming. The method begins by constructing an ellipsoid that contains the stable set polytope and has the property that the upper bound obtained by optimising over it is equal to the Lovász t...

Laurent and Poljak introduced a very general class of valid linear inequalities, called gap inequalities, for the max-cut problem. We show that an analogous class of inequalities can be defined for general non-convex mixed-integer quadratic programs. These inequalities dominate some inequalities arising from a natural semidefinite relaxation.

Many problems arising in OR/MS can be formulated as Mixed-Integer Linear Programs (MILPs); see entry #1.4.1.1. If one wishes to solve a class of MILPs to proven optimality, or near optimality, it is often useful to have a class of cutting planes available. Cutting planes are linear inequalities that are satisfied by all feasible solutions to the MI...

We present a new procedure for generating cutting planes for the max-cut problem. The procedure consists of three steps. First, we generate a violated (or near-violated) linear inequality that is valid for the semidefinite programming (SDP) relaxation of the max-cut problem. This can be done by computing the minimum eigenvalue of a certain matrix....

We consider the following natural heuristic for the Symmetric Traveling Salesman Problem: solve the subtour relaxation, yielding a solution x* , and then find the best tour x that is compatible with x*, where compatible means that every subtour elimination constraint that is satised at equality at x* is also satised at equality at x. We prove that...

Minimum Linear Arrangement is a classical basic combinatorial optimization problem from the 1960s, which turns out to be extremely challenging in practice. In particular, for most of its benchmark instances, even the order of magnitude of the optimal solution value is unknown, as testifled by the surveys on the problem that contain tables in which...

The Traveling Salesman Problem or TSP is a fundamental and well-known problem in combinatorial optimization. At present, the most successful algorithms for solving large-scale instances of the TSP to proven (near-)optimality are based on integer programming. This entry introduces the main theoretical and algorithmic tools involved. Topics covered i...

Please note that some material from this report eventually appeared in the following journal article:
L. Galli & A.N. Letchford (2014) A compact variant of the QCR method for quadratically constrained quadratic 0-1 programs. Optim. Lett., 8(4), 1213-1224.
Other material from the report may eventually appear in a future journal article.

We examine the metrics that arise when a finite set of points is embedded in
the real line, in such a way that the distance between each pair of points is
at least 1. These metrics are closely related to some other known metrics in
the literature, and also to a class of combinatorial optimization problems
known as graph layout problems. We prove se...

Minimum Linear Arrangement is a classical basic combinatorial optimization problem from the 1960s, which turns out to be extremely challenging in practice. In particular, for most of its benchmark instances, even the order of magnitude of the optimal solution value is unknown, as testified by the surveys on the problem that contain tables in which...

Valid inequalities for 0-1 knapsack polytopes often prove useful
when tackling hard 0-1 Linear Programming problems. To generate
such inequalities, one needs separation algorithms for them, i.e., rou-
tines for detecting when they are violated. We present new exact
and heuristic separation algorithms for several classes of inequalities,
namely lift...

Combinatorial optimization games form an important subclass of cooperative games. In recent years, increased attention has been given to the issue of nding good cost shares for such games. In this paper, we dene a very general class of games, called integer minimization games, which includes the combinatorial optimization games in the literature as...

This paper introduces two fundamental families of 'quasi-polyhedra' - polyhedra with a countably infinite number of facets - that arise in the context of integer quadratic programming. It is shown that any integer quadratic program can be reduced to the minimisation of a linear function over a quasi-polyhedron in the first family. Some fundamental...

We compute a complete linear description of the bipartite subgraph polytope, for up to seven nodes, and a conjectured complete description for eight nodes. We then show how these descriptions were used to compute the integrality ratio of various relaxations of the max-cut problem, again for up to eight nodes.

Caprara and Fischetti introduced a class of cutting planes, called {0, 1/2}-cuts, which are valid for arbitrary integer linear programs. They also showed that the associated separation problem is strongly NP-hard. We show that separation remains strongly NP-hard, even when all integer variables are binary, even when the integer linear program is a...

Given a graph G=(V,E) on n vertices, the Minimum Linear Arrangement Problem (MinLA) calls for a one-to-one function ψ:V→{1,…,n} which minimizes ∑{i,j}∈E|ψ(i)−ψ(j)|. MinLA is strongly NP-hard and very difficult to solve to optimality in practice. One of the reasons for this difficulty is the lack of good lower bounds. In this paper, we take a polyhe...

Although the lift-and-project operators of Lovász and Schrijver have been the subject of intense study, their M(K, K) operator has received little attention. We consider an application of this operator to the stable set problem. We begin with an initial linear programming (LP) relaxation consisting of clique and non-negativity inequalities, and the...

The capacitated arc routing problem (CARP) is a well-known and fundamental vehicle routing problem. A promising exact solution approach to the CARP is to model it as a set covering problem and solve it via branch-cut-and-price. The bottleneck in this approach is the pricing (column generation) routine. In this paper, we note that most CARP instance...

For a fixed finite set $\{1,...,n\}$, we consider the set of metrics for
which the metric space can be isometrically embedded in the real line. The
convex hull of those metrics, $Q_n$, and its closure $\close{Q_n}$ are the main
objects of this paper. We first study structural properties of $Q_n$ showing
how the set of metrics is contained in this c...

Non-Convex Quadratic Programming with Box Constraints is a fundamental NP-hard global optimisation problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterise their extreme points and verti...

Non-Convex Quadratic Programming with Box Constraints is a fun-damental N P-hard global optimisation problem. Recently, some au-thors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterise their extreme points and ve...

The famous Padberg-Rao separation algorithm for b-matching polyhedra can be implemented to run in O(vertical bar V vertical bar(2) vertical bar E vertical bar log(vertical bar V vertical bar(2)/vertical bar E vertical bar)) time in the uncapacitated case, and in O(vertical bar V vertical bar vertical bar E vertical bar(2) log (vertical bar V vertic...

We consider the positive semidefinite (psd) matrices with binary entries. We give a characterisation of such matrices, along with a graphical representation. We then move on to consider the associated integer polytopes. Several important and well-known integer polytopes — the cut, boolean quadric, multicut and clique partitioning polytopes — are sh...