Gautam Appa

On L1 and Chebyshev estimation

The problem considered here is that of fitting a linear function to a set of points. The criterion normally used for this is least squares. We consider two alternatives, viz., least sum of absolute deviations (called the L1 criterion) and the least maximum absolute deviation (called the Chebyshev criterion). Each of these criteria give rise to a linear program. We develop some theoretical properties of the solutions and in the light of these, examine the suitability of these criteria for linear estimation. Some of the estimates obtained by using them are shown to be counter-intuitive.

Handbook on modelling for discrete optimization

Methods.- The Formulation and Solution of Discrete Optimisation Models.- Continuous Approaches for Solving Discrete Optimization Problems.- Logic-Based Modeling.- Modelling for Feasibility - the Case of Mutually Orthogonal Latin Squares Problem.- Network Modelling.- Modeling and Optimization of Vehicle Routing and Arc Routing Problems.- Applications.- Radio Resource Management.- Strategic and Tactical Planning Models for Supply Chain: An Application of Stochastic Mixed Integer Programming.- Logic Inference and a Decomposition Algorithm for the Resource-Constrained Scheduling of Testing Tasks in the Development of New Pharmaceutical and Agrochemical Products.- A Mixed-Integer Nonlinear Programming Approach to the Optimal Planning of Offshore Oilfield Infrastructures.- Radiation Treatment Planning: Mixed Integer Programming Formulations and Approaches.- Multiple Hypothesis Correlation in Track-to-Track Fusion Management.- Computational Molecular Biology.

On the system of two all_different predicates

Numerous real-life problems require certain variables to be assigned different values. This requirement is encapsulated in constraints of difference. If x1, x2 denote two problem variables, the (nonlinear) constraint of difference is x1 6= x2. Given that variables x1,..., xn must all be pairwise different, a constraint of the type all_different(x1, ..., xn) can be used to formulate in a compact manner the n(n−1) 2 binary constraints of difference. Such an n-ary constraint is also called a predicate because it imposes a logical condition on its variable set. Constraint Programming (CP) makes use of such elaborate predicates in order to provide a natural modelling framework ([2]). Such models are solved by CP techniques designed to produce feasible solutions. Alternatively, Integer Programming (IP) methods can be employed in cases where a logic predicate can be represented by linear inequalities involving integer variables ([1]). Apparently, such representations are important not only because they enrich the arsenal of resolution techniques but also because they motivate the integration of CP and IP in a unified modelling and algorithmic framework (see [3]). An efficient representation of a predicate must be tight, i.e. it must include facet-defining inequalities of the convex hull of integer solutions satisfying the predicate. Such representations have been proposed for the all_different predicate ([8]), for cardinality rules ([6]) and for the sum constraint ([7]). A next step would be to derive such representations for sets of more that one predicates. The current paper works towards this direction by studying a system of two all_different constraints which may share a number of variables. In particular, we examine the polytope defined by the convex hull of integer vectors satisfying the system of the two all_different predicates. The dimension of this polytope is established and subsequently two classes of facet-defining inequalities are exhibited. These classes are of exponential size, a fact that ∗Corresponding address: D. MAGOS, 30 Theodorou Geometrou Str., Athens 11743, Greece. Email:dmagos@teiath.gr

Rational and integral k-regular matrices

In this paper we examine two possible generalisations of total unimodularity, viz., total k-modularity and k-regularity. Total k-modularity extends the permitted values for the subdeterminants of an integral matrix to the powers of k, while k-regularity sets requirements on the inverses of non-singular submatrices of a rational matrix. It is shown that the advantageous properties of totally unimodular matrices with respect to integral polyhedra can be carried over to rational k-regular matrices, namely we prove that a matrix A is k-regular if and only if the polyhedron P(A,b)={x:x?0,Ax≤b} is integral for all integral vectors b the components of which have a common divisor k. Furthermore, we show that the k-regularity of an integral matrix A is equivalent to the fact that for any integral vector b all the rank-1 Chvatal-Gomory cuts for P(A,b) are dominated by mod-k cuts. We present some results on totally k-modular and k-regular matrices, as well as give non-trivial examples of 1- and 2-regular matrices. In particular, we define binet matrices, a generalisation of network matrices for bidirected graphs.

A new framework for the solution of DEA models

We provide an alternative framework for solving data envelopment analysis (DEA) models which, in comparison with the standard linear programming (LP) based approach that solves one LP for each decision making unit (DMU), delivers much more information. By projecting out all the variables which are common to all LP runs, we obtain a formula into which we can substitute the inputs and outputs of each DMU in turn in order to obtain its efficiency number and all possible primal and dual optimal solutions. The method of projection, which we use, is Fourier–Motzkin (F–M) elimination. This provides us with the finite number of extreme rays of the elimination cone. These rays give the dual multipliers which can be interpreted as weights which will apply to the inputs and outputs for particular DMUs. As the approach provides all the extreme rays of the cone, multiple sets of weights, when they exist, are explicitly provided. Several applications are presented. It is shown that the output from the F–M method improves on existing methods of (i) establishing the returns to scale status of each DMU, (ii) calculating cross-efficiencies and (iii) dealing with weight flexibility. The method also demonstrates that the same weightings will apply to all DMUs having the same comparators. In addition it is possible to construct the skeleton of the efficient frontier of efficient DMUs. Finally, our experiments clearly indicate that the extra computational burden is not excessive for most practical problems.

On the uniqueness of solutions to linear programs

We provide a constructive method of checking whether a linear programming problem (LPP) has a unique feasible or a unique optimal solution. Our method requires the solution of only one extra LPP such that the original problem has alternative solutions if and only if the optimal value of the new LPP is positive. If the original solution is not unique, an alternative solution is displayed. Possible applications are discussed.

A polyhedral approach to the alldifferent system

This paper examines the facial structure of the convex hull of integer vectors satisfying a system of alldifferent predicates, also called an alldifferent system. The underlying analysis is based on a property, called inclusion, pertinent to such a system. For the alldifferent systems for which this property holds, we present two families of facet-defining inequalities, establish that they completely describe the convex hull and show that they can be separated in polynomial time. Consequently, the inclusion property characterises a group of alldifferent systems for which the linear optimization problem (i.e. the problem of optimizing a linear function over that system) can be solved in polynomial time. Furthermore, we establish that, for systems with three predicates, the inclusion property is also a necessary condition for the convex hull to be described by those two families of inequalities. For the alldifferent systems that do not possess that property, we establish another family of facet-defining inequalities and an accompanied polynomial-time separation algorithm. All the separation algorithms are incorporated within a cutting-plane scheme and computational experience on a set of randomly generated instances is reported. In concluding, we show that the pertinence of the inclusion property can be decided in polynomial time.

Integrating Constraint and Integer Programming for the Orthogonal Latin Squares Problem

We consider the problem of Mutually Orthogonal Latin Squares and propose two algorithms which integrate Integer Programming (IP) and Constraint Programming (CP). Their behaviour is examined and compared to traditional CP and IP algorithms. The results assess the quality of inference achieved by the CP and IP, mainly in terms of early identification of infeasible subproblems. It is clearly illustrated that the integration of CP and IP is beneficial and that one hybrid algorithm exhibits the best performance as the problem size grows. An approach for reducing the search by excluding isomorphic cases is also presented.

Searching for mutually orthogonal latin squares via integer and constraint programming

This paper applies algorithms integrating Integer Programming (IP) and Constraint Programming (CP) to the Mutually Orthogonal Latin Squares (MOLS) problem. We investigate the behaviour of these algorithms against traditional IP and CP schemes. Computational results are obtained with respect to various aspects of the algorithms, using instances of the 2 MOLS and 3 MOLS problems. The benefits of integrating IP with CP on this feasibility problem are clearly exhibited, especially in large problem instances.

A new class of facets for the Latin square polytope

Latin squares of order n have a 1-1 correspondence with the feasible solutions of the 3-index planar assignment problem (3PAPn). In this paper, we present a new class of facets for the associated polytope, induced by odd-hole inequalities.