We introduce function variables to constraint programs (CP), variables whose values are one of (exponentially many) possible functions between two sets. Such variables are useful for modelling problems from domains such as configuration, planning, scheduling, etc. We show that a function variable can be mapped into different representations in terms of integer and set variables, and illustrate ...

This work deals with the application of a variable neighborhood search (VNS) to the capacitated location-routing problem (LRP) as well as to the more general periodic LRP (PLRP). For this, previous successful VNS algorithms for related problems are considered and accordingly adapted as well as extended. The VNS is subsequently combined with three very large neighborhood searches (VLNS) based on...

Branch-and-Cut is the most commonly used algorithm for solving Integer and Mixed-Integer Linear Programs. In order to reduce the number of nodes that have to be enumerated before optimality of a solution can be proven, branching on general disjunctions (i.e. split disjunctions involving more than one variable, as opposed to branching on simple disjunctions defined on one variable only) was show...

We consider generalizations of parity polytopes whose variables, in addition to a parity constraint, satisfy certain ordering constraints. More precisely, the variable domain is partitioned into k contiguous groups, and within each group, we require xi ≥ xi+1 for all relevant i. Such constraints are used to break symmetry after replacing an integer variable by a sum of binary variables, so-call...

This paper develops an algorithm based on the Modular Approach to solve singly constrained separable discrete optimization problems (Nonlinear Knapsack Problems). The Modular Approach uses fathoming and integration techniques repeatedly. The fathoming reduces the decision space of variables. The integration reduces the number of variables in the problem by combining several variables into one v...

We denote by x a real variable and by n a positive integer variable. The reference measure on the real line R is the Lebesgue measure. In this note we will use only basic properties of the Lebesgue measure and integral on R. It is well known that the fact that a function tends to zero at infinity is a condition neither necessary nor sufficient for this function to be integrable. However, we hav...

In this paper, we consider mixed integer linear programming (MIP) formulations for piecewise linear functions (PLFs) that are evaluated when an indicator variable is turned on. We describe modifications to standard MIP formulations for PLFs with desirable theoretical properties and superior computational performance in this context. © 2013 Elsevier B.V. All rights reserved.

In this paper we summarize our results for two classes of hierarchical multi-scale models that exploit contextual information for detection of structure in mammographic imagery. The first model, the hierarchical pyramid neural network (HPNN), is a discriminative model which is capable of integrating information either coarse-to-fine or fine-tocoarse for microcalcification and mass detection. Th...

In this paper we present two major approaches to solve the car sequencing problem, in which the goal is to find an optimal arrangement of commissioned vehicles along a production line with respect to constraints of the form “no more than lc cars are allowed to require a component c in any subsequence of mc consecutive cars”. The first method is an exact one based on integer linear programming (...

Let Ω(n,Q) be the set of partitions of n into summands that are elements of the set A = { Q(k) : k ∈ Z } . Here Q ∈ Z[x] is a fixed polynomial of degree d > 1 which is increasing on R, and such that Q(m) is a non– negative integer for every integer m ≥ 0. For every λ ∈ Ω(n,Q), letMn(λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on Ω(n,Q), and r...