A Tabu Search Method for Interval Constraints

This article presents an extension of the Tabu Search (TS) metaheuristic to continuous CSPs, where the domains are represented by floating point-bounded intervals. This leads to redefine the usual TS operators to take into account the special features of interval constraints: real variables encoded in floating points domains, high cardinality of the domains, nature of the CSP where constraints may be partially satisfied. To illustrate the expressiveness of the framework, we instantiate this method to compute an inner-approximation of a set of inequalities. Metaheuristics, in particular Tabu Search (TS, [6,8]), have largely proven their efficiency on discrete optimization or constraint problems. In this article, we propose a framework to extend TS to continous problems on real variables. There exist a variety of optimization techniques to solve such problems, but they are usually dedicated to particular types of constraints (linear, polynomial, differentiable) and do not offer the guarantees of interval approaches [4,5]. Interval-based solvers like Realpaver [7] are dedicated to compute rigourous inner and outerapproximations of continuous problems, but often fail in efficiently computing a first solution due to the complete nature of the search process. We provide a unified way to express continuous CSPs in a TS framework that shares the reliability of interval techniques, and tackle a problem not well adressed by classical tools: the computation of a single solution for interval CSPs or optimisation problems, whatever the type of the constraints. 1 Interval Constraints Interval arithmetics [9] offers a reliable solution to avoid rounding errors due to finite representation of real values (see IEEE754 norm). Real values are encompassed within floating-point intervals : a real value r may be represented by any interval I, with floating-point bounds, containing r. The set of all intervals is I. The Cartesian product of intervals is called a box. The classical operators over R can be redefined over I by enforcing the fundamental property of containment[9]: the interval extension of a binary operator is given by the smallest interval containing the results of the application of . This construct L. Perron and M. Trick (Eds.): CPAIOR 2008, LNCS 5015, pp. 372–376, 2008. c © Springer-Verlag Berlin Heidelberg 2008 A Tabu Search Method for Interval Constraints 373 guarantees that no values are lost, but can lead to an over-estimation of the result. Real functions are extended on I in the same way. A CSP defined over continuous domains is translated into intervals by taking interval variables over interval domains. The constraints are extended over I in the same way as functions or operators. The goal is to find a box B ⊆ D1× . . .×Dn such that the constraints are satisfied. An interval constraint C on a box B can be certainly satisfied (every real vector in B satisfies C), certainly not satisfied (no real vector in B satisfies C). If neither of those two properties can be computed, C is said partially satisfied. Several metaheuristics have been transposed to CSP on real variables. Some of them are adapted from TS [4,5,2]. However, these algorithms are based on real configurations, and employed to tackle optimization problems. Our method is closer to that of [3] for interval CSPs with non-linear differentiable constraints, or to [1] on multi-objective problems. 2 Tabu Search on Intervals The basic components of a TS algorithm on discrete domains are well-known: a penalty function f : D1 × . . . × Dn → N, counting the number of violated constraints most of the times, a neighborhood function, and the exact definition of a tabu mechanism (basic tabu behaviour with size t, dynamic tabu, aspiration, ...). However, adapting the algorithm to new types of domains leads to a number of obstacles. Firstly, TS algorithms use many operators which are trivial over discrete configurations (equality, membership, random choice) but require specific attention over interval domains. Secondly, an interval CSP has a search space with specific properties: the domains are huge, and they are defined by two floating point numbers but include an infinite set of reals. Hence we dissect the algorithm into more precise atomic components.