Gilles Chabert

Contractor programming

This paper describes a solver programming method, called contractor programming, that copes with two issues related to constraint processing over the reals. First, continuous constraints involve an inevitable step of solver design. Existing softwares provide an insufficient answer by restricting users to choose among a list of fixed strategies. Our first contribution is to give more freedom in solver design by introducing programming concepts where only configuration parameters were previously available. Programming consists in applying operators (intersection, composition, etc.) on algorithms called contractors that are somehow similar to propagators. Second, many problems with real variables cannot be cast as the search for vectors simultaneously satisfying the set of constraints, but a large variety of different outputs may be demanded from a set of constraints (e.g., a paving with boxes inside and outside of the solution set). These outputs can actually be viewed as the result of different contractors working concurrently on the same search space, with a bisection procedure intervening in case of deadlock. Such algorithms (which are not strictly speaking solvers) will be made easy to build thanks to a new branch & prune system, called paver. Thus, this paper gives a way to deal harmoniously with a larger set of problems while giving a fine control on the solving mechanisms. The contractor formalism and the paver system are the two contributions. The approach is motivated and justified through different cases of study. An implementation of this framework named Quimper is also presented.

Constructive interval disjunction

This paper presents two new filtering operators for numerical CSPs (systems with constraints over the reals) based on constructive disjunction, as well as a new splitting heuristic. The fist operator (CID) is a generic algorithm enforcing constructive disjunction with intervals. The second one (3BCID) is a hybrid algorithm mixing constructive disjunction and shaving, another technique already used with numerical CSPs through the algorithm 3B. Finally, the splitting strategy learns from the CID filtering step the next variable to be split, with no overhead. Experiments have been conducted with 20 benchmarks. On several benchmarks, CID and 3BCID produce a gain in performance of orders of magnitude over a standard strategy. CID compares advantageously to the 3B operator while being simpler to implement. Experiments suggest to fix the CID-related parameter in 3BCID, offering thus to the user a promising variant of 3B.

Hull consistency under monotonicity

We prove that hull consistency for a system of equations or inequalities can be achieved in polynomial time providing that the underlying functions are monotone with respect to each variable. This result holds including when variables have multiple occurrences in the expressions of the functions, which is usually a pitfall for interval-based contractors. For a given constraint, an optimal contractor can thus be enforced quickly under monotonicity and the practical significance of this theoretical result is illustrated on a simple example.

A constraint on the number of distinct vectors with application to localization

This paper introduces a generalization of the nvalue constraint that bounds the number of distinct values taken by a set of variables. The generalized constraint (called nvector) bounds the number of distinct (multi-dimensional) vectors. The first contribution of this paper is to show that this global constraint has a significant role to play with continuous domains, by taking the example of simultaneous localization and map building (SLAM). This type of problem arises in the context of mobile robotics. The second contribution is to prove that enforcing bound consistency on this constraint is NP-complete. A simple contractor (or propagator) is proposed and applied on a real application.

On the Approximation of Linear AE-Solution Sets

When considering systems of equations, it often happens that parameters are known with some uncertainties. This leads to continua of solutions that are usually approximated using the interval theory. A wider set of useful situations can be modeled if one allows furthermore different quantifications of the parameters in their domains. In particular, quantified solution sets where universal quantifiers are constrained to precede existential quantifiers are called AE-solution sets. A state of the art on the approximation of linear AE- solution sets in the framework of generalized intervals (intervals whose bounds are not constrained to be ordered increasingly) is presented in a new and unifying way. Then two new generalized interval operators dedicated to the approximation of quantified linear interval systems are proposed and investigated.

Certified Calibration of a Cable-Driven Robot Using Interval Contractor Programming

In this paper, an interval based approach is proposed to rigorously identify the model parameters of a parallel cable-driven robot. The studied manipulator follows a parallel architecture having 8 cables to control the 6 DOFs of its mobile platform. This robot is complex to model, mainly due to the cable behavior. To simplify it, some hypotheses on cable properties (no mass and no elasticity) are done.An interval approach can take into account the maximal error between this model and the real one. This allows us to work with a simplified although guaranteed interval model. In addition, a specific interval operator makes it possible to manage outliers. A complete experiment validates our method for robot parameter certified identification and leads to interesting observations.

Sweeping with continuous domains

The geost constraint has been proposed to model and solve discrete placement problems involving multi-dimensional boxes (packing in space and time). The filtering technique is based on a sweeping algorithm that requires the ability for each constraint to compute a forbidden box around a given fixed point and within a surrounding area. Several cases have been studied so far, including integer linear inequalities. Motivated by the placement of objects with curved shapes, this paper shows how to implement this service for continuous constraints with arbitrary mathematical expressions. The approach relies on symbolic processing and defines a new interval arithmetic.

A generalized interval LU decomposition for the solution of interval linear systems

Generalized intervals (intervals whose bounds are not constrained to be increasingly ordered) extend classical intervals providing better algebraic properties. In particular, the generalized interval arithmetic is a group for addition and for multiplication of zero free intervals. These properties allow one constructing a LU decomposition of a generalized interval matrix A: the two computed generalized interval matrices L and U satisfy A = LU with equality instead of the weaker inclusion obtained in the context of classical intervals. Some potential applications of this generalized interval LU decomposition are investigated.

Box-set consistency for interval-based constraint problems

As opposed to finite domain CSPs, arc consistency cannot be enforced, in general, on CSPs over the reals, including very simple instances. In contrast, a stronger property, the so-called box-set consistency, that requires a no-split condition in addition to arc consistency, can be obtained on a much larger number of problems.To obtain this property, we devise a lazy algorithm that combines hull consistency filtering, interval union projection, and intelligent domain splitting. It can be applied to any numerical CSP, and achieves box-set consistency if constraints are redundancy-free in terms of variables. This holds even if the problem is not intervalconvex. The main contribution of our approach lies in the way we bypass the non-convexity issue, which so far was a synonym for either a loss of accuracy or an unbounded growth of label size.We prove the correctness of our algorithm and through experimental results, we show that, as compared to a strategy based on a standard bisection, it may lead to gains while never producing an overhead.

Computing the Pessimism of Inclusion Functions

Abstract“Computing the pessimism” means bounding the overestimation produced by an inclusion function. There are two important distinctions with classical error analysis. First, we do not consider the image by an inclusion function but the distance between this image and the exact image (in the set-theoretical sense). Second, the bound is computed over a infinite set of intervals.To our knowledge, this issue is not covered in the literature and may have a potential of applications. We first motivate and define the concept of pessimism. An algorithm is then provided for computing the pessimism, in the univariate case. This algorithm is general-purpose and works with any inclusion function. Next, we prove that the algorithm converges to the optimal bound under mild assumptions. Finally, we derive a second algorithm for automatically controlling the pessimism, i.e., determining where an inclusion function is accurate.