Laurent Granvilliers

Algorithm 852: RealPaver: an interval solver using constraint satisfaction techniques

RealPaver is an interval software for modeling and solving nonlinear systems. Reliable approximations of continuous or discrete solution sets are computed using Cartesian products of intervals. Systems are given by sets of equations or inequality constraints over integer and real variables. Moreover, they may have different natures, being square or nonsquare, sparse or dense, linear, polynomial, or involving transcendental functions.The modeling language permits stating constraint models and tuning parameters of solving algorithms which efficiently combine interval methods and constraint satisfaction techniques. Several consistency techniques (box, hull, and 3B) are implemented. The distribution includes C sources, executables for different machine architectures, documentation, and benchmarks. The portability is ensured by the GNU C compiler.

Continuous and Interval Constraints

Publisher Summary This chapter reviews that continuous constraint solving has been widely studied in several fields of applied mathematics and computer science. In computer algebra, continuous constraints are viewed as formulas from first-order logic interpreted over the real numbers. The symbolic algorithms transform the constraint systems within the same equivalence class in the interpretation domain according to some simplification ordering. The chapter also discusses the interval analysis, which is a set extension of numerical analysis such that the floating-point numbers are replaced with the intervals. The interval approximations are defined so as to enclose the computed real quantities and the algorithms are said to be complete. In constraint programming, continuous constraints are viewed as relations. The complete solving of nonlinear systems is implemented by exhaustive search techniques that compute solution space coverings by means of multi-dimensional boxes. The search is commonly accelerated through propagation-based algorithms. It reviews that continuous and interval constraints are generally contrasted with non negative integer or more generally discrete constraints. These last constraints, sometimes also called finite domain constraints, are studied in the constraint satisfaction problems (CSP) framework and are basic components of most current constraint-based languages.

Symbolic-interval cooperation in constraint programming

This paper surveys the field of cooperative constraint solving for a constraint programming perspective with an emphasis on combinations of symbolic and interval methods. On the one hand, symbolic methods provide adapted representations of the constraint expressions. On the other hand, interval methods compute verified enclosures of solution sets. Using cooperation of solvers, one can take advantage of both techniques in a unified framework: symbolic algorithms generally need to be combined with root extraction methods, and the efficiency of interval algorithms strongly depends on constraint expressions.

On the Combination of Interval Constraint Solvers

This paper tackles the combination of interval methods for solving nonlinear systems. A cooperative strategy of application of elementary solvers is designed in order to accelerate the whole computation while weakening the local domain contractions. It is implemented in a prototype solver which efficiently combines interval-based local consistencies and the multidimensional interval Newton method. A set of experiments shows a gain of one order of magnitude on average with respect to Numerica.

Novel Approaches to Numerical Software with Result Verification

Traditional design of numerical software with result verification is based on the assumption that we know the algorithm f(x 1,...,x n ) that transforms inputs x 1,...,x n into the output y=f(x 1,...,x n ), and we know the intervals of possible values of the inputs. Many real-life problems go beyond this paradigm. In some cases, we do not have an algorithm f, we only know some relation (constraints) between x i and y. In other cases, in addition to knowing the intervals x i , we may know some relations between x i ; we may have some information about the probabilities of different values of x i , and we may know the exact values of some of the inputs (e.g., we may know that x 1 = π/2). In this paper, we describe the approaches for solving these real-life problems. In Section 2, we describe interval consistency techniques related to handling constraints; in Section 3, we describe techniques that take probabilistic information into consideration, and in Section 4, we overview techniques for processing exact real numbers.

Model-driven constraint programming

Constraint programming can definitely be seen as a model-driven paradigm. The users write programs for modeling problems. These programs are mapped to executable models to calculate the solutions. This paper focuses on efficient model management (definition and transformation). From this point of view, we propose to revisit the design of constraint-programming systems. A model-driven architecture is introduced to map solving-independent constraint models to solving-dependent decision models. Several important questions are examined, such as the need for a visual highlevel modeling language, and the quality of metamodeling techniques to implement the transformations. A main result is the s-COMMA platform that efficiently implements the chain from modeling to solving constraint problems

Progress in the Solving of a Circuit Design Problem

This paper describes a new global branch-and-prune algorithm dedicated to the solving of nonlinear systems. The pruning technique combines a multidimensional interval Newton method with HC4, a state of the art constraint satisfaction algorithm recently proposed by the authors. From an algorithmic point of view, the main contributions of this paper are the design of a fine-grained interaction between both algorithms which avoids some unnecessary computation and the description of HC4 in terms of a chain rule for constraint projections. Our algorithm is experimentally compared, on a particular circuit design problem proposed by Ebers and Moll in 1954, with two global methods proposed in the last ten years by Ratschek and Rokne and by Puget and Van Hentenryck. This comparison shows an improvement factor of five with respect to the fastest of these previous implementations on the same machine.

Search heuristics for constraint-aided embodiment design

Abstract Embodiment design (ED) is an early phase of product development. ED problems consist of finding solution principles that satisfy product requirements such as physics behaviors and interactions between components. Constraint satisfaction techniques are useful to solve constraint-based models that are often partial, heterogeneous, and uncertain in ED. This paper proposes new constraint satisfaction techniques to tackle piecewise-defined physics phenomena or skill-based rules and multiple categories of variables arising in design applications. New search heuristics and a global piecewise constraint are introduced in the branch and prune framework. The capabilities of these techniques are illustrated with both academic and real-world problems. Complete models of the latter are presented.

Automatic Generation of Numerical Redundancies for Non-Linear Constraint Solving

In this paper we present a framework for the cooperation of symbolic and propagation-based numerical solvers over the real numbers. This cooperation is expressed in terms of fixed points of closure operators over a complete lattice of constraint systems. In a second part we instantiate this framework to a particular cooperation scheme, where propagation is associated to pruning operators implementing interval algorithms enclosing the possible solutions of constraint systems, whereas symbolic methods are mainly devoted to generate redundant constraints. When carefully chosen, it is well known that the addition of redundant constraint drastically improve the performances of systems based on local consistency (e.g. Prolog IV or Newton). We propose here a method which computes sets of redundant polynomials called partial Gröbner bases and show on some benchmarks the advantages of such computations.

Modeling Camera Control with Constrained Hypertubes

In this paper, we introduce a high-level modeling approach to camera control. The aim is to determine the path of a camera that verifies given declarative properties on the desired image, e.g., location or orientation of objects on the screen at a given time. The path is composed of elementary movements called hypertubes, based on established cinematographic techniques. Hypertubes are connected by relations that guarantee smooth transitions. Interval consistency techniques and quantified constraint solving algorithms are used to compute and propagate solutions between consecutive hypertubes. Preliminary experimental results from a prototype show a great improvement in time and quality of animations with respect to former approaches.