Non-linear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifier-free equivalent form by means of Cylindrical Algebraic Decomposition (CAD). However, CAD restricts its input to be conjunctions and disjunctions of polynomial constraints with rational coefficients, while some applications such as camera control involve systems with arbitrary forms where time is the only universally quantified variable. In this paper, the handling of universally quantified variables is first related to the computation of inner-approximation of real relations.

Abstract. Traditional design of numerical software with result verification is based on the assumption that we know the algorithm ¦¨§� © ©���� £��������� � that transforms input © ©�� into �� � £��������� � ©���� the output, 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 ¦, we only know some relation (constraints) between ©� � and. In other cases, in addition to knowing the intervals, we may know some relations between; we may have some information about the probabilities of different values of © � , and we may know the exact values of some of the inputs (e.g., we may know that © £ ���¨�� �). 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. 1

This paper shows how to use constraint programming techniques for solving first-order constraints over the reals (i.e., formulas in the first-order predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of consistency for atomic constraints over the reals (e.g., box-consistency), the paper provides a narrowing operator for firstorder constraints that implements a corresponding notion of first-order consistency, and a solver based on such a narrowing operator. As a consequence, this solver can take over various favorable properties from the field of constraint programming.

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques. 1

A. NEUMAIER [1] has given the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation. We show in this paper how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) enabling the flexible choice of the interval inclusion functions during propagation. We then propose a new simple algorithm which coordinates constraint propagation and exhaustive search for solving numerical constraint satisfaction problems. The experiments carried out on different problems show that the new approach outperforms previously available propagation techniques by an order of magnitude or more in speed, while being roughly the same quality w.r.t. enclosure properties. I.

Numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques, local consistencies, and interval methods. This paper first recalls the principles of Quad, a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints. Then, it introduces a generalization of Quad to polynomial constraint systems. It also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system. Different linearization techniques are investigated to limit the number of generated constraints. QuadSolver, a new branch and prune algorithm that combines Quad, local consistencies, and interval methods, is introduced. QuadSolver has been evaluated on a variety of benchmarks from kinematics, mechanics, and robotics. On these benchmarks, it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state-of-the-art optimization solvers.

system, a new Constraint Logic Programming system for nonlinear constraints over the reals, based on interval techniques. A first evaluation shows that we can improve on other systems in a number of areas. INCLP(R) is written in Prolog using Constraint Handling Rules and is the first nonlinear CLP system implemented using this technology. Directions for future research are given.

A new branch-and-prune algorithm for globally solving nonlinear systems is proposed. The pruning technique combines a multidimensional interval Newton method with the constraint satisfaction algorithm HC4 [1]. The main contributions of this paper are the fine-grained interaction between both algorithms which avoids some unnecessary computation,and the description of HC4 in terms of a chain rule for constraints’ projections. Our algorithm is experimentally compared with two global methods from Ratschek and Rokne [17] and from Puget and Van Hentenryck [16] on Ebers and Moll’ circuit design problem [6]. An interval enclosure of the solution with a precision of twelve significant digits is computed in four minutes, providing an improvement factor of five on the same machine.

Bounded-error estimation is the estimation of the parameter or state vector of a model from experimental data, under the assumption that some suitabl y de...ned errors shoul d bel ong to some prior feasibl e sets. When the model outputs arel inear in the vector to be estimated, a number of methods are avail#0 l e to encl ose al# estimates that are consistent with the data into simpl# sets such as el# ipsoids, orthotopes or paral#0xP90O es, thereby providing guaranteed set estimates. In the nonl#x]30 case, the situation is muchl#O4 devel#O ed and there are very few methods that produce such guaranteed estimates. In this paper, the discrete-time probl em is cast into the more general framework of constraint satisfaction probl ems.Al# orithms rathercl assical in the area of interval constraint propagation are extended by repl acing interva l# by moregeneral subsets of real vector spaces. This makes it possibl# to propose a new al#9Oq30 m that contracts the feasibl e domains for each uncertain variabl# optimal#O (i.e., no smal# er domain coul d be obtained) and ecientl# . The resul ting methodol#03 isil#34 trated on discrete-time nonl#O0O7 state estimation. The state at time k is estimated either from past measurement onl y or from al l measurements assumed to be avai l#bl# from the start. Even in the causal case, prior information on the future val# e of the state and output vectors, due for instance to physical constraints, is readil y taken into account.

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 non square, 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, 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.