Introduction Many science and engineering applications require the user to find solutions to systems of nonlinear constraints over real numbers or to optimize a nonlinear function subject to nonlinear constraints. This includes applications such the modeling of chemical engineering processes and of electrical circuits, robot kinematics, chemical equilibrium problems, and design problems (e.g., nuclear reactor design). The field of global optimization is the study of methods to find all solutions to systems of nonlinear constraints and all global optima to optimization problems. Nonlinear problems raise many issues from a computation standpoint. On the one hand, deciding if a set of polynomial constraints has a solution is NP-hard. In fact, Canny [ Canny, 1988 ] and Renegar [ Renegar, 1988 ] have shown that the problem is in PSPACE and it is not known whether the problem lies in NP. Nonlinear programming problems can be so hard that some methods are designed only to solve probl

Constraint programming is newly flowering in industry. Several companies have recently started up to exploit the technology, and the number of industrial applications is now growing very quickly. This survey will seek, by examples,

Most CLP languages designed in the past few years feature at least some combination of constraint solving capabilities. These combinations can take multiple forms since they achieve either the mixing of di erent domains or the use of di erent algorithms over the same domain. These solvers are also very di erent in nature. Some of them perform complete constraint solving while others are based on propagation methods. This paper is an attempt to design a uni ed framework describing the cooperation of constraint solving methods. Most techniques used in constraint-based systems are shown to be implementations of operators called constraint narrowing operators. A generalized notion of arc-consistency, called weak arc-consistency is proposed and is used to model heterogeneous constraint solving. We provide conditions on the constraint solving algorithms which guarantee termination, correctness and con uence of the resulting combined solver. This framework is shown to be general enough to describe the operational semantics of the basic constraint solving mechanisms in a number of current CLP systems. 1

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In this paper, a new search technique over numeric csps is presented: dynamic domain splitting. The usual search technique over numeric csps is a dichotomic search interleaved with a consistency filtering, which is called domain splitting. This paper proposes to replace chronological backtracking at the core of domain splitting by a non destructive backtracking technique.

A classical circuit-design problem from Ebers and Moll [6] features a system of nine nonlinear equations in nine variables that is very challenging both for local and global methods. This system was solved globally using an interval method by Ratschek and Rokne [23] in the box [0; 10] 9 . Their algorithm had enormous costs (i.e., over 14 months using a network of 30 Sun Sparc-1 workstations) but they state that "at this time, we know no other method which has been applied to this circuit design problem and which has led to the same guaranteed result of locating exactly one solution in this huge domain, completed with a reliable error estimate." The present paper gives a novel branch-and-prune algorithm that obtains a unique safe box for the above system within reasonable computation times. The algorithm combines traditional interval techniques with an adaptation of discrete constraint-satisfaction techniques to continuous problems. Of particular interest is the simplicity o...

. Constraint query languages are natural extensions of relational database query languages. A framework for their declarative specification (constraint calculi) and efficient implementation (low data complexity and secondary storage indexing) was presented in Kanellakis et al., 1995. Constraint query algebras form a procedural language layer between high-level declarative calculi and low-level indexing methods. Just like the relational algebra, this intermediate layer can be very useful for program optimization. In this paper, we study properties of constraint query algebras, which we present through three concrete examples. The dense order constraint algebra illustrates how the appropriate canonical form can simplify expensive operations, such as projection, and facilitate interaction with updates. The monotone two-variable linear constraint algebra illustrates the concept of strongly polynomial operations. Finally, the lazy evaluation of (non)linear constraint algebras illustrates ho...

Global search algorithms have been widely used in the constraint programming framework to solve constraint systems over continuous domains. This paper precisely states the relations among the different partial consistencies which are main emphasis of these algorithms. The

In 1992, V. Weispfenning proved the existence of Comprehensive Gröbner Bases (CGB) and gave an algorithm to compute one. That algorithm was not very efficient and not canonical. Using his suggestions, A. Montes obtained in 2002 a more efficient algorithm (DISPGB) for Discussing Parametric Gröbner Bases. Inspired in its philosophy, V. Weispfenning defined, in 2002, how to obtain a Canonical Comprehensive Gröbner Basis (CCGB) for parametric polynomial ideals, and provided a constructive method. In this paper we use Weispfenning’s CCGB ideas to make substantial improvements on Montes DISPGB algorithm. It now includes rewriting of the discussion tree using the Discriminant Ideal and provides a compact and effective discussion. We also describe the new algorithms in the DPGB library containing the improved DISPGB as well as new routines to check whether a given basis is a CGB or not, and to obtain a CGB. Examples and tests are also provided. Key words: discriminant ideal, comprehensive Gröbner bases, parametric polynomial system.

this paper is on the first problem. It shows that there is a strong connection between the existence of cyclic phenomena and slow convergence. The main goal is to dynamically identify cyclic phenomena while executing algorithm