Numerical Constraint Satisfaction Problems Research Articles

Enhancing interval constraint propagation by identifying and filtering n-ary subsystems

When interval branch and bound solvers are used for solving numerical constraint satisfaction problems, constraint propagation algorithms are commonly used for filtering/contracting the variable domains. However, these algorithms suffer from the locality problem which is related to the reduced scope of local consistencies. In this work we propose a preprocessing and a filtering technique to reduce the locality problem and to enhance the contraction power of constraint propagation algorithms. The preprocessing consists in constructing a directed acyclic graph (DAG) by merging equivalent nodes (or common subexpressions) and identifying subsystems of n-ary sums in the DAG. The filtering technique consists in applying iteratively HC4 and an ad-hoc technique for contracting the subsystems until reaching a fixed point. Experiments show that the new approach outperforms state-of-the-art strategies using a well known set of benchmark instances.

Adaptive filtering strategy for numerical constraint satisfaction problems

The reliability and increasing performance of search-tree-based interval solvers for solving numerical systems of constraints make them applicable to various expert system domains. Filtering methods are applied in each node of the search tree to reduce the variable domains without the loss of solutions. Current interval-based solvers generally leave it up to the solver designer to decide which set of filtering methods to apply to solve a particular problem. In this work, we propose an adaptive strategy to dynamically determine the set of filtering methods that will be applied in each node of the search tree. Our goal is twofold: first, we want to simplify the task of the solver designer, and second, we believe that an adaptive strategy may improve the average performance of the current state-of-the-art strategies.The proposed adaptive mechanism attempts to avoid calling costly filtering methods when their probability of filtering domains is low. We assume that fruitful filtering occurs in nearby revisions or clusters. Thus, the decision about whether or not to apply a filtering method is based on a cluster detection mechanism. When a cluster is detected, the associated methods are consecutively applied in order to exploit the cluster. Alternately, in zones without clusters, only a cheap method is applied, thus reducing the filtering effort in large portions of the search. We compare our approach with state-of-the-art strategies, demonstrating its effectiveness.

Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques

We investigate solution techniques for numerical constraint-satisfaction problems and validated numerical set integration methods for computing reachable sets of nonlinear hybrid dynamical systems in the presence of uncertainty. To use interval simulation tools with higher-dimensional hybrid systems, while assuming large domains for either initial continuous state or model parameter vectors, we need to solve the problem of flow/sets intersection in an effective and reliable way. The main idea developed in this paper is first to derive an analytical expression for the boundaries of continuous flows, using interval Taylor methods and techniques for controlling the wrapping effect. Then, the event detection and localization problems underlying flow/sets intersection are expressed as numerical constraint-satisfaction problems, which are solved using global search methods based on branch-and-prune algorithms, interval analysis and consistency techniques. The method is illustrated with hybrid systems with uncertain nonlinear continuous dynamics and nonlinear invariants and guards.

Enhancing numerical constraint propagation using multiple inclusion representations

Building tight and conservative enclosures of the solution set is of crucial importance in the design of efficient complete solvers for numerical constraint satisfaction problems (NCSPs). This paper proposes a novel generic algorithm enabling the cooperative use, during constraint propagation, of multiple enclosure techniques. The new algorithm brings into the constraint propagation framework the strength of techniques coming from different areas such as interval arithmetic, affine arithmetic, and mathematical programming. It is based on the directed acyclic graph (DAG) representation of NCSPs whose flexibility and expressiveness facilitates the design of fine-grained combination strategies for general factorable systems. The paper presents several possible combination strategies for creating practical instances of the generic algorithm. The experiments reported on a particular instance using interval constraint propagation, interval arithmetic, affine arithmetic, and linear programming illustrate the flexibility and efficiency of the approach.

Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint propagation techniques

We investigate solution techniques for numerical constraint satisfaction problems and validated numerical set integration methods for computing reachable sets of nonlinear hybrid dynamical systems in presence of uncertainty. To use interval simulation tools with higher dimensional hybrid systems, while assuming large domains for either initial continuous state or model parameter vectors, we need to solve the problem of flow/sets intersection in an effective and reliable way. The main idea developed in this paper is first to derive an analytical expression for the boundaries of continuous flows, using interval Taylor methods and techniques for controlling the wrapping effect. Then, the event detection and localization problems underlying flow/sets intersection are expressed as numerical constraint satisfaction problems, which are solved using global search methods based on branch-and-prune algorithms, interval analysis and consistency techniques. The method is illustrated with a hybrid system with uncertain nonlinear continuous dynamics and nonlinear invariants and guards.

Interval propagation and search on directed acyclic graphs for numerical constraint solving

The fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation have recently been proposed in Schichl and Neumaier (J. Global Optim. 33, 541–562, 2005). For representing numerical problems, the authors use DAGs whose nodes are subexpressions and whose directed edges are computational flows. Compared to tree-based representations [Benhamou et al. Proceedings of the International Conference on Logic Programming (ICLP’99), pp. 230–244. Las Cruces, USA (1999)], DAGs offer the essential advantage of more accurately handling the influence of subexpressions shared by several constraints on the overall system during propagation. In this paper we show how interval constraint propagation and search on DAGs can be made practical and efficient by: (1) flexibly choosing the nodes on which propagations must be performed, and (2) working with partial subgraphs of the initial DAG rather than with the entire graph. We propose a new interval constraint propagation technique which exploits the influence of subexpressions on all the constraints together rather than on individual constraints. We then show how the new propagation technique can be integrated into branch-and-prune search to solve numerical constraint satisfaction problems. This algorithm is able to outperform its obvious contenders, as shown by the experiments.

Accelerating filtering techniques for numeric CSPs

Search algorithms for solving Numeric CSPs (Constraint Satisfaction Problems) make an extensive use of filtering techniques. In this paper 1 1 This paper is an extended version of [31]. we show how those filtering techniques can be accelerated by discovering and exploiting some regularities during the filtering process. Two kinds of regularities are discussed, cyclic phenomena in the propagation queue and numeric regularities of the domains of the variables. We also present in this paper an attempt to unify numeric CSPs solving methods from two distinct communities, that of CSP in artificial intelligence, and that of interval analysis.