Abstract. A benchmarking suite describing over 1000 optimization problems and constraint satisfaction problems covering problems from different traditions is described, annotated with best known solutions, and accompanied by recommended benchmarking protocols for comparing test results.
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). (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.
In recent years, interval constraint-based solvers have shown their ability to efficiently solve complex instances of non-linear numerical CSPs. However, most of the working systems are designed to deliver point-wise solutions with an arbitrary accuracy. This works generally well for systems with isolated solutions but less well when there is a continuum of feasible points (e.g. underconstrained problems, problems with inequalities). In many practical applications, such large sets of solutions express equally relevant alternatives which need to be identified as completely as possible. In this paper, we address the issue of constructing concise inner and outer approximations of the complete solution set for non-linear CSPs. We propose a technique which combines the extreme vertex representation of orthogonal polyhedra [1-3], as defined in computational geometry, with adapted splitting strategies [4] to construct the approximations as unions of interval boxes. This allows for compacting the explicit representation of the complete solution set and improves efficiency.
The pioneering paper H. Schichl and A. Neumaier [1] has founded the fundamentals of interval analysis on DAGs for global optimization, including a fundamental of constraint propagation. In this paper, we extend the constraint propagation technique for solving numerical constraint satisfaction problems. In particular, we propose an advanced constraint propagation technique, which makes the constraint propagation practical and efficient, and a method to coordinate constraint propagation and exhaustive search, which uses a single DAG for each problem. The experiments carried out on various 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.
Most of the working solvers for numerical constraint satisfaction problems (NCSPs) are designed to delivering point-wise solutions with an arbitrary accuracy. When there is a continuum of feasible points this might lead to prohibitively verbose representations of the output. In many practical applications, such large sets of solutions express equally relevant alternatives which need to be identified as completely as possible. The goal of this paper is to show that by using appropriate approximation techniques, explicit representations of the solution sets, preserving both accuracy and completeness, can still be proposed for NCSPs with continuum of solutions. We present a technique for constructing concise inner and outer approximations as unions of interval boxes. The proposed technique combines a new splitting strategy with the extreme vertex representation of orthogonal polyhedra [1-3], as defined in computational geometry. This allows for compacting the representation of the approximations and improves efficiency.
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