Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
A box represents the Cartesian product of intervals. An interval\({\varvec{x}}_i = [{\underline{x}}_i,{\overline{x}}_i]\) defines the set of reals \(x_i\) s.t. \({\underline{x}}_i \le x_i \le {\overline{x}}_i\). \({\mathbb {IR}}\) denotes the set of all intervals. The size or width of \({\varvec{x}}_i\) is defined as \(\text{ wid }({\varvec{x}}_i)=\overline{x}_i-\underline{x}_i\). \(\text{ mid }({\varvec{x}}_i)=\frac{\overline{x}_i+\underline{x}_i}{2}\) denotes the midpoint of \({\varvec{x}}_i\).
It also may occur with non-monotonic unary operators such as sqr. For instance, consider the primitive constraint \(w=x^2\) where \({\varvec{x}}\in [-1,1]\) and w is an intermediate node. If, in the backward phase, \({\varvec{w}}\) is reduced to [1, 1], then \({\varvec{x}}\) cannot be reduced because its bounds satisfy the constraint: \((-1)^2 = 1^2 \in [1,1]\). Then, in the next evaluation, the reduction over \({\varvec{w}}\) is not retrieved: \({\varvec{w}}\leftarrow {\varvec{x}}^2=[-1,1]^2=[0,1]\).
The subsystems of n-ary products can be linearized by applying the absolute value and the logarithm operators. However, due to the results were not very promising (only a few instances of the test suite have several influential n-ary products), we discarded this idea for the moment.
The procedure \({\mathrm {generate\_product}}\) is analogous to \({\mathrm {generate\_sum}}\). Children nodes are split into a constant exponent and a variable node. For instance \(n_1^2\) is split into the exponent 2 and the variable node \(n_1\). Nodes that are not powers are not split and consider a exponent equal to 1 (or \(-1\) if they are involved into a division).
We did not include trigonometric functions because they generally lead to thousands of solutions.
References
Araya, I., Reyes, V.: Interval branch-and-bound algorithms for optimization and constraint satisfaction: a survey and prospects. J. Glob. Optim. 65(4), 837–866 (2016)
Mackworth, A.K.: Consistency in networks of relations. Artif. Intell. 8(1), 99–118 (1977)
Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.-F.: Revising hull and box consistency. In: International Conference on Logic Programming, Citeseer (1999)
Moore, R.: Interval analysis. Series in automatic computation. Prentice-Hall, Englewood Cliff, N.J. (1966)
Hansen, E.: Global optimization using interval analysis—the multi-dimensional case. Numer. Math. 34(3), 247–270 (1980)
Araya, I., Neveu, B., Trombettoni, G.: Exploiting common subexpressions in numerical CSPs. In: Principles and Practice of Constraint Programming (CP 2008). Springer, pp. 342–357 (2008)
Schichl, H., Neumaier, A.: Interval analysis on directed acyclic graphs for global optimization. J. Glob. Optim. 33(4), 541–562 (2005)
Ceberio, M., Granvilliers, L.: Solving nonlinear equations by abstraction, Gaussian elimination, and interval methods. In: Frontiers of Combining Systems. Springer, pp. 117–131 (2002)
Neumaier, A., Shcherbina, O.: Safe bounds in linear and mixed-integer linear programming. Math. Program. 99(2), 283–296 (2004)
Baharev, A., Achterberg, T., Rév, E.: Computation of an extractive distillation column with affine arithmetic. AIChE J. 55(7), 1695–1704 (2009)
Chabert, G., Jaulin, L.: Contractor programming. Artif. Intell. 173, 1079–1100 (2009)
Neveu, B., Trombettoni, G., et al.: Adaptive constructive interval disjunction. In: International Conference on Tools with Artificial Intelligence (ICTAI), pp. 900–906 (2013)
Hansen, E.: Global Optimization Using Interval Analysis. Marcel Dekker, New York (1992)
Trombettoni, G., Araya, I., Neveu, B., Chabert, G.: Inner regions and interval linearizations for global optimization. In: AAAI Conference on Artificial Intelligence, pp. 99–104 (2011)
Ninin, J., Messine, F., Hansen, P.: A reliable affine relaxation method for global optimization. 4OR, vol. 13, no. 3, pp. 247–277 (2015)
Araya, I., Trombettoni, G., Neveu, B.: A contractor based on convex interval Taylor. In: Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. Springer, pp. 1–16 (2012)
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)
Acknowledgements
This work is supported by the Fondecyt Project 1160224. Victor Reyes is supported by the Grant Postgrado PUCV 2017.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Araya, I., Reyes, V. Enhancing interval constraint propagation by identifying and filtering n-ary subsystems. J Glob Optim 74, 1–20 (2019). https://doi.org/10.1007/s10898-019-00738-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-019-00738-5