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# Distribution Optimization in Constraint Programming

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LIPIcs.CP.2023.29.pdf
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## Cite As

Guillaume Perez, Gaël Glorian, Wijnand Suijlen, and Arnaud Lallouet. Distribution Optimization in Constraint Programming. In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 280, pp. 29:1-29:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CP.2023.29

## Abstract

Stochastic Constraint Programming introduces stochastic variables following a probability distribution to model uncertainty. In the classical setting, probability distributions are given and constant. We propose a framework in which random variables are given a set of possible distributions and only one should be selected. A solution is obtained when all variable distributions are assigned, and all decision variables are assigned too. In such a setting, a constraint on random variables limits the possible distributions its random variables may take. We generalize the notion of chance as the probability of satisfaction of a constraint, called probabilization, given variable distributions. Probabilization can be seen as a generalization of reification in a random setting whose result is a random variable. We define minimal arithmetic to work with stochastic variables having a variable distribution. Using the introduced representation, our framework can in theory save an exponential number of decisions, and represents problems that were previously not representable with finite integer domains. Finally, we model and solve two industrial problems that require this extension - virtual network configuration and assignment of chemical delivery - and show improvement in terms of quality of solution and speed.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Constraint and logic programming
• Mathematics of computing → Solvers
• Computing methodologies → Probabilistic reasoning
##### Keywords
• Constraint Programming
• Optimization
• Stochastic Optimization

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## References

1. Eric Baum and Frank Wilczek. Supervised learning of probability distributions by neural networks. In Neural information processing systems, 1987.
2. N Bharanidharan and Harikumar Rajaguru. Improved chicken swarm optimization to classify dementia mri images using a novel controlled randomness optimization algorithm. International Journal of Imaging Systems and Technology, 30(3):605-620, 2020.
3. Giuseppe Calafiore and Marco C Campi. Uncertain convex programs: randomized solutions and confidence levels. Mathematical Programming, 102(1):25-46, 2005.
4. Gilles Chabert et al. Ibex, an interval-based explorer, 2007.
5. Gilles Chabert and Luc Jaulin. Contractor programming. Artificial Intelligence, 173(11):1079-1100, 2009.
6. Abraham Charnes and William W Cooper. Chance-constrained programming. Management science, 6(1):73-79, 1959.
7. Abraham Charnes and William W Cooper. Deterministic equivalents for optimizing and satisficing under chance constraints. Operations research, 11(1):18-39, 1963.
8. Abraham Charnes, William Wager Cooper, and MJL Kirby. Chance-constrained programming: an extension of statistical method. In Optimizing methods in statistics, pages 391-402. Elsevier, 1971.
9. Constantinos Daskalakis, Ilias Diakonikolas, and Rocco A Servedio. Learning poisson binomial distributions. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 709-728, 2012.
10. Constantinos Daskalakis and Gautam Kamath. Faster and sample near-optimal algorithms for proper learning mixtures of gaussians. In Conference on Learning Theory, pages 1183-1213. PMLR, 2014.
11. Augustin Delecluse, Pierre Schaus, and Pascal Van Hentenryck. Sequence variables for routing problems. In 28th International Conference on Principles and Practice of Constraint Programming (CP 2022), 2022.
12. Mihai Dobrescu, Katerina Argyraki, and Sylvia Ratnasamy. Toward predictable performance in software Packet-Processing platforms. In 9th USENIX Symposium on Networked Systems Design and Implementation (NSDI 12), pages 141-154, 2012.
13. Paul Emmerich, Daniel Raumer, Florian Wohlfart, and Georg Carle. Assessing soft-and hardware bottlenecks in pc-based packet forwarding systems. ICN 2015, 90, 2015.
14. Hélène Fargier and Jérôme Lang. Uncertainty in constraint satisfaction problems: a probabilistic approach. In Symbolic and Quantitative Approaches to Reasoning and Uncertainty: European Conference ECSQARU'93 Granada, Spain, November 8-10, 1993 Proceedings 2, pages 97-104. Springer, 1993.
15. Jakob Feldtkeller, David Knichel, Pascal Sasdrich, Amir Moradi, and Tim Güneysu. Randomness optimization for gadget compositions in higher-order masking. Cryptology ePrint Archive, 2022.
16. Juliver Gil Herrera and Juan Felipe Botero. Resource allocation in nfv: A comprehensive survey. IEEE Transactions on Network and Service Management, 13(3):518-532, 2016.
17. Brahim Hnich, Roberto Rossi, S Armagan Tarim, and Steven Prestwich. Synthesizing filtering algorithms for global chance-constraints. In Principles and Practice of Constraint Programming-CP 2009: 15th International Conference, CP 2009 Lisbon, Portugal, September 20-24, 2009 Proceedings 15, pages 439-453. Springer, 2009.
18. Brahim Hnich, Roberto Rossi, S Armagan Tarim, and Steven Prestwich. Filtering algorithms for global chance constraints. Artificial Intelligence, 189:69-94, 2012.
19. JN Hooker. Stochastic decision diagrams. In International Conference on Integration of Constraint Programming, Artificial Intelligence, and Operations Research, pages 138-154. Springer, 2022.
20. Mark H Houck. A chance constrained optimization model for reservoir design and operation. Water Resources Research, 15(5):1011-1016, 1979.
21. Bahareh Kargar, Mir Saman Pishvaee, Hamed Jahani, and Jiuh-Biing Sheu. Organ transportation and allocation problem under medical uncertainty: A real case study of liver transplantation. Transportation Research Part E: Logistics and Transportation Review, 134:101841, 2020.
22. Michael Kearns, Yishay Mansour, Dana Ron, Ronitt Rubinfeld, Robert E Schapire, and Linda Sellie. On the learnability of discrete distributions. In Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pages 273-282, 1994.
23. Lin Kemeng, Wang Xiaoyan, Xia Weijie, Zhang Jiaming, et al. Optimization of the randomness in einstein which based on monte carlo algorithms. In 2019 Chinese Control And Decision Conference (CCDC), pages 6305-6309. IEEE, 2019.
24. Stefan Kern, Sibylle D Müller, Nikolaus Hansen, Dirk Büche, Jiri Ocenasek, and Petros Koumoutsakos. Learning probability distributions in continuous evolutionary algorithms-a comparative review. Natural Computing, 3:77-112, 2004.
25. Minh Thanh Khong, Christophe Lecoutre, Pierre Schaus, and Yves Deville. Soft-regular with a prefix-size violation measure. In International Conference on the Integration of Constraint Programming, Artificial Intelligence, and Operations Research, pages 333-343. Springer, 2018.
26. Daphne Koller and Nir Friedman. Probabilistic graphical models: principles and techniques. MIT press, 2009.
27. Anna LD Latour, Behrouz Babaki, Daniël Fokkinga, Marie Anastacio, Holger H Hoos, and Siegfried Nijssen. Exact stochastic constraint optimisation with applications in network analysis. Artificial Intelligence, 304:103650, 2022.
28. Pu Li, Harvey Arellano-Garcia, and Günter Wozny. Chance constrained programming approach to process optimization under uncertainty. Computers & chemical engineering, 32(1-2):25-45, 2008.
29. Xiaoxia Lin, Stacy L Janak, and Christodoulos A Floudas. A new robust optimization approach for scheduling under uncertainty:: I. bounded uncertainty. Computers & chemical engineering, 28(6-7):1069-1085, 2004.
30. Alexandre Mercier-Aubin, Ludwig Dumetz, Jonathan Gaudreault, and Claude-Guy Quimper. The confidence constraint: A step towards stochastic cp solvers. In International Conference on Principles and Practice of Constraint Programming, pages 759-773. Springer, 2020.
31. Arkadi Nemirovski and Alexander Shapiro. Convex approximations of chance constrained programs. SIAM Journal on Optimization, 17(4):969-996, 2007.
32. François Pachet, Pierre Roy, Alexandre Papadopoulos, and Jason Sakellariou. Generating 1/f noise sequences as constraint satisfaction: The voss constraint. In Twenty-Fourth International Joint Conference on Artificial Intelligence, 2015.
33. Bernardo K Pagnoncelli, Shabbir Ahmed, and Alexander Shapiro. Sample average approximation method for chance constrained programming: theory and applications. Journal of optimization theory and applications, 142(2):399-416, 2009.
34. M Arenas Parra, A Bilbao Terol, B Pérez Gladish, and MV Rodrıguez Urıa. Solving a multiobjective possibilistic problem through compromise programming. European Journal of Operational Research, 164(3):748-759, 2005.
35. Judea Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan kaufmann, 1988.
36. Guillaume Perez, Steve Malalel, Gael Glorian, Victor Jung, Alexandre Papadopoulos, Marie Pelleau, Wijnand Suijlen, Jean-Charles Régin, and Arnaud Lallouet. Generalized confidence constraints. In Proceedings of the AAAI Conference on Artificial Intelligence, 2023.
37. Guillaume Perez, Brendan Rappazzo, and Carla Gomes. Extending the capacity of 1/f noise generation. In International Conference on Principles and Practice of Constraint Programming, pages 601-610. Springer, 2018.
38. Guillaume Perez and Jean-Charles Régin. MDDs are efficient modeling tools: An application to dispersion constraints. In Integration of AI and OR Techniques in Constraint Programming, 2017.
39. Guillaume Perez and Jean-Charles Régin. Mdds: Sampling and probability constraints. In International Conference on Principles and Practice of Constraint Programming, pages 226-242. Springer, 2017.
40. Gilles Pesant. Achieving domain consistency and counting solutions for dispersion constraints. INFORMS Journal on Computing, 27(4):690-703, 2015. URL: https://doi.org/10.1287/ijoc.2015.0654.
41. Warren B Powell. A unified framework for stochastic optimization. European Journal of Operational Research, 275(3):795-821, 2019.
42. Steven D Prestwich, Roberto Rossi, and S Armagan Tarim. Randomness as a constraint. In Principles and Practice of Constraint Programming: 21st International Conference, CP 2015, Cork, Ireland, August 31-September 4, 2015, Proceedings 21, pages 351-366. Springer, 2015.
43. Charles Prud’homme, Jean-Guillaume Fages, and Xavier Lorca. Choco solver documentation. TASC, INRIA Rennes, LINA CNRS UMR, 6241:13-42, 2016.
44. Jean-Charles Régin. Arc consistency for global cardinality constraints with costs. In International Conference on Principles and Practice of Constraint Programming, pages 390-404. Springer, 1999.
45. Francesca Rossi, Peter Van Beek, and Toby Walsh. Handbook of constraint programming. Elsevier, 2006.
46. Roberto Rossi, Brahim Hnich, S Armagan Tarim, and Steven Prestwich. Confidence-based reasoning in stochastic constraint programming. Artificial Intelligence, 228:129-152, 2015.
47. Christian Schulte and Guido Tack. View-based propagator derivation. Constraints, 18(1):75-107, 2013.
48. Kalika Suksomboon, Nobutaka Matsumoto, Shuichi Okamoto, Michiaki Hayashi, and Yusheng Ji. Configuring a software router by the erlang-k-based packet latency prediction. IEEE Journal on Selected Areas in Communications, 36(3):422-437, 2018.
49. S Armagan Tarim, Suresh Manandhar, and Toby Walsh. Stochastic constraint programming: A scenario-based approach. Constraints, 11:53-80, 2006.
50. Joseph A Tatman and Ross D Shachter. Dynamic programming and influence diagrams. IEEE transactions on systems, man, and cybernetics, 20(2):365-379, 1990.
51. Pascal Van Hentenryck and Laurent Michel. Domain views for constraint programming. In International Conference on Principles and Practice of Constraint Programming, pages 705-720. Springer, 2014.
52. Willem-Jan Van Hoeve, Gilles Pesant, and Louis-Martin Rousseau. On global warming: Flow-based soft global constraints. Journal of Heuristics, 12(4):347-373, 2006.
53. Toby Walsh. Stochastic constraint programming. In ECAI, volume 2, pages 111-115, 2002.
54. Weijun Xie and Shabbir Ahmed. On deterministic reformulations of distributionally robust joint chance constrained optimization problems. SIAM Journal on Optimization, 28(2):1151-1182, 2018.
55. Hui Zhang and Pu Li. Chance constrained programming for optimal power flow under uncertainty. IEEE Transactions on Power Systems, 26(4):2417-2424, 2011.
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