Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Addressing Unboundedness in Quadratically-Constrained Mixed-Integer Problems (2405.05978v2)

Published 6 May 2024 in math.OC, cs.AI, and cs.NE

Abstract: Mixed-integer (MI) quadratic models subject to quadratic constraints, known as All-Quadratic MI Programs, constitute a challenging class of NP-complete optimization problems. The particular scenario of unbounded integers defines a subclass that holds the distinction of being even undecidable [Jeroslow, 1973]. This complexity suggests a possible soft-spot for Mathematical Programming (MP) techniques, which otherwise constitute a good choice to treat MI problems. We consider the task of minimizing MI convex quadratic objective and constraint functions with unbounded decision variables. Given the theoretical weakness of white-box MP solvers to handle such models, we turn to black-box meta-heuristics of the Evolution Strategies (ESs) family, and question their capacity to solve this challenge. Through an empirical assessment of all-quadratic test-cases, across varying Hessian forms and condition numbers, we compare the performance of the CPLEX solver to modern MI ESs, which handle constraints by penalty. Our systematic investigation begins where the CPLEX solver encounters difficulties (timeouts as the search-space dimensionality increases, D>=30), and we report in detail on the D=64 case. Overall, the empirical observations confirm that black-box and white-box solvers can be competitive, where CPLEX is evidently outperformed on 13% of the cases. This trend is flipped when unboundedness is amplified by a significant translation of the optima, leading to a totally inferior performance of CPLEX at 83% of the cases. We also conclude that conditioning and separability are not intuitive factors in determining the hardness degree of this class of MI problems.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (46)
  1. R. G. Jeroslow, “There cannot be any algorithm for integer programming with quadratic constraints,” Operations Research, vol. 21, no. 1, pp. 221–224, 1973.
  2. New York: Cambridge University Press, 2004.
  3. P. M. Pardalos and S. A. Vavasis, “Quadratic programming with one negative eigenvalue is NP-hard,” Journal of Global optimization, vol. 1, no. 1, pp. 15–22, 1991.
  4. D. Whitley, “Next generation genetic algorithms,” in Proceedings of the Genetic and Evolutionary Computation Conference Companion, GECCO ’17, (New York, NY, USA), pp. 922–941, ACM, 2017.
  5. C. Blum and G. R. Raidl, Hybrid Metaheuristics: Powerful Tools for Optimization. Artificial Intelligence: Foundations, Theory, and Algorithms, Switzerland: Springer International Publishing, 2016.
  6. Y. Hong and D. Arnold, “Evolutionary mixed-integer optimization with explicit constraints,” in Proceedings of the Genetic and Evolutionary Computation Conference, GECCO ’23, (New York, NY, USA), p. 822–830, Association for Computing Machinery, 2023.
  7. T. P. Papalexopoulos, C. Tjandraatmadja, R. Anderson, J. P. Vielma, and D. Belanger, “Constrained discrete black-box optimization using mixed-integer programming,” in Proceedings of the 39th International Conference on Machine Learning (K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu, and S. Sabato, eds.), vol. 162 of Proceedings of Machine Learning Research, pp. 17295–17322, PMLR, 17–23 Jul 2022.
  8. C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Dover Books on Computer Science, Mineola, NY, USA: Dover Publications, 1998.
  9. C. Moore and S. Mertens, The Nature of Computation. Oxford, UK: Oxford University Press, 2011.
  10. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004.
  11. Natural Computing Series, Springer-Verlag Berlin Heidelberg, 2013.
  12. Cham: Springer International Publishing, 2018.
  13. C. A. Floudas, Nonlinear and mixed-integer optimization: fundamentals and applications. New York, NY, USA: Oxford University Press, 1995.
  14. N. V. Sahinidis, “Mixed-integer nonlinear programming 2018,” 2019.
  15. R. Li, M. T. Emmerich, J. Eggermont, T. Bäck, M. Schütz, J. Dijkstra, and J. H. Reiber, “Mixed integer evolution strategies for parameter optimization,” Evolutionary Computation, vol. 21, no. 1, pp. 29–64, 2013.
  16. J. F. Franco, M. J. Rider, and R. Romero, “A mixed-integer quadratically-constrained programming model for the distribution system expansion planning,” International Journal of Electrical Power & Energy Systems, vol. 62, pp. 265–272, 2014.
  17. New York, NY, USA: John Wiley and Sons, 2011.
  18. Berlin-Heidelberg, Germany: Springer-Verlag, 2012.
  19. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015.
  20. Switzeland: Springer International Publishing, 2018.
  21. Reading, MA: Addison Wesley, 1989.
  22. R. Hamano, S. Saito, M. Nomura, and S. Shirakawa, “CMA-ES with margin: lower-bounding marginal probability for mixed-integer black-box optimization,” in Proceedings of the Genetic and Evolutionary Computation Conference, GECCO ’22, (New York, NY, USA), p. 639–647, Association for Computing Machinery, 2022.
  23. G. Rudolph, “An Evolutionary Algorithm for Integer Programming,” in Parallel Problem Solving from Nature-PPSN III, pp. 139–148, Springer, 1994.
  24. T. Bäck and M. Schütz, “Evolution Strategies for Mixed Integer Optimization of Optical Multilayer Systems,” in Evolutionary Programming IV – Proc. Fourth Annual Conf. Evolutionary Programming, pp. 33–51, The MIT Press, 1995.
  25. R. Li, J. Eggermont, O. M. Shir, M. Emmerich, T. Bäck, J. Dijkstra, and J. Reiber, “Mixed-Integer Evolution Strategies with Dynamic Niching,” in Parallel Problem Solving from Nature - PPSN X, vol. 5199 of Lecture Notes in Computer Science, pp. 246–255, Springer, 2008.
  26. E. Reehuis and T. Bäck, “Mixed-integer evolution strategy using multiobjective selection applied to warehouse design optimization,” in Proceedings of the 12th annual conference on Genetic and evolutionary computation, GECCO ’10, (New York, NY, USA), pp. 1187–1194, ACM, 2010.
  27. T. Tušar, D. Brockhoff, and N. Hansen, “Mixed-integer benchmark problems for single- and bi-objective optimization,” in Proceedings of the Genetic and Evolutionary Computation Conference, GECCO ’19, (New York, NY, USA), p. 718–726, Association for Computing Machinery, 2019.
  28. T. Marty, Y. Semet, A. Auger, S. Héron, and N. Hansen, “Benchmarking CMA-ES with basic integer handling on a mixed-integer test problem suite,” in Proceedings of the Companion Conference on Genetic and Evolutionary Computation, GECCO ’23 Companion, (New York, NY, USA), p. 1628–1635, Association for Computing Machinery, 2023.
  29. J. Gondzio and E. A. Yildirim, “Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations,” Journal of Global Optimization, vol. 81, pp. 293–321, 2021.
  30. S. A. Vavasis, “Quadratic programming is in NP,” Information Processing Letters, vol. 36, no. 2, pp. 73–77, 1990.
  31. C. Bliek, P. Bonami, and A. Lodi, “Solving Mixed-Integer Quadratic Programming problems with IBM-CPLEX: a progress report,” in Proceedings of the 26th RAMP Symposium, Hosei University, Tokyo, 2014.
  32. E. R. Bixby, M. Fenelon, Z. Gu, E. Rothberg, and R. Wunderling, “MIP: Theory and practice — closing the gap,” in System Modelling and Optimization (M. J. D. Powell and S. Scholtes, eds.), (Boston, MA), pp. 19–49, Springer US, 2000.
  33. P. Belotti, C. Kirches, S. Leyffer, J. Linderoth, J. Luedtke, and A. Mahajan, “Mixed-integer nonlinear optimization,” Acta Numerica, vol. 22, p. 1–131, 2013.
  34. U. Raber, “A Simplicial Branch-and-Bound Method for Solving Nonconvex All-Quadratic Programs,” Journal of Global Optimization, vol. 13, pp. 417–432, 1998.
  35. Y. Zhao and S. Liu, “Global optimization algorithm for mixed integer quadratically constrained quadratic program,” Journal of Computational and Applied Mathematics, vol. 319, pp. 159–169, 2017.
  36. H. Markowitz, “Portfolio selection,” The Journal of Finance, vol. 7, no. 1, pp. 77–91, 1952.
  37. P. Bonami and M. A. Lejeune, “An exact solution approach for portfolio optimization problems under stochastic and integer constraints,” Operations Research, vol. 57, no. 3, pp. 650–670, 2009.
  38. T. Ibaraki and N. Katoh, Resource Allocation Problems: Algorithmic Approaches. Cambridge, MA, USA: MIT Press, 1988.
  39. J. Kingman, “A mathematical problem in population genetics,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 57, no. 3, pp. 574––582, 1961.
  40. I. M. Bomze, “Regularity versus degeneracy in dynamics, games, and optimization: A unified approach to different aspects,” SIAM Rev., vol. 44, no. 3, pp. 394–414, 2002.
  41. W. P. Adams, R. J. Forrester, and F. W. Glover, “Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs,” Discrete Optimization, vol. 1, no. 2, pp. 99–120, 2004.
  42. M. Asghari, A. M. Fathollahi-Fard, S. M. J. Mirzapour Al-e hashem, and M. A. Dulebenets, “Transformation and linearization techniques in optimization: A state-of-the-art survey,” Mathematics, vol. 10, p. 283, Jan 2022.
  43. A. Billionnet, S. Elloumi, and A. Lambert, “Linear reformulations of integer quadratic programs,” in Modelling, Computation and Optimization in Information Systems and Management Sciences (H. A. Le Thi, P. Bouvry, and T. Pham Dinh, eds.), (Berlin, Heidelberg), pp. 43–51, Springer Berlin Heidelberg, 2008.
  44. IBM ILOG, “MIQCP: mixed integer programs with quadratic terms in the constraints.” https://www.ibm.com/docs/, 2021.
  45. M. J. Best, Quadratic Programming with Computer Programs. Advances in Applied Mathematics, Boca Raton FL, USA: CRC Press, 2017.
  46. E. Rothberg, “An Evolutionary Algorithm for Polishing Mixed Integer Programming Solutions,” INFORMS Journal on Computing, vol. 19, no. 4, pp. 534–541, 2007.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets