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Optimality-based domain reduction for inequality-constrained NLP and MINLP problems

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Abstract

In spatial branch-and-bound algorithms, optimality-based domain reduction is normally performed after solving a node and relies on duality information to reduce ranges of variables. In this work, we propose novel optimality conditions for NLP and MINLP problems and apply them for domain reduction prior to solving a node in branch-and-bound. The conditions apply to nonconvex inequality-constrained problems for which we exploit monotonicity properties of objectives and constraints. We develop three separate reduction algorithms for unconstrained, one-constraint, and multi-constraint problems. We use the optimality conditions to reduce ranges of variables through forward and backward bound propagation of gradients respective to each decision variable. We describe an efficient implementation of these techniques in the branch-and-bound solver BARON. The implementation dynamically recognizes and ignores inactive constraints at each node of the search tree. Our computations demonstrate that the proposed techniques often reduce the solution time and total number of nodes for continuous problems; they are less effective for mixed-integer programs.

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References

  1. Achterberg, T., Wunderling, R.: Mixed integer programming: analyzing 12 years of progress. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization, pp. 449–481. Springer, Berlin (2013)

    MATH  Google Scholar 

  2. Balakrishnan, V., Boyd, S.: Global optimization in control system analysis and design. In: Leondes, C.T. (ed.) Control and Dynamic Systems, Advances in Theory and Applications. Academic Press, New York (1992)

    Google Scholar 

  3. Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24, 597–634 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Bixby, R., Rothberg, E.: Progress in computational mixed integer programming-a look back from the other side of the tipping point. Ann. Oper. Res. 149, 37–41 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Bliek, C., Spellucci, P., Vicente, L. N., Neumaier, A., Granvilliers, L., Huens, E., Hentenryck, P., Sam-Haroud, D., Faltings, B.: Algorithms for solving nonlinear constrained and optimization problems: the state of the art (2001). https://www.mat.univie.ac.at/~neum/ms/StArt.pdf

  6. Brearley, A.L., Mitra, G., Williams, H.P.: Analysis of mathematical programming problems prior to applying the simplex algorithm. Math. Program. 8, 54–83 (1975)

    MathSciNet  MATH  Google Scholar 

  7. Bussieck, M.R., Drud, A.S., Meeraus, A.: MINLPLib-A collection of test models for mixed-integer nonlinear programming. INFORMS J. Comput. 15, 114–119 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Catalão, J.P.S., Pousinho, H.M.I., Mendes, V.M.F.: Hydro energy systems management in Portugal: profit-based evaluation of a mixed-integer nonlinear approach. Energy 36, 500–507 (2011)

    Google Scholar 

  9. Cozad, A., Sahinidis, N.V.: A global MINLP approach to symbolic regression. Math. Program. 170, 97–119 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Faria, D.C., Bagajewicz, M.J.: A new approach for global optimization of a class of MINLP problems with applications to water management and pooling problems. AIChE J. 58, 2320–2335 (2012)

    Google Scholar 

  11. GLOBAL Library. http://www.gamsworld.org/global/globallib.htm

  12. Gondzio, J.: Presolve analysis of linear programs prior to applying an interior point method. INFORMS J. Comput. 9, 73–91 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Grossmann, I.E., Caballero, J.A., Yeomans, H.: Advances in mathematical programming for the synthesis of process systems. Lat. Am. Appl. Res. 30, 263–284 (2000)

    Google Scholar 

  14. Harjunkoski, I., Westerlund, T., Pörn, R., Skrifvars, H.: Different transformations for solving non-convex trim-loss problems by MINLP. Eur. J. Oper. Res. 105, 594–603 (1998)

    MATH  Google Scholar 

  15. Heinz, S., Schulz, J., Beck, J.C.: Using dual presolving reductions to reformulate cumulative constraints. Constraints 18, 166–201 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Hoffman, K.L., Padberg, M.: Improving LP-representations of zero-one linear programs for branch-and-cut. ORSA J. Comput. 3, 121–134 (1991)

    MATH  Google Scholar 

  17. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, Third edn. Springer, Berlin (1996)

    MATH  Google Scholar 

  18. Imbert, J., Van Hentenryck, P.: Redundancy elimination with a lexicographic solved form. Ann. Math. Artif. Intell. 17, 85–106 (1996)

    MathSciNet  MATH  Google Scholar 

  19. Jezowski, J.: Review of water network design methods with literature annotations. Ind. Eng. Chem. Res. 49, 4475–4516 (2010)

    Google Scholar 

  20. Khajavirad, A., Michalek, J.J., Sahinidis, N.V.: Relaxations of factorable functions with convex-transformable intermediates. Math. Program. 144, 107–140 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Khajavirad, A., Sahinidis, N.V.: A hybrid LP/NLP paradigm for global optimization relaxations. Math. Program. Comput. 10, 383–421 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Lin, Y., Schrage, L.: The global solver in the LINDO API. Optim. Methods Softw. 24, 657–668 (2009)

    MathSciNet  MATH  Google Scholar 

  23. Mahajan, A.I.: Presolving mixed-integer linear programs. In: Cochran, J.J., Cox, L.A., Keskinocak, P., Kharoufeh, J.P., Smith, J.C. (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2010)

    Google Scholar 

  24. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I-convex underestimating problems. Math. Program. 10, 147–175 (1976)

    MATH  Google Scholar 

  25. McCormick, G.P.: Nonlinear Programming: Theory, Algorithms and Applications. Wiley, Hoboken (1983)

    MATH  Google Scholar 

  26. MINLP2 Library. http://www.minlplib.org

  27. Misener, R., Floudas, ChA: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59, 503–526 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Mitsos, A., Chachuat, B., Barton, P.I.: Mccormick-based relaxations of algorithms. SIAM J. Optim. 20, 573–601 (2009)

    MathSciNet  MATH  Google Scholar 

  29. Nemhauser, G.L., Savelsbergh, M.P., Sigismondi, G.C.: MINTO, a mixed INTeger optimizer. Oper. Res. Lett. 15, 47–58 (1994)

    MathSciNet  MATH  Google Scholar 

  30. Neumaier, A.: Molecular modeling of proteins and mathematical prediction of protein structure. SIAM Rev. 39, 407–460 (1997)

    MathSciNet  MATH  Google Scholar 

  31. Princeton Library. http://www.gamsworld.org/performance/princetonlib/princetonlib.htm

  32. Puranik, Y., Sahinidis, N.V.: Bounds tightening based on optimality conditions for nonconvex box-constrained optimization. J. Glob. Optim. 67, 59–77 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Puranik, Y., Sahinidis, N.V.: Domain reduction techniques for global NLP and MINLP optimization. Constraints 22, 338–376 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Ryoo, H.S., Sahinidis, N.V.: Global optimization of nonconvex NLPs and MINLPs with applications in process design. Comput. Chem. Eng. 19, 551–566 (1995)

    Google Scholar 

  35. Ryoo, H.S., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8, 107–139 (1996)

    MathSciNet  MATH  Google Scholar 

  36. Sahinidis, N. V.: Global optimization and constraint satisfaction: the branch-and-reduce approach. In: Bliek, C., Jermann, C., Neumaier, A., (eds.) Global Optimization and Constraint Satisfaction, Lecture Notes in Computer Science, vol. 2861, pp. 1–16. Springer, Berlin (2003)

    MATH  Google Scholar 

  37. Sahinidis, N.V.: Mixed-integer nonlinear programming 2018. Optim. Eng. 20, 301–306 (2019)

    MathSciNet  Google Scholar 

  38. Savelsbergh, M.W.P.: Preprocessing and probing for mixed integer programming problems. ORSA J. Comput. 6, 445–454 (1994)

    MathSciNet  MATH  Google Scholar 

  39. Schichl, H., Neumaier, A.: Interval analysis on directed acyclic graphs for global optimization. J. Glob. Optim. 33, 541–562 (2005)

    MathSciNet  MATH  Google Scholar 

  40. Shectman, J.P., Sahinidis, N.V.: A finite algorithm for global minimization of separable concave programs. J. Glob. Optim. 12, 1–36 (1998)

    MathSciNet  MATH  Google Scholar 

  41. Sherali, H.D., Wang, H.: Global optimization of nonconvex factorable programming problems. Math. Program. 89, 459–478 (2001)

    MathSciNet  MATH  Google Scholar 

  42. Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99, 563–591 (2004)

    MathSciNet  MATH  Google Scholar 

  43. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)

    MathSciNet  MATH  Google Scholar 

  44. Van Roy, T.J., Wolsey, L.A.: Solving mixed integer programming problems using automatic reformulation. Oper. Res. 35, 45–57 (1987)

    MathSciNet  MATH  Google Scholar 

  45. Vigerske, S., Gleixner, A.: SCIP: global optimization of mixed-integer nonlinear programs in a branch-and-cut framework. Optim. Methods Softw. 33(3), 563–593 (2018)

    MathSciNet  MATH  Google Scholar 

  46. Vigerske, S.: Decomposition of multistage stochastic programs and a constraint integer programming approach to mixed-integer nonlinear programming. PhD thesis, Humboldt Universität zu Berlin (2013)

  47. Wilson, Z.T., Sahinidis, N.V.: The ALAMO approach to machine learning. Comput. Chem. Eng. 106, 785–795 (2017)

    Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, China

    Yi Zhang & Gang Rong

  2. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA

    Nikolaos V. Sahinidis & Carlos Nohra

Authors
  1. Yi Zhang
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  2. Nikolaos V. Sahinidis
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  3. Carlos Nohra
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  4. Gang Rong
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Correspondence to Nikolaos V. Sahinidis.

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This research was supported by the Smart Manufacturing Comprehensive Standardization and New Mode Application Project of MIIT (No. 2017-ZJ-003), Science Fund for Creative Research Groups of NSFC (Grant No. 61621002), and China Scholarships Council (CSC).

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Zhang, Y., Sahinidis, N.V., Nohra, C. et al. Optimality-based domain reduction for inequality-constrained NLP and MINLP problems. J Glob Optim 77, 425–454 (2020). https://doi.org/10.1007/s10898-020-00886-z

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  • DOI: https://doi.org/10.1007/s10898-020-00886-z

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