A multiplicative weights update algorithm for MINLP

Original Paper


We discuss an application of the well-known multiplicative weights update (MWU) algorithm to non-convex and mixed-integer non-linear programming. We present applications to: (a) the distance geometry problem, which arises in the positioning of mobile sensors and in protein conformation; (b) a hydro unit commitment problem arising in the energy industry, and (c) a class of Markowitz’ portfolio selection problems. The interest of the MWU with respect to one of its closest competitors (classic multi-start) is that it provides a relative approximation guarantee on a certain quality measure of the solution.



We are very grateful to Dr. Pascale Bendotti (EDF) for useful suggestions about the HUC problem. Luca Mencarelli is sponsored by a Ph.D. Fellowship from the FP7 Marie Curie ITN “MINO” project. Youcef Sahraoui is sponsored by a CIFRE Ph.D. Fellowship with Éléctricité De France (EDF). Leo Liberti was partly sponsored by the ANR Bip:Bip project under contract ANR-10-BINF-0003, and completed this work during a visiting term at IMECC, University of Campinas (SP), Brazil, sponsored by the Chaires Françaises dans l’état de São Paulo (CFSP) program, a collaboration between the French Consulate in São Paulo, and the three main universities in São Paulo: UNICAMP, USP, and UNESP.


  1. Arora S, Hazan E, Kale S (2005) Fast algorithms for approximate semidefinite programming using the multiplicative weights update method. In: Foundations of Computer Science, vol 46. FOCS, IEEE, New York, pp 339–348Google Scholar
  2. Arora S, Hazan E, Kale S (2012) The multiplicative weights update method: a meta-algorithm and applications. Theory Comput 8:121–164CrossRefGoogle Scholar
  3. Bahr A, Leonard J, Fallon M (2009) Cooperative localization for autonomous underwater vehicles. Int J Robot Res 28(6):714–728CrossRefGoogle Scholar
  4. Beasley JE (1990) OR-Library: distributing test problems by electronic mail. J Oper Res Soc 41(11):1069–1072CrossRefGoogle Scholar
  5. Beasley JE (1996) Obtaining test problems via internet. J Glob Optim 8(4):429–433CrossRefGoogle Scholar
  6. Beeker N, Gaubert S, Glusa C, Liberti L (2013) Is the distance geometry problem in NP? In: Mucherino A, Lavor C, Liberti L, Maculan N (eds) Distance geometry: theory, methods, and applications. Springer, New YorkGoogle Scholar
  7. Berman H, Westbrook J, Feng Z, Gilliland G, Bhat T, Weissig H, Shindyalov IN, Bourne P (2000) The protein data bank. Nucl Acid Res 28:235–242CrossRefGoogle Scholar
  8. Bienstock D (1996) Computational study of a family of mixed-integer quadratic programming problems. Math Program 74(2):121–140CrossRefGoogle Scholar
  9. Bonami P, Lee J (2007) BONMIN user’s manual. Technical report, IBM CorporationGoogle Scholar
  10. Bonami P, Lee J, Leyffer S, Waecher A (2011) More Branch-and-Bound experiments in convex nonlinear integer programming. Preprint ANL/MCS-P1949-0911. Argonne National Laboratory, Mathematics and Computer Science DivisionGoogle Scholar
  11. Borghetti A, D’Ambrosio C, Lodi A, Martello S (2015) Optimal scheduling of a multiunit hydro power station in a short-term planning horizon. In: Murty KG (ed) Case studies in operations research. International series in operations research & management science, vol 212, pp 167–181. Springer, New YorkGoogle Scholar
  12. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  13. Chang T-J, Meade N, Beasley JE, Sharaiha YM (2000) Heuristics for cardinality constrained portfolio optimization. Comput Oper Res 27(13):1271–1302CrossRefGoogle Scholar
  14. COIN-OR (2006) Introduction to IPOPT: a tutorial for downloading, installing, and using IPOPTGoogle Scholar
  15. Costa A, Hansen P, Liberti L (2010) Formulation symmetries incircle packing. In: Mahjoub R (ed) Proceedings of the international symposium on combinatorial optimization. Electronic notes in discrete mathematics, vol 36. Elsevier, Amsterdam, pp 1303–1310Google Scholar
  16. D’Ambrosio C, Ky Vu, Lavor C, Liberti L, Maculan N (2014) Solving distance geometry problems with interval data using formulation-based methods. Technical report, LIX Ecole Polytechnique (working paper)Google Scholar
  17. D’Ambrosio C, Mencarelli L (2014) Complex portfolio selection via convex mixed-integer quadratic approaches: a survey. Technical report, LIX, École Polytechnique (working paper)Google Scholar
  18. Ding Y, Krislock N, Qian J, Wolkowicz H (2010) Sensor network localization, Euclidean distance matrix completions, and graph realization. Optim Eng 11:45–66CrossRefGoogle Scholar
  19. Du H, Alechina N, Stock K, Jackson M (2013) The logic of NEAR andFAR. In: Tenbrink T et al (ed) COSIT. LNCS, vol 8116. Springer, Switzerland, pp 475–494Google Scholar
  20. Fischetti M, Lodi A (2003) Local branching. Math Program Ser B 98(1–3):23–47CrossRefGoogle Scholar
  21. Frangioni A, Gentile C (2006) Perspective cuts for a class of convex 0–1 mixed integer programs. Math Program Ser A 106(2):225–236CrossRefGoogle Scholar
  22. Gupta OK, Ravindran A (1985) Branch-and-Bound experiments in convex nonlinear integer programming. Manag Sci 31(12):1533–1546CrossRefGoogle Scholar
  23. Hansen P, Mladenović N (2001) Variable neighbourhood search: principles and applications. Eur J Oper Res 130:449–467CrossRefGoogle Scholar
  24. IBM (2010) ILOG CPLEX 12.2 User’s Manual, IBMGoogle Scholar
  25. Kannan R, Monma CL (1978) On the computational complexity of integer programming problems. In: Henn R, Korte B, Oettli W (eds) Optimization and operations research. Lecture notes in economics and mathematical systems, vol 157, pp 161–172. Springer, BerlinGoogle Scholar
  26. Konno H, Wijayanayake A (2001) Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Math Program Ser B 89(2):233–250CrossRefGoogle Scholar
  27. Lavor C, Liberti L, Maculan N (2006) Computational experience with the molecular distance geometry problem. In: Pintér J (ed) Global optimization: scientific and engineering case studies. Springer, Berlin, pp 213–225CrossRefGoogle Scholar
  28. Liberti L (2009) Reformulations in mathematical programming: definitions and systematics. RAIRO-RO 43(1):55–86CrossRefGoogle Scholar
  29. Liberti L, Lavor C, Maculan N, Mucherino A (2014) Euclidean distance geometry and applications. SIAM Rev 56(1):3–69CrossRefGoogle Scholar
  30. Malliavin T, Mucherino A, Nilges M (2013) Distance geometry in structural biology. In: Mucherino A, Lavor C, Liberti L, Maculan N (eds) Distance geometry: theory, methods, and applications. Springer, New YorkGoogle Scholar
  31. Maniezzo V, Stützle T, Voß S (eds) (2009) Hybridizing metaheuristics and mathematical programming. Annals of information systems, vol 10. Springer, New YorkGoogle Scholar
  32. Markowitz HM (1952) Portfolio selection. J Finan 7(1):77–91Google Scholar
  33. Plotkin S, Shmoys D, Tardos É (1995) Fast approximation algorithm for fractional packing and covering problems. Math Oper Res 20:257–301CrossRefGoogle Scholar
  34. Saxe J (1979) Embeddability of weighted graphs in \(k\)-space is strongly NP-hard. In: Proceedings of 17th Allerton conference in communications, control and computing, pp 480–489Google Scholar
  35. Scherer B, Martin D (2005) Introduction to modern portfolio optimization. Springer, BerlinCrossRefGoogle Scholar
  36. Schlick T (2002) Molecular modelling and simulation: an interdisciplinary guide. Springer, New YorkCrossRefGoogle Scholar
  37. Shaw DK, Liu S, Kopman L (2008) Lagrangian relaxation procedure for cardinality-constrained portfolio optimization. Optim Methods Softw 23(3):411–420CrossRefGoogle Scholar
  38. Singer A (2011) Angular synchronization by eigenvectors and semidefinite programming. Appl Comput Harmonic Anal 30:20–36CrossRefGoogle Scholar
  39. Sun X, Zheng X, Li D (2013) Recent advances in mathematical programming with semi-continuous variables and cardinality constraint. J Oper Res Soc China 1(1):55–77CrossRefGoogle Scholar
  40. Tahanan M, van Ackooij W, Frangioni A, Lacalandra F (2015) Large-scale unit commitment under uncertainty: a literature survey. 4OR 13:115–171CrossRefGoogle Scholar
  41. Xue H-G, Xu G-X, Feng Z-X (2006) Mean-variance portfolio optimal problem under concave transaction cost. Appl Math Comput 174(1):1–12Google Scholar

Copyright information

© EURO - The Association of European Operational Research Societies 2016

Authors and Affiliations

  • Luca Mencarelli
    • 1
  • Youcef Sahraoui
    • 1
    • 2
  • Leo Liberti
    • 1
  1. 1.CNRS LIX, École PolytechniquePalaiseauFrance
  2. 2.OSIRIS, EDF R&DClamartFrance

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