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The unconstrained binary quadratic programming problem: a survey

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Abstract

In recent years the unconstrained binary quadratic program (UBQP) has grown in importance in the field of combinatorial optimization due to its application potential and its computational challenge. Research on UBQP has generated a wide range of solution techniques for this basic model that encompasses a rich collection of problem types. In this paper we survey the literature on this important model, providing an overview of the applications and solution methods.

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References

  • Alidaee B, Glover F, Kochenberger GA, Rego C (2005) A new modeling and solution approach for the number partitioning problem. J Appl Math Decis Sci 2005(2):113–121. doi:10.1155/JAMDS.2005.113

    Article  MATH  MathSciNet  Google Scholar 

  • Alidaee B, Kochenberger G, Lewis K, Lewis M, Wang H (2008) A new approach for modeling and solving set packing problems. Eur J Oper Res 186(2):504–512. doi:10.1016/j.ejor.2006.12.068

    Article  MATH  MathSciNet  Google Scholar 

  • Alidaee B, Kochenberger GA, Ahmadian A (1994) 0–1 Quadratic programming approach for optimum solutions of two scheduling problems. Int J Syst Sci 25(2):401–408. doi:10.1080/00207729408928968

    Article  MATH  MathSciNet  Google Scholar 

  • Alkhamis TM, Hasan M, Ahmed MA (1998) Simulated annealing for the unconstrained quadratic pseudo-Boolean function. Eur J Oper Res 108(3):641–652. doi:10.1016/S0377-2217(97)00130-6

    Article  MATH  Google Scholar 

  • Amini MM, Alidaee B, Kochenberger GA (eds) (1999) A scatter search approach to unconstrained quadratic binary programs. New ideas in optimization. McGraw-Hill Ltd., London

  • Barahona F (1986) A solvable case of quadratic 0–1 programming. Discret Appl Math 13(1):23–26. doi:10.1016/0166-218X(86)90065-X

    Article  MATH  MathSciNet  Google Scholar 

  • Barahona F, Grotschel M, Junger M, Reinelt G (1988) An application of combinatorial optimization to statistical. Oper Res 36(3):493

    Article  MATH  Google Scholar 

  • Barahona F, Junger M, Reinelt G (1989) Experiments in quadratic 0–1 programming. Math Program 44:127–137

    Article  MATH  MathSciNet  Google Scholar 

  • Beck A, Teboulle M (2000) Global optimality conditions for quadratic optimization problems with binary constraints. SIAM J Optim 11(1):179–188

    Article  MATH  MathSciNet  Google Scholar 

  • Beasley JE (1998) Heuristic algorithms for the unconstrained binary quadratic programming problem. PhD thesis, Imperial College, England

  • Billionnet A, Elloumi S (2007) Using a mixed integer quadratic programming solver for the unconstrained quadratic 0–1 problem. Math Program 109(1):55–68

    Article  MATH  MathSciNet  Google Scholar 

  • Billionnet A, Sutter A (1994) Minimization of a quadratic pseudo-Boolean function. Eur J Oper Res 78(1):106–115. doi:10.1016/0377-2217(94)90125-2

    Article  MATH  Google Scholar 

  • Bomze IM, Budinich M, Pardalos PM, Pelillo M (1999) The maximum clique problem. In: Handbook of combinatorial optimization. Springer, Berlin, pp 1–74

  • Boros E, Hammer P, Sun X (1989) The DDT method for quadratic 0–1 minimization. RUTCOR Research Center, RRR:39–89

  • Boros E, Hammer PL (1991) The max-cut problem and quadratic 0–1 optimization polyhedral aspects, relaxations and bounds. Ann Oper Res 33(1–4):151–180

    Article  MATH  MathSciNet  Google Scholar 

  • Boros E, Hammer PL (2002) Pseudo-Boolean optimization. Discret Appl Math 123(1–3):155–225. doi:10.1016/S0166-218X(01)00341-9

    Article  MATH  MathSciNet  Google Scholar 

  • Boros E, Hammer PL, Tavares G (2006) Preprocessing of Unconstrained Quadratic Binary Optimization. Rutcor Research Report, vol 13

  • Boros E, Hammer PL, Tavares G (2007) Local search heuristics for quadratic unconstrained binary optimization (QUBO). J Heuristics 13(2):99–132

    Article  Google Scholar 

  • Cai Y, Wang J, Yin J, Zhou Y (2011) Memetic clonal selection algorithm with EDA vaccination for unconstrained binary quadratic programming problems. Expert Syst Appl 38(6):7817–7827. doi:10.1016/j.eswa.2010.12.124

    Article  Google Scholar 

  • Carraesi P, Malucelli F, Farinaccio F (1995) Testing optimality for quadratic 0-1 unconstrained problems. ZOR-Math Methods Oper Res 42:295–311

    Article  MathSciNet  Google Scholar 

  • Carraesi P, Farinaccio F, Malucelli F (1999) Testing optimality for quadratic 0-1 problems. Math Program 85:407–421

    Article  MathSciNet  Google Scholar 

  • Carter MW (1984) The indefinite zero-one quadratic problem. Discret Appl Math 7(1):23–44

    Article  MATH  Google Scholar 

  • De Simone C, Diehl M, Jünger M, Mutzel P, Reinelt G, Rinaldi G (1995) Exact ground states of Ising spin glasses: new experimental results with a branch-and-cut algorithm. J Stat Phys 80(1–2):487–496

    Article  MATH  Google Scholar 

  • Douiri SM, Elbernouss S (2012) The unconstrained binary quadratic programming for the sum coloring problem. Mod Appl Sci 6(9):26–33. doi:10.5539/mas.v6n9p26

    Article  Google Scholar 

  • Gao D, Ruan N (2010) Solutions to quadratic minimization problems with box and integer constraints. J Global Optim 47:463–484. doi:10.1007/s10898-009-9469-0

    Article  MATH  MathSciNet  Google Scholar 

  • Glover F, Alidaee B, Rego C, Kochenberger G (2002) One-pass heuristics for large-scale unconstrained binary quadratic problems. Eur J Oper Res 137(2):272–287. doi:10.1016/S0377-2217(01)00209-0

    Article  MATH  MathSciNet  Google Scholar 

  • Glover F, Kochenberger G, Alidaee B, Amini M (1999) Tabu search with critical event memory: an enhanced application for binary quadratic programs. In: Meta-Heuristics. Springer, Berlin, pp 93–109

  • Glover F, Kochenberger GA, Alidaee B (1998) Adaptive memory tabu search for binary quadratic programs. Manag Sci 44(3):336–345

    Article  MATH  Google Scholar 

  • Glover F, Lü Z, Hao J-K (2010) Diversification-driven tabu search for unconstrained binary quadratic problems. 4OR 8(3):239–253

    Article  MATH  MathSciNet  Google Scholar 

  • Gueye S, Michelon P (2009) A linearization framework for unconstrained quadratic (0-1) problems. Discret Appl Math 157(6):1255–1266. doi:10.1016/j.dam.2008.01.028

    Article  MATH  MathSciNet  Google Scholar 

  • Gulati VP, Gupta SK, Mittal AK (1984) Unconstrained quadratic bivalent programming problem. Eur J Oper Res 15(1):121–125. doi:10.1016/0377-2217(84)90055-9

    Article  MATH  MathSciNet  Google Scholar 

  • Hammer P, Shlifer E (1971) Applications of pseudo-Boolean methods to economic problems. Theor Decis 1(3):296–308. doi:10.1007/BF00139572

    Article  MATH  Google Scholar 

  • Hammer PL, Rudeanu S (1968) Boolean methods in operations research and related areas, vol 5. Springer, Berlin

    Book  MATH  Google Scholar 

  • Hanafi S, Rebai AR, Vasquez M (2013) Several versions of the devour digest tidy-up heuristic for unconstrained binary quadratic problems. J Heuristics 19(4):645–677

    Article  Google Scholar 

  • Hansen P (1979) Methods of nonlinear 0-1 programming. Ann Discret Math 5:53–70. doi:10.1016/S0167-5060(08)70343-1

    Article  MATH  Google Scholar 

  • Hansen P, Jaumard B (1990) Algorithms for the maximum satisfiability problem. Computing 44(4):279–303

    Article  MATH  MathSciNet  Google Scholar 

  • Hansen P, Jaumard B, Mathon V (1993) State-of-the-art survey-constrained nonlinear 0-1 programming. ORSA J Comput 5(2):97–119

    Article  MATH  MathSciNet  Google Scholar 

  • Hansen P, Jaumard B, Meyer C (2000) Exact sequential algorithms for additive clustering. Groupe d’études et de recherche en analyse des décisions, Montréal

    Google Scholar 

  • Helmberg C, Rendl F (1998) Solving quadratic (0, 1)-problems by semidefinite programs and cutting planes. Math Program Ser B 82(3):291–315

    Article  MATH  MathSciNet  Google Scholar 

  • Huang H-X, Pardalos PM, Prokopyev OA (2006) Lower bound improvement and forcing rule for quadratic binary programming. Comput Optim Appl 33(2–3):187–208

    Article  MATH  MathSciNet  Google Scholar 

  • Iasemidis L, Pardalos P, Sackellares J, Shiau D-S (2001) Quadratic binary programming and dynamical system approach to determine the predictability of epileptic seizures. J Comb Optim 5(1):9–26

    Article  MATH  MathSciNet  Google Scholar 

  • Jeyakumar V, Rubinov AM, Wu ZY (2007) Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Math Program Ser A 110:521–541. doi:10.1007/s10107-006-0012-5

    Article  MATH  MathSciNet  Google Scholar 

  • Kalantari B, Bagchi A (1990) An algorithm for quadratic zero-one programs. Naval Res Logist (NRL) 37(4):527–538

    Article  MathSciNet  Google Scholar 

  • Katayama K, Narihisa H (2001) Performance of simulated annealing-based heuristic for the unconstrained binary quadratic programming problem. Eur J Oper Res 134(1):103–119. doi:10.1016/S0377-2217(00)00242-3

    Article  MATH  MathSciNet  Google Scholar 

  • Katayama K, Tani M, Narihisa H (2000) Solving large binary quadratic programming problems by effective genetic local search algorithm. In: Proceedings of 2000 Genetic and Evolutionary Computation Conference, pp 643–650

  • Kernighan B, Lin S (1970) An eflicient heuristic procedure for partitioning graphs. Bell Syst Tech J 49:291–307

    Article  MATH  Google Scholar 

  • Kochenberger G, Alidaee B, Glover F, Wang H (2007) An effective modeling and solution approach for the generalized independent set problem. Optim Lett 1(1):111–117

    Article  MATH  MathSciNet  Google Scholar 

  • Kochenberger G, Glover F, Alidaee B, Lewis K (2005a) Using the unconstrained quadratic program to model and solve Max 2-SAT problems. Int J Oper Res 1(1):89–100

    Article  MATH  MathSciNet  Google Scholar 

  • Kochenberger G, Glover F, Alidaee B, Rego C (2005b) An unconstrained quadratic binary programming approach to the vertex coloring problem. Ann Oper Res 139(1–4):229–241. doi:10.1007/s10479-005-3449-7

    Article  MATH  MathSciNet  Google Scholar 

  • Kochenberger G, Glover F, Alidaee B, Wang H (2005c) Clustering of microarray data via clique partitioning. J Comb Optim 10(1):77–92

    Article  MATH  MathSciNet  Google Scholar 

  • Kochenberger GA, Hao J-K, Lü Z, Wang H, Glover F (2013) Solving large scale max cut problems via tabu search. J Heuristics 19(4):565–571

    Article  Google Scholar 

  • Krarup J, Pruzan P (1978) Computer-aided layout design. In: Balinski ML, Lemarechal C (eds) Mathematical Programming in Use, vol 9. Mathematical Programming Studies. Springer, Berlin, pp 75–94. doi:10.1007/BFb0120827

  • Laughhunn D (1970) Quadratic binary programming with application to capital-budgeting problems. Oper Res 18(3):454–461

    Article  MATH  Google Scholar 

  • Lewis M, Alidaee B, Glover F, Kochenberger G (2009) A note on xQx as a modelling and solution framework for the Linear Ordering Problem. Int J Oper Res 5(2):152–162

    Article  MATH  Google Scholar 

  • Lewis M, Alidaee B, Kochenberger G (2005) Using xQx to model and solve the uncapacitated task allocation problem. Oper Res Lett 33(2):176–182. doi:10.1016/j.orl.2004.04.014

    Article  MATH  MathSciNet  Google Scholar 

  • Lewis M, Kochenberger G, Alidaee B (2008) A new modeling and solution approach for the set-partitioning problem. Comput Oper Res 35(3):807–813. doi:10.1016/j.cor.2006.04.002

    Article  MATH  MathSciNet  Google Scholar 

  • Lewis M, Kochenberger G, Wang H, Glover F (2013) Exact Solutions to Generalized Vertex Covering Problems: A Comparison of Two Models. working paper

  • Li D, Sun XL, Liu CL (2012) An exact solution method for unconstrained quadratic 0 1 programming: a geometric approach. J Global Optim 52(4):797–829

    Article  MATH  MathSciNet  Google Scholar 

  • Li G (2012) Global quadratic minimization over bivalent constraints: necessary and sufficient global optimality condition. J Optim Theory 52:710–726. doi:10.1007/s10957-011-9930-3

    Article  Google Scholar 

  • Lodi A, Allemand K, Liebling TM (1999) An evolutionary heuristic for quadratic 0–1 programming. Eur J Oper Res 119(3):662–670. doi:10.1016/S0377-2217(98)00359-2

    Article  MATH  Google Scholar 

  • Lu C, Fang A, Jin Q, Wang Z, Xing W (2011) KKT solution and conic relaxation for solving quadratically constrained quadratic programming problems. SIAM J Optim 21(4):1475–1490

    Article  MATH  MathSciNet  Google Scholar 

  • Lü Z, Glover F, Hao J-K (2010a) A hybrid metaheuristic approach to solving the UBQP problem. Eur J Oper Res 207(3):1254–1262. doi:10.1016/j.ejor.2010.06.039

    Article  MATH  Google Scholar 

  • Lü Z, Hao J-K, Glover F (2010b) A study of memetic search with multi-parent combination for UBQP. In: Evolutionary Computation in Combinatorial Optimization. Springer, Berlin, pp 154–165

  • Lü Z, Hao J-K, Glover F (2011) Neighborhood analysis: a case study on curriculum-based course timetabling. J Heuristics 17(2):97–118. doi:10.1007/s10732-010-9128-0

    Article  Google Scholar 

  • Mahdavi Pajouh F, Balasundaram B, Prokopyev OA (2013) On characterization of maximal independent sets via quadratic optimization. J Heuristics 19(4):629–644

    Article  Google Scholar 

  • Mauri GR, Lorena LAN (2011) Lagrangean decompositions for the unconstrained binary quadratic programming problem. Int Trans Oper Res 18(2):257–270. doi:10.1111/j.1475-3995.2009.00743.x

    Article  MATH  MathSciNet  Google Scholar 

  • Mauri GR, Lorena LAN (2012a) A column generation approach for the unconstrained binary quadratic programming problem. Eur J Oper Res 217(1):69–74. doi:10.1016/j.ejor.2011.09.016

    Article  MATH  MathSciNet  Google Scholar 

  • Mauri GR, Lorena LAN (2012b) Improving a Lagrangian decomposition for the unconstrained binary quadratic programming problem. Comput Oper Res 39(7):1577–1581. doi:10.1016/j.cor.2011.09.008

    Article  MATH  MathSciNet  Google Scholar 

  • Merz P, Freisleben B (1999) Genetic algorithms for binary quadratic programming. In: Proceedings of the genetic and evolutionary computation conference, Citeseer, pp 417–424

  • Merz P, Freisleben B (2002) Greedy and local search heuristics for unconstrained binary quadratic programming. J Heuristics 8(2):197–213

    Article  MATH  Google Scholar 

  • Merz P, Katayama K (2004) Memetic algorithms for the unconstrained binary quadratic programming problem. Biosystems 78(1–3):99–118. doi:10.1016/j.biosystems.2004.08.002

    Article  Google Scholar 

  • Neven H, Rose G, Macready WG (2008) Image recognition with an adiabatic quantum computer I. Mapping to quadratic unconstrained binary optimization. arXiv:0804.4457

  • Oosten M, Rutten J, Spieksma F (2001) The Clique partitioning problem: facets and patching facets. Networks 38(4):209–226

    Article  MATH  MathSciNet  Google Scholar 

  • Palubeckis G (1995) A heuristic-based branch and bound algorithm for unconstrained quadratic zero-one programming. Computing 54(4):283–301. doi:10.1007/BF02238228

    Article  MATH  MathSciNet  Google Scholar 

  • Palubeckis G (2004) Multistart tabu search strategies for the unconstrained binary quadratic optimization problem. Ann Oper Res 131(1–4):259–282. doi:10.1023/B:ANOR.0000039522.58036.68

    Article  MATH  MathSciNet  Google Scholar 

  • Palubeckis G (2006) Iterated tabu search for the unconstrained binary quadratic optimization problem. Informatica 17(2):279–296

    MATH  MathSciNet  Google Scholar 

  • Palubeckis G, Tomkevicius A (2002) GRASP implementations for the unconstrained binary quadratic optimization problem. Inf Technol Control 24:14–20

    Google Scholar 

  • Pan S, Tan T, Jiang Y (2008) A global continuation algorithm for solving binary quadratic programming problems. Comput Optim Appl 41(3):349–362. doi:10.1007/s10589-007-9110-4

    Article  MATH  MathSciNet  Google Scholar 

  • Pardalos PM, Jha S (1991) Graph separation techniques for quadratic zero-one programming. Comput Math Appl 21(6–7):107–113. doi:10.1016/0898-1221(91)90165-Z

    Article  MATH  MathSciNet  Google Scholar 

  • Pardalos PM, Jha S (1992) Complexity of uniqueness and local search in quadratic 0-1 programming. Oper Res Lett 11(2):119–123. doi:10.1016/0167-6377(92)90043-3

    Article  MATH  MathSciNet  Google Scholar 

  • Pardalos PM, Prokopyev OA, Busygin S (2006) Continuous approaches for solving discrete optimization problems. In: Handbook on modelling for discrete optimization. Springer, Berlin, pp 39–60

  • Pardalos PM, Rodgers GP (1990a) Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45(2):131–144

    Article  MATH  MathSciNet  Google Scholar 

  • Pardalos PM, Rodgers GP (1990b) Parallel branch and bound algorithms for quadratic zero-one programs on the hypercube architecture. Ann Oper Res 22(1–4):271–292

    Article  MATH  MathSciNet  Google Scholar 

  • Pardalos PM, Rodgers GP (1992) A branch and bound algorithm for the maximum clique problem. Comput Oper Res 19(5):363–375. doi:10.1016/0305-0548(92)90067-F

    Article  MATH  Google Scholar 

  • Pardalos PM, Xue J (1994) The maximum clique problem. J Global Optim 4(3):301–328

    Article  MATH  MathSciNet  Google Scholar 

  • Pham Dinh T, Nguyen Canh N, Le Thi HA (2010) An efficient combined DCA and B&B using DC/SDP relaxation for globally solving binary quadratic programs. J Global Optim 48(4):595–632

    Article  MATH  MathSciNet  Google Scholar 

  • Picard J-C (1976) Maximal closure of a graph and applications to combinatorial problems. Manag Sci 22(11):1268–1272. doi:10.2307/2630227

    Article  MATH  MathSciNet  Google Scholar 

  • Pinar MC (2004) Sufficient global optimality conditions for bivalent quadratic optimization. J Optim Theory Appl 122(2):433–440

    Article  MATH  MathSciNet  Google Scholar 

  • Rao MR (1971) Cluster analysis and mathematical programming. J Am Stat Assoc 66(335):622–626. doi:10.1080/01621459.1971.10482319

    Article  MATH  Google Scholar 

  • Rhys J (1970) A selection problem of shared fixed costs and network flows. Manag Sci 17(3):200–207

    Article  MATH  Google Scholar 

  • Shylo V, Shylo O (2011) Systems analysis solving unconstrained binary quadratic programming problem by global equilibrium search. Cybern Syst Anal 47(6):889–897. doi:10.1007/s10559-011-9368-5

    Article  Google Scholar 

  • Sun XL, Liu CL, Li D, Gao JJ (2012) On duality gap in binary quadratic programming. J Global Optim 53:255–269. doi:10.1007/s10898-011-9683-4

    Article  MATH  MathSciNet  Google Scholar 

  • Wang F, Xu Z (2013) Metaheuristics for robust graph coloring. J Heuristics 19(4):529–548

    Article  Google Scholar 

  • Wang H, Alidaee B, Glover F, Kochenberger G (2006) Solving group technology problems via clique partitioning. Int J Flex Manuf Syst 18(2):77–77

    Article  MATH  Google Scholar 

  • Wang J, Zhou Y, Yin J (2011) Combining tabu Hopfield network and estimation of distribution for unconstrained binary quadratic programming problem. Expert Syst Appl 38(12):14870–14881. doi:10.1016/j.eswa.2011.05.060

    Article  Google Scholar 

  • Wang Y, Lü Z, Glover F, Hao J-K (2012a) A multilevel algorithm for large unconstrained binary quadratic optimization. In: Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems. Springer, Berlin, pp 395–408

  • Wang Y, Lü Z, Glover F, Hao J-K (2012b) Path relinking for unconstrained binary quadratic programming. Eur J Oper Res 223(3):595–604. doi:10.1016/j.ejor.2012.07.012

    Article  Google Scholar 

  • Wang Y, Lü Z, Glover F, Hao J-K (2012c) Probabilistic GRASP-tabu search algorithms for the UBQP problem. Comput Oper Res 40:3100–3107

    Article  Google Scholar 

  • Williams HP (1985) Model building in linear and integer programming. In: Schittkowski K (ed) Computational mathematical programming, vol 15. NATO ASI Series. Springer, Berlin, pp 25–53. doi:10.1007/978-3-642-82450-0_2

  • Witzgall C (1975) Mathematical methods of site selection for Electronic Message Systems (EMS). NBS Internal report, NBS

  • Xia Y (2009) New optimality conditions for quadratic optimization problems with binary constraints. Optim Lett 3:253–263. doi:10.1007/s11590-008-0105-6

    Article  MATH  MathSciNet  Google Scholar 

  • Zheng XJ, Sun XL, Li D, Xu YF (2012) On zero duality gap in nonconvex quadratic programming problems. J Global Optim 52:229–242. doi:10.1007/s10898-011-9660-y

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Business Administration, University of Colorado at Denver, Denver, CO , 80217, USA

    Gary Kochenberger

  2. LERIA, Université d’Angers, 2 Boulevard Lavoisier, 49045 , Angers, France

    Jin-Kao Hao

  3. OptTek Inc., Boulder, CO , 80302, USA

    Fred Glover

  4. Craig School of Business, Missouri Western State University, Saint Joseph, MO , 64507, USA

    Mark Lewis

  5. School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan , 430074, China

    Zhipeng Lü

  6. Sanchez School of Business, Texas A&M International University, Laredo, TX , 78041, USA

    Haibo Wang

  7. School of Management, Northwestern Polytechnical University, 127 Youyi West Road, Xi’an , 710072, China

    Yang Wang

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  1. Gary Kochenberger
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Kochenberger, G., Hao, JK., Glover, F. et al. The unconstrained binary quadratic programming problem: a survey. J Comb Optim 28, 58–81 (2014). https://doi.org/10.1007/s10878-014-9734-0

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