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Efficiently solving linear bilevel programming problems using off-the-shelf optimization software

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Abstract

Many optimization models in engineering are formulated as bilevel problems. Bilevel optimization problems are mathematical programs where a subset of variables is constrained to be an optimal solution of another mathematical program. Due to the lack of optimization software that can directly handle and solve bilevel problems, most existing solution methods reformulate the bilevel problem as a mathematical program with complementarity conditions (MPCC) by replacing the lower-level problem with its necessary and sufficient optimality conditions. MPCCs are single-level non-convex optimization problems that do not satisfy the standard constraint qualifications and therefore, nonlinear solvers may fail to provide even local optimal solutions. In this paper we propose a method that first solves iteratively a set of regularized MPCCs using an off-the-shelf nonlinear solver to find a local optimal solution. Local optimal information is then used to reduce the computational burden of solving the Fortuny-Amat reformulation of the MPCC to global optimality using off-the-shelf mixed-integer solvers. This method is tested using a wide range of randomly generated examples. The results show that our method outperforms existing general-purpose methods in terms of computational burden and global optimality.

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References

  • Bard JF (1991) Some properties of the bilevel programming problem. J Optim Theory Appl 68(2):371–378

    Article  MathSciNet  MATH  Google Scholar 

  • Bard JF (1998) practical bilevel optimization: algorithms and applications. Springer, Berlin

    Book  MATH  Google Scholar 

  • Bard JF, Falk JE (1982) An explicit solution to the multi-level programming problem. Comput Oper Res 9(1):77–100

    Article  MathSciNet  Google Scholar 

  • Bard J, Moore J (1990) A branch and bound algorithm for the bilevel programming problem. SIAM J Sci Stat Comput 11(2):281–292

    Article  MathSciNet  MATH  Google Scholar 

  • Baringo L, Conejo AJ (2011) Wind power investment within a market environment. Appl Energy 88(9):3239–3247

    Article  Google Scholar 

  • Baringo L, Conejo AJ (2012) Wind power investment: a benders decomposition approach. IEEE Trans Power Syst 27(1):433–441

    Article  Google Scholar 

  • Baringo L, Conejo AJ (2013) Risk-constrained multi-stage wind power investment. IEEE Trans Power Syst 28(1):401–411

    Article  Google Scholar 

  • Baringo L, Conejo AJ (2014) Strategic wind power investment. IEEE Trans Power Syst 29(3):1250–1260

    Article  Google Scholar 

  • Ben-Ayed O, Blair CE (1990) Computational difficulties of bilevel linear programming. Oper Res 38:556–560

    Article  MathSciNet  MATH  Google Scholar 

  • Bialas WF, Karwan MH (1984) Two-level linear programming. Manag Sci 30:1004–1020

    Article  MathSciNet  MATH  Google Scholar 

  • Calvete HI, Galé C, Mateo PM (2008) A new approach for solving linear bilevel problems using genetic algorithms. Eur J Oper Res 188(1):14–28

    Article  MathSciNet  MATH  Google Scholar 

  • Candler W, Townsley R (1982) A linear two-level programming problem. Comput Oper Res 9(1):59–76

    Article  MathSciNet  Google Scholar 

  • Colson B, Marcotte P, Savard G (2005) Bilevel programming: a survey. 4OR 3(2):87–107

    Article  MathSciNet  MATH  Google Scholar 

  • Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153(1):235–256

    Article  MathSciNet  MATH  Google Scholar 

  • Dempe S (2002) Foundations of bilevel programming. Springer, Berlin

    MATH  Google Scholar 

  • Dempe S (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3):333–359

    Article  MathSciNet  MATH  Google Scholar 

  • Dempe S, Dutta J (2010) Is bilevel programming a special case of a mathematical program with complementarity constraints? Math Program 131(1–2):37–48

    MathSciNet  MATH  Google Scholar 

  • Dempe S, Franke S (2014) Solution algorithm for an optimistic linear Stackelberg problem. Comput Oper Res 41:277–281

    Article  MathSciNet  MATH  Google Scholar 

  • Dempe S, Zemkoho AB (2012) The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math Program 138(1–2):447–473

    MathSciNet  MATH  Google Scholar 

  • Dempe S, Dutta J, Mordukhovich BS (2007) New necessary optimality conditions in optimistic bilevel programming. Optimization 56(5–6):577–604

    Article  MathSciNet  MATH  Google Scholar 

  • Dempe S, Mordukhovich BS, Zemkoho AB (2014) Necessary optimality conditions in pessimistic bilevel programming. Optimization 63(4):505–533

    Article  MathSciNet  MATH  Google Scholar 

  • Dempe S, Kalashnikov V, Pérez-Valdés GA, Kalashnikova N (2015) Bilevel programming problems: theory, algorithms and applications to energy networks. Energy systems. Springer, Berlin

    Book  MATH  Google Scholar 

  • Fletcher R, Leyffer S (2002) Numerical experience with solving MPECs as NLPs. Technical report, Department of Mathematics and Computer Science, University of Dundee, Dundee

  • Fletcher R, Leyffer S (2004) Solving mathematical programs with complementarity constraints as nonlinear programs. Optim Methods Softw 19(1):15–40

    Article  MathSciNet  MATH  Google Scholar 

  • Fortuny-Amat J, McCarl B (1981) A representation and economic interpretation of a two-level programming problem. J Oper Res Soc 32(9):783–792

    Article  MathSciNet  MATH  Google Scholar 

  • Gabriel SA, Leuthold FU (2010) Solving discretely-constrained MPEC problems with applications in electric power markets. Energy Econ 32(1):3–14

    Article  Google Scholar 

  • Garces LP, Conejo AJJ, Garcia-Bertrand R, Romero R, Garcés LP (2009) A bilevel approach to transmission expansion planning within a market environment. IEEE Trans Power Syst 24(3):1513–1522

    Article  Google Scholar 

  • Hansen P, Jaumard B, Savard G (1992) New branch-and-bound rules for linear bilevel programming. SIAM J Sci Stat Comput 13(5):1194–1217

    Article  MathSciNet  MATH  Google Scholar 

  • Hasan E, Galiana FD, Conejo AJ (2008) Electricity markets cleared by merit order—part I: finding the market outcomes supported by pure strategy nash equilibria. IEEE Trans Power Syst 23(2):361–371

    Article  Google Scholar 

  • Hejazi SR, Memariani A, Jahanshahloo G, Sepehri MM (2002) Linear bilevel programming solution by genetic algorithm. Comput Oper Res 29(13):1913–1925

    Article  MathSciNet  MATH  Google Scholar 

  • Hu XM, Ralph D (2004) Convergence of a penalty method for mathematical programming with complementarity constraints. J Optim Theory Appl 123(2):365–390

    Article  MathSciNet  Google Scholar 

  • Jenabi M, Ghomi SM, Smeers Y (2013) Bi-level game approaches for coordination of generation and transmission expansion planning within a market environment. IEEE Trans Power Syst 28(3):2639–2650

    Article  Google Scholar 

  • Jeroslow RG (1985) The polynomial hierarchy and a simple model for competitive analysis. Math Program 32(2):146–164

    Article  MathSciNet  MATH  Google Scholar 

  • Jiang Y, Li X, Huang C, Xianing W (2013) Application of particle swarm optimization based on CHKS smoothing function for solving nonlinear bilevel programming problem. Appl Math Comput 219(9):4332–4339

    MathSciNet  MATH  Google Scholar 

  • Kazempour SJ, Conejo AJ (2012) Strategic generation investment under uncertainty via benders decomposition. IEEE Trans Power Syst 27(1):424–432

    Article  Google Scholar 

  • Kazempour SJ, Conejo AJ, Ruiz C (2011) Strategic generation investment using a complementarity approach. IEEE Trans Power Syst 26(2):940–948

    Article  Google Scholar 

  • Kazempour SJ, Conejo AJ, Ruiz C (2012) Strategic generation investment considering futures and spot markets. IEEE Trans Power Syst 27(3):1467–1476

    Article  Google Scholar 

  • Li H, Fang L (2012) An evolutionary algorithm for solving bilevel programming problems using duality conditions. Math Probl Eng. https://doi.org/10.1155/2012/471952

    MathSciNet  MATH  Google Scholar 

  • Lorenczik S, Malischek R, , Trüby J (2014) Modeling strategic investment decisions in spatial markets. Technical Report 14/09, Köln

  • Lv Y, Tiesong H, Wang G, Wan Z (2007) A penalty function method based on Kuhn–Tucker condition for solving linear bilevel programming. Appl Math Comput 188(1):808–813

    MathSciNet  MATH  Google Scholar 

  • Maurovich-Horvat L, Boomsma TK, Siddiqui AS (2014) Transmission and wind investment in a deregulated electricity industry. IEEE Trans Power 30(3):1633–1643

    Article  Google Scholar 

  • Mersha AG, Dempe S (2006) Linear bilevel programming with upper level constraints depending on the lower level solution. Appl Math Comput 180(1):247–254

    MathSciNet  MATH  Google Scholar 

  • Moiseeva E, Hesamzadeh MR, Biggar DR (2015) Exercise of market power on ramp rate in wind-integrated power systems. IEEE Trans Power Syst 30(3):1614–1623

    Article  Google Scholar 

  • Morales JM, Zugno M, Pineda S, Pinson P (2014) Electricity market clearing with improved scheduling of stochastic production. Eur J Oper Res 235(3):765–774

    Article  MathSciNet  MATH  Google Scholar 

  • Motto ALL, Arroyo JMM, Galiana FDD (2005) A mixed-integer LP procedure for the analysis of electric grid security under disruptive threat. IEEE Trans Power Syst 20(3):1357–1365

    Article  Google Scholar 

  • Outrata J (2000) On mathematical programs with complementarity constraints. Optim Methods Softw 14(1):117–137

    Article  MathSciNet  MATH  Google Scholar 

  • Pisciella P, Bertocchi M, Vespucci MT (2016) A leader-followers model of power transmission capacity expansion in a market driven environment. Comput Manag Sci 13:87–118

    Article  MathSciNet  Google Scholar 

  • Pozo D, Contreras J (2011) Finding multiple nash equilibria in pool-based markets: a stochastic EPEC approach. IEEE Trans Power Syst 26(3):1744–1752

    Article  Google Scholar 

  • Pozo D, Sauma E, Contreras J (2013) A three-level static MILP model for generation and transmission expansion planning. IEEE Trans Power Syst 28(1):202–210

    Article  Google Scholar 

  • Ralph D, Wright SJ (2004) Some properties of regularization and penalization schemes for MPECs. Optim Methods Softw 19(5):527–556

    Article  MathSciNet  MATH  Google Scholar 

  • Ruiz C, Conejo AJ (2009) Pool strategy of a producer with endogenous formation of locational marginal prices. IEEE Trans Power Syst 24(4):1855–1866

    Article  Google Scholar 

  • Ruiz C, Conejo AJ (2014) Robust transmission expansion planning. Eur J Oper Res 242:390–401

    Article  Google Scholar 

  • Ruiz C, Conejo AJ, Smeers Y (2012) Equilibria in an oligopolistic electricity pool with stepwise offer curves. IEEE Trans Power Syst 27(2):752–761

    Article  Google Scholar 

  • Scheel H, Scholtes S (2000) Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math Oper Res 25(1):1–22

    Article  MathSciNet  MATH  Google Scholar 

  • Scholtes S (2001) Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J Optim 11(4):918–936

    Article  MathSciNet  MATH  Google Scholar 

  • Shi C, Jie L, Zhang G (2005a) An extended Kuhn–Tucker approach for linear bilevel programming. Appl Math Comput 162(1):51–63

    MathSciNet  MATH  Google Scholar 

  • Shi C, Jie L, Zhang G (2005b) An extended Kth-best approach for linear bilevel programming. Appl Math Comput 164(3):843–855

    MathSciNet  MATH  Google Scholar 

  • Shi C, Zhang G, Jie L (2005c) On the definition of linear bilevel programming solution. Appl Math Comput 160(1):169–176

    MathSciNet  MATH  Google Scholar 

  • Shi C, Jie L, Zhang G, Zhou H (2006) An extended branch and bound algorithm for linear bilevel programming. Appl Math Comput 180(2):529–537

    MathSciNet  MATH  Google Scholar 

  • Siddiqui S, Gabriel SA (2012) An SOS1-based approach for solving MPECs with a natural gas market application. Netw Spat Econ 13(2):205–227

    Article  MathSciNet  MATH  Google Scholar 

  • Sinha A, Malo P, Deb K (2013) Efficient evolutionary algorithm for single-objective bilevel optimization. arXiv:1303.3901

  • Strekalovsky AS, Orlov AV, Malyshev AV (2010a) On computational search for optimistic solutions in bilevel problems. J Glob Optim 48(1):159–172

    Article  MathSciNet  MATH  Google Scholar 

  • Strekalovsky AS, Orlov AV, Malyshev AV (2010b) Numerical solution of a class of bilevel programming problems. Numer Anal Appl 3(2):165–173

    Article  Google Scholar 

  • The ILOG CPLEX (2015) http://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/index.html

  • Valinejad J, Barforoushi T (2015) Generation expansion planning in electricity markets: a novel framework based on dynamic stochastic MPEC. Int J Electr Power Energy Syst 70:108–117

    Article  Google Scholar 

  • Von Stackelberg H (1952) The theory of the market economy. Oxford University Press, Oxford

    Google Scholar 

  • White DJ, Anandalingam G (1993) A penalty function approach for solving bi-level linear programs. J Glob Optim 3(4):397–419

    Article  MathSciNet  MATH  Google Scholar 

  • Wogrin S, Centeno E, Barquín J (2011) Generation capacity expansion in liberalized electricity markets: a stochastic MPEC approach. IEEE Trans Power Syst 26(4):2526–2532

    Article  Google Scholar 

  • Wogrin S, Barquin J, Centeno E (2013) Capacity expansion equilibria in liberalized electricity markets: an EPEC approach. IEEE Trans Power Syst 28(2):1531–1539

    Article  Google Scholar 

  • Zhang G, Lu J, Gao Y (2015) Multi-level decision making: models, methods and applications. Intelligent systems reference library. Springer, Berlin

    Book  Google Scholar 

  • Zugno M, Morales JM, Pinson P, Madsen H (2013) Pool strategy of a price-maker wind power producer. IEEE Trans Power Syst 28(3):3440–3450

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the Spanish Ministry of Economy, Industry and Competitiveness through Project ENE2016-80638-R and in part by the Research Funding Program for Young Talented Researchers of the University of Málaga through Project PPIT-UMA-B1-2017/18.

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Authors and Affiliations

  1. Department of Electrical Engineering, University of Malaga, Málaga, Spain

    S. Pineda

  2. Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    H. Bylling

  3. Department of Applied Mathematics, University of Malaga, Málaga, Spain

    J. M. Morales

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  1. S. Pineda
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  2. H. Bylling
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Correspondence to S. Pineda.

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Pineda, S., Bylling, H. & Morales, J.M. Efficiently solving linear bilevel programming problems using off-the-shelf optimization software. Optim Eng 19, 187–211 (2018). https://doi.org/10.1007/s11081-017-9369-y

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  • DOI: https://doi.org/10.1007/s11081-017-9369-y

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