Abstract
This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem and a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aubin, J.-P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9(1), 87–111 (1984)
Bard, J.F.: Practical bilevel optimization: Algorithms and applications. Kluwer Academic Publishers, Dordrecht (1998)
Bonnel, H.: Optimality conditions for the semivectorial bilevel optimization problem. Pac. J. Optim. 2(3), 447–467 (2006)
Červinka, M., Matonoha, C., Outrata, J.V.: On the computation of relaxed pessimistic solutions to mpecs. Optim. Methods Softw. 28(1), 186–206 (2013)
Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper Res. 153, 235–256 (2007)
Dempe, S., bundle algorithm applied to bilevel programming problems with non-unique lower level solutions. A Comput. Optim. Appl. 15(2), 145–166 (2000)
Dempe, S.: Foundations of bilevel programming. Kluwer Academic Publishers, Dordrecht (2002)
Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3), 333–359 (2003)
Dempe, S., Bard, J.F.: Bundle trust-region algorithm for bilinear bilevel programming. J. Optim. Theory Appl. 110(2), 265–288 (2001)
Dempe, S., Dutta, J., Mordukhovich, B.S.: Variational analysis in bilevel programming. In: Neogy, S.K., Bapat, R.B., Das, A.K., Parthasarathy, T. (eds.) Mathematical programming and game theory for decision making, pp 257–277. World Scientific (2008)
Dempe, S., Franke, S.: Solution algorithm for an optimistic linear stackelberg problem. Comput. Oper Res. 41, 277–281 (2014)
Dempe, S., Gadhi, N., Zemkoho, A.B.: New optimality conditions for the semivectorial bilevel optimization problem. J. Optim Theory Appl. 157(1), 54–74 (2013)
Dempe, S., Mordukhovich, B., Zemkoho, A.B.: Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22 (4), 1309–1343 (2012)
Dempe, S., Mordukhovich, B.S., Zemkoho, A.B.: Necessary optiMality conditions in pessimistic bilevel programming. Optimization 63(4), 505–533 (2014)
Dempe, S., Schmidt, H.: On an algorithm solving two-level programming problems with nonunique lower level solutions. Comput. Optim. Appl. 6(3), 227–249 (1996)
Dempe, S., Zemkoho, A.B.: The generalized Mangasarian-Fromowitz constraint qualification and optimality conditions for bilevel programs. J. Optim. Theory Appl. 148, 433–441 (2011)
Dempe, S., Zemkoho, A.B.: On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problem. Nonlinear Anal. 75(3), 1202–1218 (2012)
Dempe, S., Zemkoho, A.B.: The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math. Program. 138(1-2), 447–473 (2013)
Dempe, S., Zemkoho, A.B.: KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization. SIAM J. Optim. 24(4), 1639–1669 (2014)
Dias, S., Smirnov, G.: On the Newton method for set-valued maps. Nonlinear Anal. 75(3), 1219–1230 (2012)
Dutta, J., Dempe, S.: Bilevel programming with convex lower level problems. In: Dempe, S., Kalashnikov, V. (eds.) Optimization with multivalued mappings, pp 51–71. Springer, New York (2006)
Fliege, J., Vicente, L.N.: Multicriteria approach to bilevel optimization. J. Optim. Theory Appl. 131(2), 209–225 (2006)
Ha, T.X.D.: OptiMality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems. Nonlinear Anal. 75, 1305–1323 (2012)
Hatz, K., Leyffer, S., Schlöder, J.P., Bock, H.G.: Regularizing bilevel nonlinear programs by lifting Preprint ANL/MCS-P4076-0613 (2013)
Huy, N.Q., Mordukhovich, B.S., Yao, J.C.: Coderivatives of frontier and solution maps in parametric multiobjective optimization. Taiwanese J. Math. 12(8), 2083–2111 (2008)
Jahn, J.: Vector optimization. Theory, applications, and extensions, 2nd ed. Springer, Berlin (2011)
Lin, G.-H., Xu, M., Ye, J.J.: On solving simple bilevel programs with a nonconvex lower level program. Math. Program. 144(1-2), 277–305 (2014)
Loridan, P., Morgan, J.: Weak via strong Stackelberg problem: new results. J. Glob. Optim. 8(3), 263–287 (1996)
Lucchetti, R., Mignanego, F., Pieri, G.: Existence theorems of equilibrium points in Stackelberg games with contraints. Optimization 18(6), 857–866 (1987)
Mersha, A.G., Dempe, S.: Direct search algorithm for bilevel programming problems. Comput. Optim Appl. 49(1), 1–15 (2011)
Mersha, A.G., Dempe, S.: Feasible direction method for bilevel programming problem. Optimization 61(5), 597–616 (2012)
Molodtsov, D.A.: The solution of a class of non antagonistic games. USSR Compt. Maths. Math. Phys. 16, 1451–1456 (1976)
Mordukhovich, B.S.: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems. Sov. Math Dokl. 22, 526–530 (1980)
Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183(1), 250–288 (1994)
Mordukhovich, B.S.: Variational analysis and generalized differentiation I: basic theory. II: Applications. Springer, Berlin (2006)
Mordukhovich, B.S., Nam, M.N., Phan, H.M.: Variational analysis of marginal function with applications to bilevel programming problems. J. Optim Theory Appl. 152(3), 557–586 (2012)
Mordukhovich, B.S., Outrata, J.V.: Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J. Optim. 18(2), 389–412 (2007)
Morgan, J., Patrone, F.: Stackelberg problems: Subgame perfect equilibria via Tikhonov regularization. In: Haurie, A., Muto, S., Petrosjan, L.A., Raghavan, T.E.S. (eds.) Advances in dynamic games, pp 209–221. Birkhäuser, Boston (2006)
Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth approach to optimization problems with equilibrium constraints. Kluwer Academic Publishers, Dordrecht (1998)
Robinson, S.M.: Generalized equations and their solutions. I: basic theory. Math. Program. Study 10, 128–141 (1979)
Ruuska, S., Miettinen, K., Wiecek, M.M.: Connections between single-level and bilevel multiobjective optimization. J. Optim. Theory Appl. 153(1), 60–74 (2012)
von Stackelberg, H.F.: Marktform und Gleichgewicht. Springer, Berlin (1934)
Wiesemann, W., Tsoukalas, A., Kleniati, P., Rustem, B.: Pessimistic bilevel optimization. SIAM J. Optim. 23(1), 353–380 (2013)
Xu, M., Ye, J.J.: A smoothing augmented Lagrangian method for solving simple bilevel programs. Comput. Optim. Appl. 59(1–2), 353–377 (2014)
Ye, J.J.: Constraint qualifications and KKT conditions for bilevel programming problems. Math. Oper Res. 31(4), 811–824 (2006)
Ye, J.J.: Necessary optimality conditions for multiobjective bilevel programs. Math. Oper. Res. 36(1), 165–184 (2011)
Ye, J.J., Zhu, D.L.: optimality conditions for bilevel programming problems. Optimization 33(1), 9–27 (1995). (with Erratum in Optimization, 39(4):361-366, 1997)
Ye, J.J., Zhu, D.L.: New necessary optimality conditions for bilevel programs by combining MPEC and the value function approach. SIAM J. Optim. 20(4), 1885–1905 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zemkoho, A.B. Solving Ill-posed Bilevel Programs. Set-Valued Var. Anal 24, 423–448 (2016). https://doi.org/10.1007/s11228-016-0371-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-016-0371-x
Keywords
- Bilevel optimization
- Multiobjective bilevel optimization
- Set-valued optimization
- Variational analysis
- Coderivative
- Optimality conditions