Abstract
In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a nonsmooth and nonconvex optimization problem. We propose a smoothing augmented Lagrangian method for solving a general class of nonsmooth and nonconvex constrained optimization problems. We show that, if the sequence of penalty parameters is bounded, then any accumulation point is a Karush-Kuch-Tucker (KKT) point of the nonsmooth optimization problem. The smoothing augmented Lagrangian method is used to solve the combined problem. Numerical experiments show that the algorithm is efficient for solving the simple bilevel program.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
ALGENCAN: http://www.ime.usp.br/~egbirgin/tango/
Bard, J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic, Norwell (1998)
Calamai, P.H., Moré, J.J.: Projected gradient method for linearly constrained problems. Math. Program. 39, 93–116 (1987)
Chen, X.: Smoothing methods for nonsmooth, nonconvex optimization. Math. Program. 134, 71–99 (2012)
Chen, X., Womersley, R.S., Ye, J.J.: Minimizing the condition number of a Gram matrix. SIAM J. Optim. 21, 127–148 (2011)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)
Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15, 751–779 (2005)
Danskin, J.M.: The Theory of Max-Min and Its Applications to Weapons Allocation Problems. Springer, New York (1967)
Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic, Norwell (2002)
Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52, 333–359 (2003)
Dunn, J.C.: Global and asymptotic convergence rate estimates for a class of projected gradient processes. SIAM J. Control Optim. 19, 368–400 (1981)
Gafni, E.M., Bertsekas, D.P.: Two-metric projection methods for constrained optimization. SIAM J. Control Optim. 22, 936–964 (1984)
Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)
Jourani, A.: Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems. J. Optim. Theory Appl. 81, 533–548 (1994)
LANCELOT. http://www.cse.scitech.ac.uk/nag/lancelot/lancelot.shtml
Lin, G.H., Xu, M., Ye, J.J.: On solving simple bilevel programs with a nonconvex lower level program. Math. Program., Ser. A (2013). doi:10.1007/s10107-013-0633-4
Liu, B.: The mathematics of principal-agent problems. M.Sc. Thesis, University of Victoria (2008). doi:10.1007/s10107-013-0633-4
Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)
Mirrlees, J.: The theory of moral hazard and unobservable behaviour: part I. Rev. Econ. Stud. 66, 3–22 (1999)
Mitsos, A., Lemonidis, P., Barton, P.: Global solution of bilevel programs with a nonconvex inner program. J. Glob. Optim. 42, 475–513 (2008)
Outrata, J.V.: On the numerical solution of a class of Stackelberg problems. ZOR, Z. Oper.-Res. 34, 255–277 (1990)
Pang, J.-S., Fukushima, M.: Complementarity constraint qualifications and simplified B-stationary conditions for mathematical programs with equilibrium constraints. Comput. Optim. Appl. 13, 111–136 (1999)
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969)
Rockafellar, R.T.: A dual approach for solving nonlinear programming problems by unconstrained optimization. Math. Program. 5, 354–373 (1973)
Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control Optim. 12, 268–285 (1974)
Shimizu, K., Ishizuka, Y., Bard, J.F.: Nondifferentiable and Two-Level Mathematical Programming. Kluwer Academic, Boston (1997)
Vicente, L.N., Calamai, P.H.: Bilevel and multilevel programming: a bibliography review. J. Glob. Optim. 5, 291–306 (1994)
Ye, J.J.: Necessary and sufficient conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)
Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995)
Ye, J.J., Zhu, D.L.: A note on optimality conditions for bilevel programming problems. Optimization 39, 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, 1885–1905 (2010)
Zhang, C., Chen, X.: Smoothing projected gradient method and its application to stochastic linear complementarity problems. SIAM J. Optim. 20, 627–649 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Masao Fukushima in honor of his 65th birthday.
The research of J.J. Ye was partially supported by NSERC.
Rights and permissions
About this article
Cite this article
Xu, M., Ye, J.J. A smoothing augmented Lagrangian method for solving simple bilevel programs. Comput Optim Appl 59, 353–377 (2014). https://doi.org/10.1007/s10589-013-9627-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-013-9627-7