Abstract
In this paper, we consider a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint and the upper level program has a convex set constraint. By using the value function of the lower level program, we reformulate the bilevel program as a single level optimization problem with a nonsmooth inequality constraint and a convex set constraint. To deal with such a nonsmooth and nonconvex optimization problem, we design a smoothing projected gradient algorithm for a general optimization problem with a nonsmooth inequality constraint and a convex set constraint. We show that, if the sequence of penalty parameters is bounded then any accumulation point is a stationary point of the nonsmooth optimization problem and, if the generated sequence is convergent and the extended Mangasarian-Fromovitz constraint qualification holds at the limit then the limit point is a stationary point of the nonsmooth optimization problem. We apply the smoothing projected gradient algorithm to the bilevel program if a calmness condition holds and to an approximate bilevel program otherwise. Preliminary numerical experiments show that the algorithm is efficient for solving the simple bilevel program.
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References
Bard, J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer, Dordrecht (1998)
Calamai, P.H., Moré J.J. : Projected gradient method for linearly constrained problems. Math. Program. 39, 93–116 (1987)
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-Interscience, 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, Dordrecht (2002)
Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52, 333–359 (2003)
Dempe, S., Dutta, J.: Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Program. 131, 37–48 (2012)
Dunn, J.C.: Global and asymptotic convergence rate estimates for a class of projected gradient processes. SIAM J. Control Optim. 19, 368–400 (1981)
Fang, S.C., Wu, S.Y.: Solving min-max problems and linear semi-infinite programs. Comput. Math. Appl. 32, 87–93 (1996)
Jourani, A.: Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems. J. Optim. Theory Appl. 81, 533–548 (1994)
Lignola, M.B., Morgan, J.: Stability of regularized bilevel programming problems. J. Optim. Theo. Appl. 93, 575–596 (1997)
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)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. 1: Basic Theory, vol. 2: Applications. Springer, Berlin (2006)
Outrata, J.V.: On the numerical solution of a class of Stackelberg problems. Z. Oper. Res. 34, 255–277 (1990)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Shimizu, K., Ishizuka, Y., Bard, J.F.: Nondifferentiable and Two-Level Mathematical Programming. Kluwer, Boston (1997)
Stein, E.M., Shakarchi, R.: Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Springer, Berlin (2005)
Vicente, L.N., Calamai, P.H.: Bilevel and multilevel programming: a bibliography review. J. Glob. Optim. 5, 291–306 (1994)
Ye, J.J.: Multiplier rules under mixed assumptions of differentiability and Lipschitz continuity. SIAM J. Control Optim. 39, 1441–1460 (2001)
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)
Zarantonello, E.H.: Contributions to Nonlinear Functional Analysis. Academic Press, New York (1971)
Zhang, C., Chen, X.: Smoothing projected gradient method and its application to stochastic linear complementarity problems. SIAM J. Optim. 20, 627–649 (2009)
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The authors are grateful to the two anonymous referees for their helpful comments and suggestions.
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The first and second author’s work was supported in part by NSFC Grant #11071028. The third author’s work was supported in part by NSERC.
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Lin, GH., Xu, M. & Ye, J.J. On solving simple bilevel programs with a nonconvex lower level program. Math. Program. 144, 277–305 (2014). https://doi.org/10.1007/s10107-013-0633-4
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DOI: https://doi.org/10.1007/s10107-013-0633-4
Keywords
- Bilevel program
- Value function
- Partial calmness
- Smoothing function
- Gradient consistent property
- Integral entropy function
- Smoothing projected gradient algorithm