Abstract
In this paper, we study the application of a class of direct search methods to bilevel programming with convex lower level problems with strongly stable optimal solutions. In those methods, directions of descent in each iterations are selected within a finite set of directions. To guarantee the existence of such a finite set, we investigate the relation between the aperture of a descent cone at a non stationary point and the vector density of a finite set of directions. It is shown that the direct search method converges to a Clarke stationary point of the bilevel programming problem.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramson, M.A.: Second-order behavior of pattern search. SIAM J. Optim. 16, 515–530 (2005)
Abramson, M.A., Audet, C.: Convergence of mesh adaptive direct search to second-order stationary points. SIAM J. Optim. 17, 606–619 (2006)
Audet, C.: Convergence results for generalized pattern search algorithms are tight. Optim. Eng. 5, 101–122 (2004)
Audet, C., Dennis, J.E. Jr.: Analysis of generalized pattern searches. SIAM J. Optim. 13, 889–903 (2003)
Audet, C., Dennis, J.E. Jr.: A pattern search filter method for nonlinear programming with out derivatives. SIAM J. Optim. 14, 980–1010 (2004)
Audet, C., Dennis, J.E. Jr.: Mesh adaptive direct search algorithms for constrained optimization. SIAM J. Optim. 17, 188–217 (2006)
Bard, J.F.: Some properties of the bilevel programming problem. J. Optim. Theory Appl. 68, 371–378 (1991)
Bard, J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic, Dordrecht (1998)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Dempe, S.: A necessary and a sufficient optimality condition for bilevel programming problems. Optimization 25, 341–354 (1992)
Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic, Dordrecht (2002)
Hager, W.W.: Lipschitz continuity for constrained processes. SIAM J. Control Optim. 17, 321–269 (1979)
Kolda, T.G., Lewis, R.M., Torczon, V.: Optimization by direct search: New perspectives on some classical and modern methods. SIAM Rev. 45, 385–482 (2003)
Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)
Popović, D., Teel, A.R.: Direct search methods for nonsmooth optimization. In: 43rd IEEE Conference on Decision and Control, Vol. 3, pp. 3173–3178, Atlantis, Paradise Island, Bahamas, December 14–17, 2004
Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Math. Program. 70, 159–172 (1995)
Scholtes, S.: Introduction to piecewise differentiable equations. Preprint 53-1994, Universität Karlsruhe, Karlsruhe (1994)
Torczon, V.: On the convergence of the multidirectional search algorithm. SIAM J. Optim. 1, 123–145 (1991)
Torczon, V.: On the convergence of pattern search algorithms. SIAM J. Optim. 7, 1–25 (1997)
Von Stackelberg, H.: Marktform und Gleichgewicht. Springer, Berlin (1934). Engl. transl.: The Theory of the Market Economy. Oxford University Press (1952)
Author information
Authors and Affiliations
Corresponding author
Additional information
Work of the first author was supported by DAAD (Deutscher Akademischer Austausch Dienst) with a scholarship grant.
Rights and permissions
About this article
Cite this article
Mersha, A.G., Dempe, S. Direct search algorithm for bilevel programming problems. Comput Optim Appl 49, 1–15 (2011). https://doi.org/10.1007/s10589-009-9295-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-009-9295-9