Direct search algorithm for bilevel programming problems

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.

[1]  H. Stackelberg,et al.  Marktform und Gleichgewicht , 1935 .

[2]  Jonathan F. Bard,et al.  Practical Bilevel Optimization , 1998 .

[3]  Mark A. Abramson,et al.  Second-Order Behavior of Pattern Search , 2005, SIAM J. Optim..

[4]  George B.Richardson.  The Theory of the Market Economy. , 1995, Revue économique.

[5]  Charles Audet,et al.  Mesh Adaptive Direct Search Algorithms for Constrained Optimization , 2006, SIAM J. Optim..

[6]  W. Hager.  Lipschitz Continuity for Constrained Processes , 1979 .

[7]  Charles Audet,et al.  Analysis of Generalized Pattern Searches , 2000, SIAM J. Optim..

[8]  Heinrich von Stackelberg,et al.  Stackelberg (Heinrich von) - The Theory of the Market Economy, translated from the German and with an introduction by Alan T. PEACOCK. , 1953 .

[9]  J. Bard.  Some properties of the bilevel programming problem , 1991 .

[10]  S. Scholtes.  Introduction to Piecewise Differentiable Equations , 2012 .

[11]  Virginia Torczon,et al.  On the Convergence of the Multidirectional Search Algorithm , 1991, SIAM J. Optim..

[12]  Stephan Dempe,et al.  Foundations of Bilevel Programming , 2002 .

[13]  Charles Audet,et al.  Convergence of Mesh Adaptive Direct Search to Second-Order Stationary Points , 2006, SIAM J. Optim..

[14]  Charles Audet,et al.  A Pattern Search Filter Method for Nonlinear Programming without Derivatives , 2001, SIAM J. Optim..

[15]  Tamara G. Kolda,et al.  Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods , 2003, SIAM Rev..

[16]  O. SIAMJ.,et al.  ON THE CONVERGENCE OF PATTERN SEARCH ALGORITHMS , 1997 .

[17]  R. Mifflin.  Semismooth and Semiconvex Functions in Constrained Optimization , 1977 .

[18]  S. Dempe.  A necessary and a sufficient optimality condition for bilevel programming problems , 1992 .

[19]  Andrew R. Teel,et al.  Direct search methods for nonsmooth optimization , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[20]  Stephan Dempe,et al.  Directional derivatives of the solution of a parametric nonlinear program , 1995, Math. Program..

[21]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[22]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications (Nonconvex Optimization and Its Applications) , 2006 .

[23]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications , 1998 .

[24]  Charles Audet.  Convergence Results for Generalized Pattern Search Algorithms are Tight , 2004 .