Some properties of the bilevel programming problem
Journal of Optimization Theory and Applications
Directional derivatives of the solution of a parametric nonlinear program
Mathematical Programming: Series A and B
On the Convergence of Pattern Search Algorithms
SIAM Journal on Optimization
Analysis of Generalized Pattern Searches
SIAM Journal on Optimization
A Pattern Search Filter Method for Nonlinear Programming without Derivatives
SIAM Journal on Optimization
Second-Order Behavior of Pattern Search
SIAM Journal on Optimization
Mesh Adaptive Direct Search Algorithms for Constrained Optimization
SIAM Journal on Optimization
Practical Bilevel Optimization: Algorithms and Applications (Nonconvex Optimization and Its Applications)
Convergence of Mesh Adaptive Direct Search to Second-Order Stationary Points
SIAM Journal on Optimization
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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.