A note on the optimality condition for a bilevel programming

Yue Sun,Huan Wang,Jie Zhang

doi:10.1186/s13660-015-0882-2

Abstract

The equality type Mordukhovich coderivative rule for a solution mapping to a second-order cone constrained parametric variational inequality is derived under the constraint nondegenerate condition, which improves the result published recently. The rule established is then applied to deriving a necessary and sufficient local optimality condition for a bilevel programming with a second-order cone constrained lower level problem.

Highlights

1 Introduction In this paper, we focus on the following bilevel programming (BP): min f (x, y) ( )
If the lower level problem of BP is replaced by its KKT condition, this is a mathematical program with a second-order cone complementarity problem among the constraints [ ]
The equality type representation of the coderivative of a solution mapping S ( ) is established under conditions weaker than [ ], Theorem . , and it is used to obtain a necessary and sufficient local optimality conditions for the bilevel programming ( ). This is done on the basis of an exact description of the coderivative of the normal cone operator onto the second-order cone

Summary

Introduction

We focus on the following bilevel programming (BP): min f (x, y) ( ). To establish the necessary and sufficient local optimality condition for bilevel programming ( ), a crucial step is to compute generalized differentiation for the solution mapping S(·) defined by ( ). And it is used to obtain a necessary and sufficient local optimality conditions for the bilevel programming ( ) This is done on the basis of an exact description of the coderivative of the normal cone operator onto the second-order cone. In Section , the main results are established, i.e., the equality type calculus rule of the coderivatives of a solution mapping S ( ) is established and used to derive the optimality condition of bilevel programming ( ).

Preliminaries

Conclusion