Second order optimality conditions for bilevel set optimization problems

In this work, we use the notion of the support function to the feasible set mapping to establish second order necessary and sufficient optimality conditions for the optimistic case of bilevel optimization problems. The main tools we exploit are approximate Jacobians, approximate Hessians, second order approximations and second order contingent sets.

[1]  T. Amahroq,et al.  Second order approximations and dual necessary optimality conditions , 1997 .

[2]  B. Mordukhovich,et al.  New necessary optimality conditions in optimistic bilevel programming , 2007 .

[3]  Jane J. Ye,et al.  Optimality conditions for bilevel programming problems , 1995 .

[4]  P. Pardalos,et al.  A Multivariate Partition Approach to Optimization Problems , 2002 .

[5]  N. Gadhi,et al.  Necessary Optimality Conditions for Bilevel Optimization Problems Using Convexificators , 2006, J. Glob. Optim..

[6]  S. Dempe.  First-Order Necessary Optimality Conditions for General Bilevel Programming Problems , 1997 .

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

[8]  Nazih Abderrazzak Gadhi,et al.  On the Regularity Condition for Vector Programming Problems , 2001, J. Glob. Optim..

[9]  D. T. Luc,et al.  Optimality Conditions for C1,1 Vector Optimization Problems , 2001 .

[10]  V. Jeyakumar,et al.  Approximate Jacobian Matrices for Nonsmooth Continuous Maps and C 1 -Optimization , 1998 .

[11]  J. Outrata.  Necessary optimality conditions for Stackelberg problems , 1993 .

[12]  A. Guerraggio,et al.  SECOND-ORDER OPTIMALITY CONDITIONS FOR C 1 MULTIOBJECTIVE PROGRAMMING PROBLEMS , 2001 .

[13]  S. Dempe.  Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints , 2003 .

[14]  Brahim Aghezzaf,et al.  Second-Order Optimality Conditions in Multiobjective Optimization Problems , 1999 .

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

[16]  Pham Huu Sach,et al.  Second-order optimality conditions for the extremal problem under inclusion constraints , 1989 .

[17]  N. Gadhi,et al.  Second order optimality conditions for the extremal problem under inclusion constraints , 2003 .

[18]  B. Mordukhovich.  Variational analysis and generalized differentiation , 2006 .

[19]  S. R. Arora,et al.  Optimality Conditions and an Algorithm for Linear-Quadratic Bilevel Programming , 2001 .

[20]  N. Gadhi,et al.  EPI-DIFFERENTIABILITY AND OPTIMALITY CONDITIONS FOR AN EXTREMAL PROBLEM UNDER INCLUSION CONSTRAINTS , 2003 .

[21]  Jirí V. Outrata,et al.  On Optimization Problems with Variational Inequality Constraints , 1994, SIAM J. Optim..

[22]  S. Bolintinéanu,et al.  Second-Order Efficiency Conditions and Sensitivity of Efficient Points , 1998 .

[23]  Ruoxin Zhang,et al.  Problems of Hierarchical Optimization in Finite Dimensions , 1994, SIAM J. Optim..