Optimality conditions for pessimistic semivectorial bilevel programming problems

In this paper, a class of pessimistic semivectorial bilevel programming problems is investigated. By using the scalarization method, we transform the pessimistic semivectorial bilevel programming problem into a scalar objective optimization problem with inequality constraints. Furthermore, we derive a generalized minimax optimization problem using the maximization bilevel optimal value function, of which the sensitivity analysis is constructed via the lower-level value function approach. Using the generalized differentiation calculus of Mordukhovich, the first-order necessary optimality conditions are established in the smooth setting. As an application, we take the optimality conditions of the bilevel programming problems with multiobjective lower level problem when the lower level multiobjective optimization problem is linear with respect to the lower-level variables.MSC:90C26, 90C30, 90C31, 90C46.

[1]  Gabriele Eichfelder,et al.  Multiobjective bilevel optimization , 2010, Math. Program..

[2]  Abdelmalek Aboussoror,et al.  Strong-weak Stackelberg Problems in Finite Dimensional Spaces , 1995 .

[3]  Boris Polyak,et al.  B.S. Mordukhovich. Variational Analysis and Generalized Differentiation. I. Basic Theory, II. Applications , 2009 .

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

[5]  Boris S. Mordukhovich,et al.  Variational Stability and Marginal Functions via Generalized Differentiation , 2005, Math. Oper. Res..

[6]  Matthias Ehrgott,et al.  Multicriteria Optimization (2. ed.) , 2005 .

[7]  Jacqueline Morgan,et al.  Weak via strong Stackelberg problem: New results , 1996, J. Glob. Optim..

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

[9]  Jun Li,et al.  Multiobjective optimization problems with modified objective functions and cone constraints and applications , 2011, J. Glob. Optim..

[10]  Boris S. Mordukhovich,et al.  Variational Analysis of Marginal Functions with Applications to Bilevel Programming , 2012, J. Optim. Theory Appl..

[11]  Tiesong Hu,et al.  A penalty function method for solving weak price control problem , 2007, Appl. Math. Comput..

[12]  Stephan Dempe,et al.  New Optimality Conditions for the Semivectorial Bilevel Optimization Problem , 2012, Journal of Optimization Theory and Applications.

[13]  Alain B. Zemkoho,et al.  Necessary optimality conditions in pessimistic bilevel programming , 2014 .

[14]  Jacqueline Morgan,et al.  On Strict ε- Solutions for a Two-Level Optimization Problem , 1992 .

[15]  Jacqueline Morgan,et al.  Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems , 2013 .

[16]  R. Henrion,et al.  On calmness conditions in convex bilevel programming , 2011 .

[17]  A. Aboussoror,et al.  Weak linear bilevel programming problems : Existence of solutions via a penalty method , 2005 .

[18]  Z. Ankhili,et al.  An exact penalty on bilevel programs with linear vector optimization lower level , 2009, Eur. J. Oper. Res..

[19]  Boris S. Mordukhovich,et al.  Sensitivity Analysis for Two-Level Value Functions with Applications to Bilevel Programming , 2012, SIAM J. Optim..

[20]  Samarathunga M. Dassanayaka.  Methods Of Variational Analysis In Pessimistic Bilevel Programming , 2010 .

[21]  J. Morgan,et al.  e-regularized two-level optimization problems: approximation and existence results , 1988 .

[22]  Jacqueline Morgan,et al.  Approximate solutions for two-level optimization problems , 1988 .

[23]  Zhongping Wan,et al.  A solution method for semivectorial bilevel programming problem via penalty method , 2011 .

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

[25]  Bergakademie Freiberg,et al.  BILEVEL PROGRAMMING: REFORMULATIONS, REGULARITY, AND STATIONARITY , 2012 .

[26]  René Henrion,et al.  On the Calmness of a Class of Multifunctions , 2002, SIAM J. Optim..

[27]  Berç Rustem,et al.  Pessimistic Bilevel Optimization , 2013, SIAM J. Optim..

[28]  S. M. Robinson.  Some continuity properties of polyhedral multifunctions , 1981 .

[29]  Hecheng Li,et al.  A Genetic Algorithm for Solving Weak Nonlinear Bilevel Programming Problems , 2011, 2011 Fourth International Conference on Intelligent Computation Technology and Automation.

[30]  Jacqueline Morgan,et al.  Stability of Regularized Bilevel Programming Problems , 1997 .

[31]  Herminia I. Calvete,et al.  On linear bilevel problems with multiple objectives at the lower level , 2011 .

[32]  Y. Cho,et al.  Nonsmooth multiobjective optimization problems and weak vector quasi-variational inequalities , 2013 .

[33]  On global search for pessimistic solution in bilevel problems( Bilevel Programming, Optimization Methods, and Applications to Economics) , 2013, SOCO 2013.

[34]  A. Marhfour,et al.  Mixed Solutions for Weak Stackelberg Problems: Existence and Stability Results , 2000 .

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

[36]  Jacqueline Morgan,et al.  Least-Norm Regularization For Weak Two-Level Optimization Problems , 1992 .

[37]  Boris S. Mordukhovich,et al.  Subgradients of marginal functions in parametric mathematical programming , 2008, Math. Program..

[38]  Boris S. Mordukhovich,et al.  Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs , 2010, Math. Program..

[39]  Stephan Dempe,et al.  The Generalized Mangasarian-Fromowitz Constraint Qualification and Optimality Conditions for Bilevel Programs , 2011, J. Optim. Theory Appl..

[40]  Abdelmalek Aboussoror,et al.  Existence of Solutions to Two-Level Optimization Problems with Nonunique Lower-Level Solutions , 2001 .

[41]  J. Morgan,et al.  Semivectorial Bilevel Optimization Problem: Penalty Approach , 2006 .

[42]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[43]  Alain B. Zemkoho.  OPTIMIZATION PROBLEMS WITH VALUE FUNCTION OBJECTIVES , 2011 .

[44]  Stephan Dempe,et al.  Optimality Conditions for a Simple Convex Bilevel Programming Problem , 2010 .

[45]  R. Lucchetti,et al.  Existence theorems of equilibrium points in stackelberg , 1987 .

[46]  Matthias Ehrgott,et al.  Multicriteria Optimization , 2005 .

[47]  Jianxin Zhou,et al.  Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization , 1995 .