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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 63, 2014 - Issue 4
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Articles

Necessary optimality conditions in pessimistic bilevel programming

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Pages 505-533 | Received 21 Dec 2011, Accepted 16 May 2012, Published online: 26 Jun 2012

References

  • Bao , TQ , Gupta , P and Mordukhovich , BS . 2007 . Necessary conditions in multiobjective optimization with equilibrium constraints . J. Optim. Theory Appl. , 135 : 179 – 203 .
  • Bonnel , H . 2006 . Optimality conditions for the semivectorial bilevel optimization problem . Pacif. J. Optim. , 2 : 447 – 467 .
  • Bonnel , H and Morgan , J . 2006 . Semivectorial bilevel optimization problem: A penalty approach . J. Optim. Theory Appl. , 131 : 365 – 382 .
  • Cervinka , M , Matonoha , M and Outrata , JV . On the computation of relaxed pessimistic solutions to MPECs . Optim. Math. Soft. , to appear. DOI:10.1080/10556788.2011.627585
  • Dassanayaka , S . 2010 . “ Methods of variational analysis in pessimistic bilevel programming ” . In Ph.D. thesis , Detroit : Department of Mathematics, Wayne State University .
  • Dempe , S . 2002 . Foundations of Bilevel Programming , Dordrecht : Kluwer Academic Publishers .
  • Dempe , S . 2003 . Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints . Optimization , 52 : 333 – 359 .
  • Dempe , S , Dinh , N and Dutta , J . 2010 . “ Optimality conditions for a simple convex bilevel programming problem ” . In Variational Analysis and Generalized Differentiation in Optimization and Control , Edited by: Burachik , RS and Yao , J-C . 149 – 161 . Berlin : Springer .
  • Dempe , S , Dutta , J and Mordukhovich , BS . 2007 . New necessary optimality conditions in optimistic bilevel programming . Optimization , 56 : 577 – 604 .
  • Dempe , S , Kalashnikov , VV and Kalashnykova , N . 2006 . “ Optimality conditions for bilevel programming problems ” . In Optimization with Multivalued Mappings , Edited by: Dempe , S and Kalashnikov , V . 3 – 28 . New York : Springer .
  • Dempe , S , Mordukhovich , BS and Zemkoho , AB . Sensitivity analysis for two-level value functions with applications to bilevel programming , submitted for publication .
  • Dempe , S and Zemkoho , AB . 2011 . The generalized Mangasarian–Fromowitz constraint qualification and optimality conditions for bilevel programs . J. Optim. Theory Appl. , 148 : 433 – 441 .
  • Henrion , R and Surowiec , T . 2011 . On calmness conditions in convex bilevel programming . Appl. Analysis , 90 : 951 – 970 .
  • Ishizuka , Y . 1988 . Farkas' theorem of nonconvex type and its application to a min-max problem . J. Optim. Theory Appl. , 57 : 341 – 354 .
  • Lignola , MB and Morgan , J . 1995 . Topological existence and stability for Stackelberg problems . J. Optim. Theory Appl. , 84 : 145 – 169 .
  • Lohse , S . 2010 . “ Eine spezielle klasse von zwei-ebenen-optimierungsaufgaben ” . In Ph.D. thesis Freiberg, Fakultät für Mathematik und Informatik, Technische Universität Bergakademie
  • Loridan , P and Morgan , J . 1996 . Weak via strong Stackelberg problem: New results . J. Glob. Optim. , 8 : 263 – 287 .
  • Mordukhovich , BS . 1976 . Maximum principle in problems of time optimal control with nonsmooth constraints . J. Appl. Math. Mech. , 40 : 960 – 969 .
  • Mordukhovich , BS . 1980 . Metric approximations and necessary optimality conditions for general classes of extremal problems . Soviet Math. Dokl. , 22 : 526 – 530 .
  • Mordukhovich , BS . 2004 . Necessary conditions in nonsmooth minimization via lower and upper subgradients . Set-Valued Anal. , 12 : 163 – 193 .
  • Mordukhovich , BS . 2006 . “ Variational Analysis and Generalized Differentiation ” . In I: Basic Theory. II: Applications , Berlin : Springer .
  • Mordukhovich , BS and Nam , NM . 2005 . Variational stability and marginal functions via generalized differentiation . Math. Oper. Res. , 30 : 800 – 816 .
  • Mordukhovich , BS , Nam , NM and Phan , HM . 2012 . Variational analysis of marginal functions with applications to bilevel programming problems . J. Optim. Theory Appl. , 152 : 557 – 586 .
  • Mordukhovich , BS , Nam , NM and Yen , ND . 2009 . Subgradients of marginal functions in parametric mathematical programming . Math. Program. , 116 : 369 – 396 .
  • Mordukhovich , BS and Outrata , JV . 2007 . Coderivative analysis of quasi-variational inequalities with applications to stability and optimization . SIAM J. Optim. , 18 : 389 – 412 .
  • Outrata , JV . 1990 . On the numerical solution of a class of Stackelberg problems . ZOR–Methods Models Oper. Res. , 34 : 255 – 277 .
  • Outrata , JV , Kočvara , M and Zowe , J . 1998 . Nonsmooth Approach to Optimization Problems with Equilibrium Constraints , Dordrecht , , The Netherlands : Kluwer .
  • Robinson , SM . 1979 . Generalized equations and their solutions. I: Basic theory . Math. Program. Study , 10 : 128 – 141 .
  • Rockafellar , RT and Wets , RJ-B . 1998 . Variational Analysis , Berlin : Springer .
  • Shimizu , K , Ishizuka , Y and Bard , JF . 1997 . Nondifferentiable and Two-level Mathematical Programming , Dordrecht : Kluwer Academic Publishers .
  • Ye , JJ . 1999 . Optimality conditions for optimization problems with complementarity constraints . SIAM J. Optim. , 9 : 74 – 387 .
  • Ye , JJ . 2000 . Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints . SIAM J. Optim. , 10 : 943 – 962 .
  • Ye , JJ . 2011 . Necessary optimality conditions for multiobjective bilevel programs . Math. Oper. Res. , 36 : 165 – 184 .
  • Ye , JJ and Zhu , DL . 2010 . New necessary optimality conditions for bilevel programs by combining MPEC and the value function approach . SIAM J. Optim. , 20 : 1885 – 1905 .
  • Zemkoho , AB . 2012 . A simple approach to optimality conditions in minmax programming . Optimization , to appear. DOI: 10.1080/02331934.2011.653788

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