July  2011, 7(3): 559-571. doi: 10.3934/jimo.2011.7.559

Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems

1. 

Université Cadi Ayyad, Faculté Polydisciplinaire de Safi, B.P. 4162, Sidi Bouzid, Safi, Morocco

2. 

Laboratoire XLIM UMR-CNRS 6172, Université de Limoges Département de Mathématiques, 123 Avenue Albert Thomas 87060 Limoges Cedex, France, France

Received  July 2010 Revised  March 2011 Published  June 2011

In this paper, for a class of weak bilevel programming problems we provide sufficient conditions guaranteeing the existence of global solutions. These conditions are based on the use of reverse convex and convex maximization problems.
Citation: Abdelmalek Aboussoror, Samir Adly, Vincent Jalby. Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems. Journal of Industrial & Management Optimization, 2011, 7 (3) : 559-571. doi: 10.3934/jimo.2011.7.559
References:
[1]

A. Aboussoror and P. Loridan, Existence of solutions to two-level optimization problems with nonunique lower-level solutions,, Journal of Mathematical Analysis and Applications, 254 (2001), 348.  doi: 10.1006/jmaa.2000.7001.  Google Scholar

[2]

A. Aboussoror, Weak bilevel programming problems: Existence of solutions,, in, 1 (2002), 83.   Google Scholar

[3]

A. Aboussoror and A. Mansouri, Weak linear bilevel programming problems: Existence of solutions via a penalty method,, Journal of Mathematical Analysis and Applications, 304 (2005), 399.  doi: 10.1016/j.jmaa.2004.09.033.  Google Scholar

[4]

A. Aboussoror and A. Mansouri, Existence of solutions to weak nonlinear bilevel problems via MinSup and D.C. problems,, RAIRO Operations Research, 42 (2008), 87.  doi: 10.1051/ro:2008012.  Google Scholar

[5]

A. Aboussoror, Reverse convex programs: Stability and global optimality,, Pacific Journal of Optimization, 5 (2009), 143.   Google Scholar

[6]

A. Aboussoror and S. Adly, A Fenchel-Lagrange duality approach for a bilevel programming problem with extremal-value function,, Journal of Optimization Theory and Applications, 149 (2011), 254.  doi: 10.1007/s10957-011-9831-5.  Google Scholar

[7]

D. Azé, "Eléments d'Analyse Convexe et Variationnelle,", Ellipses, (1997).   Google Scholar

[8]

M. Breton, A. Alj and A. Haurie, Sequential Stackelberg equilibria in two-person games,, Journal of Optimization Theory and Applications, 59 (1988), 71.  doi: 10.1007/BF00939867.  Google Scholar

[9]

S. Dempe, "Foundations of Bilevel Programming,", Nonconvex Optimization and its Applications, 61 (2002).   Google Scholar

[10]

M. Dür, R. Horst and M. Locatelli, Necessary and sufficient global optimality conditions for convex maximization revisited,, Journal of Mathematical Analysis and Applications, 217 (1998), 637.  doi: 10.1006/jmaa.1997.5745.  Google Scholar

[11]

J. Haberl, Maximization of generalized convex functionals in locally convex spaces,, Journal of Optimization Theory and Applications, 121 (2004), 327.  doi: 10.1023/B:JOTA.0000037408.31141.e4.  Google Scholar

[12]

J.-B. Hiriart-Urruty, From convex optimization to nonconvex optimization. Part 1: Necessary and sufficient conditions for global optimality,, in, (1989), 219.   Google Scholar

[13]

J.-B. Hiriart-Urruty, Conditions for global optimality 2,, Journal of Global Optimization, 13 (1998), 349.  doi: 10.1023/A:1008365206132.  Google Scholar

[14]

J.-B. Hiriart-Urruty and Y. S. Ledyaev, A note on the characterization of the global maxima of a (tangentially) convex function over a convex set,, Journal of Convex Analysis, 3 (1996), 55.   Google Scholar

[15]

R. Horst and H. Tuy, "Global Optimization, Deterministic Approaches," 2nd edition,, Springer-Verlag, (1993).   Google Scholar

[16]

M. Laghdir, Optimality conditions in reverse convex optimization,, Acta Mathematica Vietnamica, 28 (2003), 215.   Google Scholar

[17]

R. Lucchetti, F. Mignanego and G. Pieri, Existence theorem of equilibrium points in Stackelberg games with constraints,, Optimization, 18 (1987), 857.  doi: 10.1080/02331938708843300.  Google Scholar

[18]

C. Michelot, "Caractérisation des Minima Locaux des Fonctions de la Classe D.C.,", Technical Note, (1987).   Google Scholar

[19]

R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar

[20]

K. Shimizu, Y. Ishizuka and J. F. Bard, "Non Differentiable and Two-Level Mathematical Programming,", Kluwer Academic Publishers, (1997).   Google Scholar

[21]

K. Shimizu and Y. Ishizuka, Optimality conditions and algorithms for parameter design problems with two-level structure,, IEEE Transactions on Automatic Control, 30 (1985), 986.  doi: 10.1109/TAC.1985.1103803.  Google Scholar

[22]

A. Strekalovsky, On global maximum search of convex functions over a feasible set,, Journal of Numerical Mathematics and Mathematical Physics, 3 (1993), 349.   Google Scholar

[23]

A. Strekalovsky, Extremal problems on complements of convex sets,, Cybernetics and System Analysis, 1 (1994), 88.   Google Scholar

[24]

A. Strekalovsky, Global optimality conditions for nonconvex optimization,, Journal of Global Optimization, 12 (1998), 415.  doi: 10.1023/A:1008277314050.  Google Scholar

[25]

A. Strekalovsky, On convergence of a global search strategy for reverse convex problems,, Journal of Applied Mathematics and Decision Sciences, 2005 (): 149.  doi: 10.1155/JAMDS.2005.149.  Google Scholar

[26]

T. Tanino and T. Ogawa, An algorithm for solving two-level convex optimization problems,, Int. J. Systems Sci., 15 (1984), 163.  doi: 10.1080/00207728408926552.  Google Scholar

[27]

I. Tseveendorj, Reverse convex problems: An approach based on optimality conditions,, Journal of Applied Mathematics and Decision Sciences, 2006 (): 1.  doi: 10.1155/JAMDS/2006/29023.  Google Scholar

[28]

H. Tuy, Convex programs with an additional reverse convex constraint,, Journal of Optimization Theory and Applications, 52 (1987), 463.  doi: 10.1007/BF00938217.  Google Scholar

[29]

H. Tuy and N. V. Thuong, On the global minimization of a convex function under general non convex constraints,, Applied Mathematics and Optimization, 18 (1988), 119.  doi: 10.1007/BF01443618.  Google Scholar

[30]

H. Tuy, "Convex Analysis and Global Optimization,", Nonconvex Optimization and its Applications, 22 (1998).   Google Scholar

show all references

References:
[1]

A. Aboussoror and P. Loridan, Existence of solutions to two-level optimization problems with nonunique lower-level solutions,, Journal of Mathematical Analysis and Applications, 254 (2001), 348.  doi: 10.1006/jmaa.2000.7001.  Google Scholar

[2]

A. Aboussoror, Weak bilevel programming problems: Existence of solutions,, in, 1 (2002), 83.   Google Scholar

[3]

A. Aboussoror and A. Mansouri, Weak linear bilevel programming problems: Existence of solutions via a penalty method,, Journal of Mathematical Analysis and Applications, 304 (2005), 399.  doi: 10.1016/j.jmaa.2004.09.033.  Google Scholar

[4]

A. Aboussoror and A. Mansouri, Existence of solutions to weak nonlinear bilevel problems via MinSup and D.C. problems,, RAIRO Operations Research, 42 (2008), 87.  doi: 10.1051/ro:2008012.  Google Scholar

[5]

A. Aboussoror, Reverse convex programs: Stability and global optimality,, Pacific Journal of Optimization, 5 (2009), 143.   Google Scholar

[6]

A. Aboussoror and S. Adly, A Fenchel-Lagrange duality approach for a bilevel programming problem with extremal-value function,, Journal of Optimization Theory and Applications, 149 (2011), 254.  doi: 10.1007/s10957-011-9831-5.  Google Scholar

[7]

D. Azé, "Eléments d'Analyse Convexe et Variationnelle,", Ellipses, (1997).   Google Scholar

[8]

M. Breton, A. Alj and A. Haurie, Sequential Stackelberg equilibria in two-person games,, Journal of Optimization Theory and Applications, 59 (1988), 71.  doi: 10.1007/BF00939867.  Google Scholar

[9]

S. Dempe, "Foundations of Bilevel Programming,", Nonconvex Optimization and its Applications, 61 (2002).   Google Scholar

[10]

M. Dür, R. Horst and M. Locatelli, Necessary and sufficient global optimality conditions for convex maximization revisited,, Journal of Mathematical Analysis and Applications, 217 (1998), 637.  doi: 10.1006/jmaa.1997.5745.  Google Scholar

[11]

J. Haberl, Maximization of generalized convex functionals in locally convex spaces,, Journal of Optimization Theory and Applications, 121 (2004), 327.  doi: 10.1023/B:JOTA.0000037408.31141.e4.  Google Scholar

[12]

J.-B. Hiriart-Urruty, From convex optimization to nonconvex optimization. Part 1: Necessary and sufficient conditions for global optimality,, in, (1989), 219.   Google Scholar

[13]

J.-B. Hiriart-Urruty, Conditions for global optimality 2,, Journal of Global Optimization, 13 (1998), 349.  doi: 10.1023/A:1008365206132.  Google Scholar

[14]

J.-B. Hiriart-Urruty and Y. S. Ledyaev, A note on the characterization of the global maxima of a (tangentially) convex function over a convex set,, Journal of Convex Analysis, 3 (1996), 55.   Google Scholar

[15]

R. Horst and H. Tuy, "Global Optimization, Deterministic Approaches," 2nd edition,, Springer-Verlag, (1993).   Google Scholar

[16]

M. Laghdir, Optimality conditions in reverse convex optimization,, Acta Mathematica Vietnamica, 28 (2003), 215.   Google Scholar

[17]

R. Lucchetti, F. Mignanego and G. Pieri, Existence theorem of equilibrium points in Stackelberg games with constraints,, Optimization, 18 (1987), 857.  doi: 10.1080/02331938708843300.  Google Scholar

[18]

C. Michelot, "Caractérisation des Minima Locaux des Fonctions de la Classe D.C.,", Technical Note, (1987).   Google Scholar

[19]

R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970).   Google Scholar

[20]

K. Shimizu, Y. Ishizuka and J. F. Bard, "Non Differentiable and Two-Level Mathematical Programming,", Kluwer Academic Publishers, (1997).   Google Scholar

[21]

K. Shimizu and Y. Ishizuka, Optimality conditions and algorithms for parameter design problems with two-level structure,, IEEE Transactions on Automatic Control, 30 (1985), 986.  doi: 10.1109/TAC.1985.1103803.  Google Scholar

[22]

A. Strekalovsky, On global maximum search of convex functions over a feasible set,, Journal of Numerical Mathematics and Mathematical Physics, 3 (1993), 349.   Google Scholar

[23]

A. Strekalovsky, Extremal problems on complements of convex sets,, Cybernetics and System Analysis, 1 (1994), 88.   Google Scholar

[24]

A. Strekalovsky, Global optimality conditions for nonconvex optimization,, Journal of Global Optimization, 12 (1998), 415.  doi: 10.1023/A:1008277314050.  Google Scholar

[25]

A. Strekalovsky, On convergence of a global search strategy for reverse convex problems,, Journal of Applied Mathematics and Decision Sciences, 2005 (): 149.  doi: 10.1155/JAMDS.2005.149.  Google Scholar

[26]

T. Tanino and T. Ogawa, An algorithm for solving two-level convex optimization problems,, Int. J. Systems Sci., 15 (1984), 163.  doi: 10.1080/00207728408926552.  Google Scholar

[27]

I. Tseveendorj, Reverse convex problems: An approach based on optimality conditions,, Journal of Applied Mathematics and Decision Sciences, 2006 (): 1.  doi: 10.1155/JAMDS/2006/29023.  Google Scholar

[28]

H. Tuy, Convex programs with an additional reverse convex constraint,, Journal of Optimization Theory and Applications, 52 (1987), 463.  doi: 10.1007/BF00938217.  Google Scholar

[29]

H. Tuy and N. V. Thuong, On the global minimization of a convex function under general non convex constraints,, Applied Mathematics and Optimization, 18 (1988), 119.  doi: 10.1007/BF01443618.  Google Scholar

[30]

H. Tuy, "Convex Analysis and Global Optimization,", Nonconvex Optimization and its Applications, 22 (1998).   Google Scholar

[1]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067

[2]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[3]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1

[4]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[5]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225

[6]

Sohana Jahan. Discriminant analysis of regularized multidimensional scaling. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 255-267. doi: 10.3934/naco.2020024

[7]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379

[8]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[9]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[10]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[11]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

[12]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[13]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933

[14]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025

[15]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[16]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[17]

José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030

[18]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045

[19]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113

[20]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (9)

[Back to Top]