Back
 AM  Vol.4 No.4 , April 2013
A Note on the Guignard Constraint Qualification and the Guignard Regularity Condition in Vector Optimization
Abstract: Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions in differentiable vector optimization problems.
Cite this paper: G. Giorgi, "A Note on the Guignard Constraint Qualification and the Guignard Regularity Condition in Vector Optimization," Applied Mathematics, Vol. 4 No. 4, 2013, pp. 734-740. doi: 10.4236/am.2013.44101.
References

[1]   S. Y. Wang and F. M. Yang, “A Gap between Multiobjective Optimization and Scalar Optimization,” Journal of Optimization Theory and Applications, Vol. 68, No. 2, 1991, pp. 389-391. doi:10.1007/BF00941577

[2]   B. Aghezzaf and M. Hachimi, “On a Gap between Multiobjective Optimization and Scalar Optimization,” Journal of Optimization Theory and Applications, Vol. 109, No. 2, 2001, pp. 431-435. doi:10.1023/A:1017574608034

[3]   T. Maeda, “Constraint Qualifications in Multiobjective Optimization Problems: Differentiable Case,” Journal of Optimization Theory and Applications, Vol. 80, No. 3, 1994, pp. 483-500. doi:10.1007/BF02207776

[4]   B. Jimenez and V. Novo, “Cualificaciones de Restricciones en Problemas de Optimización Vectorial Diferenciables,” XVI CEDYA Congreso de Ecuaciones Diferen-ciales y Aplicaciones, VI CMA Congreso de Matemática Aplicada, Vol. 1, 1999, pp. 727-734.

[5]   G. Giorgi, B. Jimenez and V. Novo, “On Constraint Qualifications in Directionally Differentiable Multiobjective Optimization Problems,” RAIRO—Operations Research, Vol. 38, No. 3, 2004, pp. 255-274. doi:10.1051/ro:2004023

[6]   G. Giorgi, B. Jimenez and V. Novo, “Strong Kuhn-Tucker Conditions and Constraint Qualifications in Locally Lipschitz Multiobjective Optimization,” TOP, Vol. 17, No. 2, 2009, pp. 288-304. doi:10.1007/s11750-008-0058-z

[7]   G. Giorgi and C. Zuccotti, “Again on Regularity Conditions in Differentiable Vector Optimization,” Annals of the University of Bucharest (Mathematical Series), Vol. LX, No.2, 2011, pp. 157-177.

[8]   M. Guignard, “Generalized Kuhn-Tucker Conditions for Mathematical Programming Problems in a Banach Space,” SIAM Journal on Control, Vol. 7, No. 2, 1969, pp. 232-241. doi:10.1137/0307016

[9]   J. P. Aubin and H. Frankowska, “Set-Valued Analysis,” Birkhäuser, Boston, 1990.

[10]   M. S. Bazaraa and C. M. Shetty, “Foundations of Optimization,” Springer Verlag, Berlin, 1976. doi:10.1007/978-3-642-48294-6

[11]   G. Bigi, “Optimality and Lagrangian Regularity in Vector Optimization,” Ph.D. Thesis, University of Pisa, Pisa, 1999.

[12]   H. W. Corley, “On Optimality Conditions for Maximizations with Respect to Cones,” Journal of Optimization Theory and Applications, Vol. 46, No. 1, 1985, pp. 67-78. doi:10.1007/BF00938760

[13]   G. Giorgi and C. Zuccotti, “On the Use of Some Tangent Cones and Sets in Vector Optimization,” Report N. 169, Università di Pavia, Pavia, 2012.

[14]   T. Staib, “On Necessary and Sufficient Optimality Conditions for Multicriteria Optimization Problems,” ZOR-Methods and Models of Operations Research, Vol. 35, 1991, pp. 231-248.

[15]   M. Castellani and M. Pappalardo, “About a Gap between Multiobjective Optimization and Scalar Optimization,” Journal of Optimization Theory and Applications, Vol. 109, No. 2, 2001, pp. 437-439. doi:10.1023/A:1017526724872

[16]   G. Giorgi, “Remarks on the Fritz John Conditions for Problems with Inequality and Equality Constraints,” International Journal of Pure and Applied Mathematics, Vol. 71, 2011, pp. 643-657.

[17]   O. L. Mangasarian, “Nonlinear Programming,” McGraw-Hill, New York, 1969.

[18]   J. G. Lin, “Maximal Vectors and Multi—Objective Optimization,” Journal of Optimization Theory and Applications, Vol. 18, No. 1, 1976, pp. 41-64. doi:10.1007/BF00933793

[19]   C. Singh, “Optimality Conditions in Multiobjective Differentaible Programming,” Journal of Optimization Theory and Applications, Vol. 53, No. 1, 1987, pp. 115-123. C. Singh, “Errata Corrige,” Journal of Optimization Theory and Applications, Vol. 57, No. 2, 1988, p. 369. doi:10.1007/BF00938547

[20]   I. Marusciac, “On Fritz John Type Optimality Criterion in Multiobjective Optimization,” L’Analyse Numérique et la Théorie de l’Approximation, Vol. 11, 1982, pp. 109-114.

[21]   S. Y. Wang, “A Note on Optimality Conditions in Multiobjective Programming,” Systems Science and Mathematical Sciences, Vol. 1, 1988, pp. 184-190.

[22]   F. J. Gould and J. W. Tolle, “A Necessary and Sufficient Qualification for Constrained Optimization,” SIAM Journal on Applied Mathematics, Vol. 20, No. 2, 1971, pp. 164-172. doi:10.1137/0120021

[23]   G. Bigi and M. Pappalardo, “Regularity Conditions in Vector Optimization,” Journal of Optimization Theory and Applications, Vol. 102, No. 1, 1999, pp. 83-96. doi:10.1023/A:1021890328184

[24]   M. C. Maciel, S. A. Santos and G. N. Sottosanto, “Regularity Conditions in Differentiable Vector Optimization Revisited,” Journal of Optimization Theory and Applications, Vol. 142, No. 2, 2009, pp. 385-398. doi:10.1007/s10957-009-9519-2

[25]   K. H. Elster and J. Thierfelder, “Abstract Cone Approximations and Generalized Differentiability in Nonsmooth Optimization,” Optimization, Vol. 19, No. 3, 1988, pp. 315-341. doi:10.1080/02331938808843348

[26]   G. Giorgi and A. Guerraggio, “On the Notion of Tangent Cone in Mathematical Programming,” Optimization, Vol. 25, No. 1, 1992, pp. 11-23. doi:10.1080/02331939208843804

[27]   J. Palata, “A Survey of Conical Approximations Used in Optimization,” Optimization, Vol. 20, No. 2, 1989, pp. 147-161. doi:10.1080/02331938908843424

[28]   J. P. Penot, “A Characterization of Tangential Regularity,” Nonlinear Analysis, Methods & Applications, Vol. 5, No. 6, 1981, pp. 625-643. doi:10.1016/0362-546X(81)90079-1

[29]   Z. F. Li and S. Y. Wang, “Lagrange Multipliers and Saddle Points in Multiobjective Programming,” Journal of Optimization Theory and Applications, Vol. 83, No. 1, 1994, pp. 63-81. doi:10.1007/BF02191762

[30]   K. H. Elster and R. Nehse, “Optimality Conditions for Some Nonconvex Problems,” Lecture Notes in Control and Information Sciences, Vol. 23, Springer Verlag, Berlin, 1980, pp. 1-9.

[31]   M. Hayashi and K. Komiya, “Perfect Duality for Convexlike Programs,” Journal of Optimization Theory and Applications, Vol. 38, No. 2, 1982, pp. 179-189. doi:10.1007/BF00934081

[32]   V. Jeyakumar, “Convexlike Alternative Theorems and Mathematical Programming,” Journal of Optimization Theory and Applications, Vol. 16, No. 5, 1985, pp. 643-652. doi:10.1080/02331938508843061

 
 
Top