A Note on the Guignard Constraint Qualification and the Guignard Regularity Condition in Vector Optimization
Author(s)
Giorgio Giorgi
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.
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
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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
[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