Abstract

Necessary conditions of optimality are presented for weakly efficient solutions to multiobjective minimization problems with inequality-type constraints. These conditions are applied when the constraints do not necessarily satisfy any regularity assumptions and they are based on the concept of 2-regularity introduced by Izmailov. In general, the optimality conditions do not provide the complete weak Pareto optimal set, so 2-KKT-pseudoinvex problems are defined. This new concept of generalized convexity is both necessary and sufficient to guarantee the characterization of all weakly efficient solutions based on the optimality conditions and it is the weakest one.

Author information

Contact details are reproduced from the original publication and may be historical.

Beatriz Hernández-Jiménez

Depto Economía, Métodos Cuantitativos e Historia Económica, Area de Estadística e Investigación Operativa, Universidad Pablo de Olavide, Edificio 3 - Ctra Utrera - Km 1, 41013 Sevilla, Spain

mbherjim@upo.es

Marko A. Rojas-Medar

Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Chillán, Chile

marko@ueubiobio.cl

Rafaela Osuna-Gómez

Departamento de Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, 41012 Sevilla, Spain

rafaela@us.es

Antonio Rufián-Lizana

Departamento de Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, 41012 Sevilla, Spain

rufian@us.es

Keywords

  • Convexity
  • regularity
  • constraints qualifications
  • optimality conditions

Mathematics Subject Classification

90C29, 90C46, 26B25, 46T20, 47J20

References

  1. B. Aghezzaf, M. Hachimi: On a gap between multiobjective optimization and scalar optimization, J. Optimization Theory Appl. 109 (2001) 431–435.https://doi.org/10.1023/A:1017574608034Google Scholar
  2. A. V. Arutyunov: Optimality Conditions: Abnormal and Degenerate Problems, Mathematics and its Applications 526, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-94-015-9438-7Google Scholar
  3. E. R. Avakov, A. V. Arutyunov, A. F. Izmailov: Necessary conditions for an extremum in a mathematical programming problem, Proc. Steklov Inst. Math. 256 (2007) 2–25.https://doi.org/10.1134/S0081543807010014Google Scholar
  4. M. S. Bazaara, H. D. Sherali, C. M. Shetty: Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, New York (1993).Google Scholar
  5. G. A. Bliss: Lectures on the Calculus of Variations, University of Chicago Press, Chicago (1946).Google Scholar
  6. B. Hernández-Jiménez, M. A. Rojas-Medar, R. Osuna-Gómez, A. Beato-Moreno: Generalized convexity in non-regular programming problems with inequality-type constraints, J. Math. Anal. Appl. 352 (2009) 604–613.https://doi.org/10.1016/j.jmaa.2008.11.013Google Scholar
  7. B. D. Craven: Lagrangean conditions and quasiduality, Bull. Aust. Math. Soc. 16 (1977) 325–339.https://doi.org/10.1017/S0004972700023431Google Scholar
  8. A. F. Izmailov: Optimality conditions for degenerate extremum problems with inequality-type constraints, Comput. Math. Math. Phys. 34 (1994) 723–736.Google Scholar
  9. R. N. Kaul, S. K. Suneja, M. K. Srivastava: Optimality criteria and duality in multipleobjective optimization involving generalized invexity, J. Optimization Theory Appl. 80 (1994) 465–482.https://doi.org/10.1007/BF02207775Google Scholar
  10. J. Janh: Vector Optimization, Springer, Berlin (2004).https://doi.org/10.1007/978-3-540-24828-6Google Scholar
  11. O. L. Mangasarian: Nonlinear Programming, McGraw-Hill, New York (1969).Google Scholar
  12. D. M. Martin: The essence of invexity, J. Optimization Theory Appl. 17 (1985) 65–76.https://doi.org/10.1007/BF00941316Google Scholar
  13. K. M. Miettinen: Nonlinear Multiobjective Optimization, Kluwer, Dordrecht (1999).https://doi.org/10.1007/978-1-4615-5563-6Google Scholar
  14. M. Minoux: Mathematical Programming: Theory and Algorithms, John Wiley and Sons, Chichester (1986).Google Scholar
  15. P. M. Naccache: Connectedness of the set of the nondominated outcomes in multicriteria optimization, J. Optimization Theory Appl. 25 (1978) 459–467.https://doi.org/10.1007/BF00932907Google Scholar
  16. R. Osuna-Gómez, A. Rufián-Lizana, P. Ruiz-Canales: Invex functions and generalized convexity in multiobjective programming, J. Optimization Theory Appl. 98 (1998) 651–661.https://doi.org/10.1023/A:1022628130448Google Scholar
  17. R. Osuna-Gómez, A. Beato-Moreno, A. Rufian-Lizana: Generalized convexity in multiobjective programming, J. Math. Anal. Appl. 233 (1999) 205–220.https://doi.org/10.1006/jmaa.1999.6284Google Scholar
  18. M. G. Rueda, M. A. Hanson, C. Singh: Optimality and duality with generalized convexity, J. Optim. Theory Appl. 86 (1995) 491–500.https://doi.org/10.1007/BF02192091Google Scholar
  19. Y. Sawaragi, H. Nakayama, T. Tanino: Theory of Multiobjective Optimization, Academic Press, Orlando (1985).Google Scholar
  20. S. K. Suneja, M. K. Srivastava: Optimality and duality in nondifferentiable multiobjective optimization involving d-type I and related functions, J. Math. Anal. Appl. 206 (1997) 465–479.https://doi.org/10.1006/jmaa.1997.5238Google Scholar

Suggested citation

B. Hernández-Jiménez, M. A. Rojas-Medar, R. Osuna-Gómez, A. Rufián-Lizana. “Characterization of Weakly Efficient Solutions for Non-Regular Multiobjective Programming Problems with Inequality-Type Constraints.” Journal of Convex Analysis 18 (2011), No. 3, 749–768. https://doi.org/10.68381/jca18048

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.