Fritz John type optimality and duality in nonlinear programming under weak pseudo-invexity

Slimani, Hachem; Radjef, Mohammed Said

doi:10.1051/ro/2014046

Slimani, Hachem ¹ ; Radjef, Mohammed Said ²

¹ LaMOS Research Unit, Computer Science Department, University of Bejaia, 06000 Bejaia, Algeria.
² LaMOS Research Unit, Operational Research Department, University of Bejaia, 06000 Bejaia, Algeria.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 49 (2015) no. 3, pp. 451-472.

Résumé

In this paper, we use a generalized Fritz John condition to derive optimality conditions and duality results for a nonlinear programming with inequality constraints, under weak invexity with respect to different ${(η_{i})}_{i}$ assumption. The equivalence between saddle points and optima, and a characterization of optimal solutions are established under suitable generalized invexity requirements. Moreover, we prove weak, strong, converse and strict duality results for a Mond-Weir type dual. It is shown in this study, with examples, that the introduced generalized Fritz John condition combining with the invexity with respect to different ${(η_{i})}_{i}$ are especially easy in application and useful in the sense of sufficient optimality conditions and of characterization of solutions.

Reçu le : 2012-02-24
Accepté le : 2014-08-01

MR Zbl

DOI : 10.1051/ro/2014046

Classification : 26A51, 90C26, 90C30, 90C46
Mots-clés : Nonlinear programming, weak (FJ)-pseudo-invexity, generalized Fritz John condition, generalized Fritz John stationary point, optimality, duality, saddle point

Affiliations des auteurs :

Slimani, Hachem ¹ ; Radjef, Mohammed Said ²

¹ LaMOS Research Unit, Computer Science Department, University of Bejaia, 06000 Bejaia, Algeria.
² LaMOS Research Unit, Operational Research Department, University of Bejaia, 06000 Bejaia, Algeria.

@article{RO_2015__49_3_451_0,
     author = {Slimani, Hachem and Radjef, Mohammed Said},
     title = {Fritz {John} type optimality and duality in nonlinear programming under weak pseudo-invexity},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {451--472},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {3},
     year = {2015},
     doi = {10.1051/ro/2014046},
     mrnumber = {3349129},
     zbl = {1338.90326},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ro/2014046/}
}

TY  - JOUR
AU  - Slimani, Hachem
AU  - Radjef, Mohammed Said
TI  - Fritz John type optimality and duality in nonlinear programming under weak pseudo-invexity
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2015
SP  - 451
EP  - 472
VL  - 49
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ro/2014046/
DO  - 10.1051/ro/2014046
LA  - en
ID  - RO_2015__49_3_451_0
ER  -

%0 Journal Article
%A Slimani, Hachem
%A Radjef, Mohammed Said
%T Fritz John type optimality and duality in nonlinear programming under weak pseudo-invexity
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2015
%P 451-472
%V 49
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ro/2014046/
%R 10.1051/ro/2014046
%G en
%F RO_2015__49_3_451_0

Slimani, Hachem; Radjef, Mohammed Said. Fritz John type optimality and duality in nonlinear programming under weak pseudo-invexity. RAIRO - Operations Research - Recherche Opérationnelle, Tome 49 (2015) no. 3, pp. 451-472. doi : 10.1051/ro/2014046. http://archive.numdam.org/articles/10.1051/ro/2014046/

Bibliographie
Cité par

T. Antczak, A class of B-(p,r)-invex functions and mathematical programming. J. Math. Anal. Appl. 286 (2003) 187–206. | DOI | MR | Zbl

T. Antczak, New optimality conditions and duality results of G type in differentiable mathematical programming. Nonlinear Anal. 66 (2007) 1617–1632. | DOI | MR | Zbl

T. Antczak, On $(p, r)$ -invexity-type nonlinear programming problems. J. Math. Anal. Appl. 264 (2001) 382–397. | DOI | MR | Zbl

T. Antczak, $r$ -preinvexity and $r$ -invexity in mathemetical programming. Comp. Math. Appl. 50 (2005) 551–566. | DOI | MR | Zbl

M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, New York (2006).

C.R. Bector, S.K. Suneja and C.S. Lalitha, Generalized B-vex functions and generalized B-vex programming. J. Optim. Theor. Appl. 76 (1993) 561–576. | DOI | MR | Zbl

A. Ben-Israel and B. Mond, What is invexity ? J. Austral. Math. Soc. Ser. B 28 (1986) 1–9. | DOI | MR | Zbl

B.D. Craven, Invex functions and constrained local minima. Bull. Austral. Math. Soc. 24 (1981) 357–366. | DOI | MR | Zbl

B.D. Craven and B.M. Glover, Invex functions and duality. J. Austral. Math. Soc. Ser. A 39 (1985) 1–20. | DOI | MR | Zbl

F. Dinuzzo, C.S. Ong, P. Gehler and G. Pillonetto, Learning output kernels with block coordinate descent, in Proc. of the 28th International Conference on Machine Learning, Bellevue, WA, USA, 2011.

K.H. Elster, R. Nehse, Optimality conditions for some nonconvex problems. Springer-Verlag, New York 1980. | MR | Zbl

C. Fulga and V. Preda, Nonlinear programming with E-preinvex and local E-preinvex functions. Eur. J. Oper. Res. 192 (2009) 737–743. | DOI | MR | Zbl

M.A. Hanson, On sufficiency of the Kuhn-Tuker conditions. J. Math. Anal. Appl. 80 (1981) 445–550. | DOI | MR | Zbl

M.A. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization. Math. Program. 37 (1987) 51–58. | DOI | MR | Zbl

M. Hayashi and H. Komiya, Perfect duality for convexlike programs. J. Optim. Theor. Appl. 38 (1982) 179–189. | DOI | MR | Zbl

V. Jeyakumar and B. Mond, On generalized convex mathematical programming. J. Austral. Math. Soc. Ser. B 34 (1992) 43–53. | DOI | MR | Zbl

F. John, Extremum problems with inequalities as side conditions, in K.O. Friedrichs, O.E. Neugebauer and J.J. Stoker Eds., Studies and Essays, Courant Anniversary Volume. Wiley (Interscience), New York (1948) 187–204. | MR | Zbl

W. Karush, Minima of functions of several variables with inequalities as side conditions, Master’s Thesis. Department of Mathematics, University of Chicago, 1939. | MR

R.N. Kaul and S. Kaur, Optimality criteria in nonlinear programming involving nonconvex functions. J. Math. Anal. Appl. 105 (1985) 104–112. | DOI | MR | Zbl

R.N. Kaul, S.K. Suneja and M.K. Srivastava, Optimality criteria and duality in multiple-objective optimization involving generalized invexity. J. Optim. Theor. Appl. 80 (1994) 465–482. | DOI | MR | Zbl

H.W. Kuhn and A.W. Tucker, Nonlinear programming, in Proc. of the Second Berkeley Symposium on Mathematical Statistics and Probability, edited by J. Neyman, University of California Press, Berkeley, California (1951) 481–492. | MR | Zbl

O.L. Mangasarian, Nonlinear Programming. McGrawHill, New York (1969). | MR

D.H. Martin, The essence of invexity. J. Optim. Theor. Appl. 47 (1985) 65–76. | DOI | MR | Zbl

S.K. Mishra, R.P. Pant and J.S. Rautela, Duality in vector optimization under type I $α$ -invexity in banach spaces. Numer. Funct. Anal. Optim. 29 (2008) 1128–1139. | DOI | MR | Zbl

S.K. Mishra, S.Y. Wang and K.K. Lai, On non-smooth $α$ -invex functions and vector variational-like inequality. Optim Lett. 02 (2008) 91–98. | DOI | MR | Zbl

B. Mond and T. Weir, Generalized concavity and duality, in Generalized concavity in optimization and economics, edited by S. Schaible and W.T. Ziemba, Academic Press, New York (1981) 263–276. | Zbl

H. Nickisch and M. Seeger, Multiple kernel learning: a unifying probabilistic viewpoint. arXiv:1103.0897v2 [stat.ML] 30 March (2011).

R. Osuna-Gomez, A. Beato-Morero and A. Rufian-Lizana, Generalized convexity in multiobjective programming. J. Math. Anal. Appl. 233 (1999) 205–220. | DOI | MR | Zbl

R. Pini and C. Singh, A survey of recent [1985-1995] advances in generalized convexity with applications to duality theory and optimality conditions. Optim. 39 (1997) 311–360. | DOI | MR | Zbl

N.G. Rueda and M.A. Hanson, Optimality criteria in mathematical programming involving generalized invexity. J. Math. Anal. Appl. 130 (1988) 375–385. | DOI | MR | Zbl

H. Slimani and M.S. Radjef, Duality for nonlinear programming under generalized Kuhn-Tucker condition. Int. J. Optim. Theor. Methods Appl. 1 (2009) 75–86. | MR | Zbl

H. Slimani and M.S. Radjef, Fonctions invexes généralisées et optimisation vectorielle: optimalité, caractérisations, dualité et applications. Editions Universitaires Européennes, Saarbrücken, 2011.

H. Slimani and M.S. Radjef, Multiobjective programming under generalized invexity: optimality, duality, applications. LAP Lambert Academic Publishing, Saarbrücken, 2010. | MR

H. Slimani and M.S. Radjef, Nondifferentiable multiobjective programming under generalized $d_{I}$ -invexity. Eur. J. Oper. Res. 202 (2010) 32–41. | DOI | MR | Zbl

M. Soleimani-damaneh and M.E. Sarabi, Sufficient conditions for nonsmooth $r$ -invexity. Numer. Funct. Anal. Optim. 29 (2008) 674–686. | DOI | MR | Zbl

T. Weir and B. Mond, Pre-invex functions in multiple objective optimization. J. Math. Anal. Appl. 136 (1988) 29–38. | DOI | MR | Zbl

Cité par Sources :