**AUTHORS:**Jiagen Liao, Tingsong Du

**Download as PDF**

**ABSTRACT:**
A new class of generalized invex sets and generalized preinvex functions, called m-invex sets and
(B, m)-preinvex functions are defined by modulating the m-convexity and B-preinvexity. Furthermore, some characterizations
of (B, m)-preinvex functions are presented and the sufficient conditions of (B, m)-preinvex programming
are established. In addition the nonlinear multi-objective programming with (B, m)-preinvex functions are
considered and some relationships between vector critical point and weakly efficient solution for multi-objective
programming with (B, m)-preinvexity are researched.

**KEYWORDS:**
(B, m)-preinvex functions, m-invex sets, (B, m)-invex sets, (B, m)-preinvex programming, multiobjective
programming

**REFERENCES:**

[1] G. Bhatia and R. R. Sahay, Strict global minimizers and higher-order generalized strong invexity in multiobjective optimization, Journal of Inequalities and Applications, 2013, (2013), pp. 31, doi: 10.1186/1029-242X-2013-31.

[2] T. Emam, Roughly B-invex programming problems, Journal of Mathematical Analysis and Applications, 48, (2011), pp. 173–188, doi: 10.1007/s10092-010-0034-5.

[3] C. Fulga and V. Preda, Nonlinear programming with E-preinvex and local E-preinvex functions, European Journal of Operational Research, 192, (2009), pp. 737–743, doi: 10.1016/j.ejor.2007.11.056.

[4] A. M. Geoffrion, Proper Eficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, 22, (1968), pp, 618–630.

[5] R. Osuna-Go´mez, A. Beato-Moreno and A. Rufian-Lizana, Generalized Convexity in Multiobjective Programming, Journal of Mathematical Analysis and Applications, 233, (1999), pp. 205–220.

[6] H. H. Jiao and S. Y. Liu, Semilocal Epreinvexity and its applications in nonlinear multiple objective fractional programming, Journal of Inequalities and Applications, 2011, (2011), pp. 116, doi: 10.1186/1029-242X-2011- 116.

[7] A. Jayswal, A. K. Prasad and I. Ahmad and R. P. Agarwal, Duality for semi-infinite programming problems involving (Hp, r)-invex functions, Journal of Inequalities and Applications, 2013, (2013), pp. 200, doi: 10.1186/1029- 242X-2013-200.

[8] J. G. Liao and T. S. Du, On some characterizations of sub-b-s-convex functions, Filomat, accepted, in press, 2015.

[9] X. J. Long and J. W. Peng, Semi-B-Preinvex Functions, Journal of Optimization Theory and Applications, 131, (2006), pp. 301–305, doi: 10.1007/s10957-006-9146-0.

[10] H. Z. Luo and H. X. Wu, On the Characterization of Preinvex Functions, Journal of Optimization Theory and Applications, 138, (2008), pp. 297– 304, doi: 10.1007/s10957-008-9373-7.

[11] S. K. Mishra, S. Y. Wang and K. K. Lai, Nonsmooth Continuous-Time Multiobjective Optimization Problems with Invexity, WSEAS Transactions on Mathematics, 26, (2006), pp. 367– 371.

[12] C. Singh and C. R. Bector, B-Vex Functions, Journal of Optimization Theory and Applications, 71, 2, (1991), pp. 237–253.

[13] S. K. Suneja and C. Singh and C. R. Bector, Generalization of Preinvex and B-Vex Functions, Journal of Optimization Theory and Applications, 76, 3, (1993), pp. 577–587.

[14] G. Toader, Some generalisations of the convexity, Proceedings of the Colloquium on Approximation and Optimization, (1984), pp. 329–338.

[15] X. M. Yang and D. Li, On Properties of Preinvex Functions, Journal of Mathematical Analysis and Applications, 256, (2001), pp. 229–241, doi: 10.1006/jmaa.2000.7310.

[16] D. H. Yuan and X. L. Liu and S. Y. Yang and G. M. Lai, Nondifferentiable mathematical programming involving (G, β)-invexity, Journal of Inequalities and Applications, 2012, (2012), pp. 256, doi: 10.1186/1029-242X-2012-256.

[17] X. Y. Yang and X. Q. Yang and K. L. Teo, Explicitly B-preinvex functions, Journal of Computational and Applied Mathematics, 146,