## $$G$$-pre-invex functions in mathematical programming.(English)Zbl 1219.90126

Summary: We introduce the concept of $$G$$-pre-invex functions with respect to $$\eta$$ defined on an invex set with respect to $$\eta$$. These function unify the concepts of nondifferentiable convexity, pre-invexity and $$r$$-pre-invexity. Furthermore, relationships of $$G$$-pre-invex functions to various introduced earlier pre-invexity concepts are also discussed. Some (geometric) properties of this class of functions are also derived. Finally, optimality results are established for optimization problems under appropriate $$G$$-pre-invexity conditions.

### MSC:

 90C26 Nonconvex programming, global optimization 26B25 Convexity of real functions of several variables, generalizations
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### References:

 [1] Antczak, T., $$(p, r)$$-invex sets and functions, J. math. anal. appl., 80, 545-550, (2001) [2] Antczak, T., Relationships between pre-invex concepts, Nonlinear anal., 60, 349-367, (2005) · Zbl 1103.90398 [3] Antczak, T., Mean value in invexity analysis, Nonlinear anal., 60, 1473-1484, (2005) · Zbl 1100.26005 [4] Antczak, T., $$r$$-pre-invexity and $$r$$-invexity in mathematical programming, Comput. math. appl., 50, 551-566, (2005) · Zbl 1129.90052 [5] Antczak, T., New optimality conditions and duality results of G-type in differentiable mathematical programming, Nonlinear anal., 66, 1617-1632, (2007) · Zbl 1143.90034 [6] M. Avriel, W.E. Diewert, S. Schaible, I. Zang, Generalized Concavity, Plenum Press, New York, London, 1975. [7] Ben-Israel, A.; Mond, B., What is invexity?, J. austral. math. soc. ser. B, 28, 1-9, (1986) · Zbl 0603.90119 [8] Craven, B.D., Invex functions and constrained local minima, Bull. austral. math. soc., 24, 357-366, (1981) · Zbl 0452.90066 [9] Hanson, M.A., On sufficiency of the kuhn – tucker conditions, J. math. anal. appl., 80, 545-550, (1981) · Zbl 0463.90080 [10] Mohan, S.R.; Neogy, S.K., On invex sets and preinvex functions, J. math. anal. appl., 189, 901-908, (1995) · Zbl 0831.90097 [11] Pini, R., Invexity and generalized convexity, Optimization, 22, 513-525, (1991) · Zbl 0731.26009 [12] Suneja, S.K.; Singh, C.; Bector, C.R., Generalizations of pre-invex functions and B-vex functions, J. optim. theory appl., 76, 577-587, (1993) · Zbl 0802.49026 [13] Weir, T.; Jeyakumar, V., A class of nonconvex functions and mathematical programming, Bull. austral. math. soc., 38, 177-189, (1988) · Zbl 0639.90082 [14] Weir, T.; Mond, B., Preinvex functions in multiple objective optimization, J. math. anal. appl., 136, 29-38, (1988) · Zbl 0663.90087
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