New optimality conditions and duality results of \(G\) type in differentiable mathematical programming. (English) Zbl 1143.90034

Antczak, Tadeusz

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 7, 1617-1632 (2007).

Summary: A new class of differentiable functions, called \(G\)-invex functions with respect to \(\eta\) , is introduced by extending the definition of invex functions. New necessary optimality conditions of \(G\)-F. John and \(G\)-Karush-Kuhn-Tucker type are obtained for differentiable constrained mathematical programming problems. The \(G\)-invexity concept introduced is used to prove the sufficiency of these necessary optimality conditions. Further, a so-called \(G\)-Mond-Weir-type dual is formulated and various duality results are also established by assuming the functions involved to be \(G\)-invex with respect to the same function \(\eta\) .

MSC:

90C46	Optimality conditions and duality in mathematical programming
90C26	Nonconvex programming, global optimization
26B25	Convexity of real functions of several variables, generalizations

Keywords:

\(G\)-invex function with respect to \(\eta\); \(G\)-F. John necessary optimality conditions; \(G\)-Karush-Kuhn-Tucker optimality conditions; \(G\)-type constraint qualification; duality

References:

[1]	Antczak, T., \((p, r)\)-invex sets and functions, Journal of mathematical analysis and applications, 80, 545-550, (2001)
[2]	Antczak, T., Relationships between pre-invex concepts, Nonlinear analysis, 60, 349-367, (2005) · Zbl 1103.90398
[3]	Antczak, T., \(r\)-pre-invexity and \(r\)-invexity in mathematical programming, Computers and mathematics with applications, 50, 551-566, (2005) · Zbl 1129.90052
[4]	Antczak, T., An \(\eta\)-approximation method in mathematical programming problems, Numerical functional analysis and optimization, 25, 5-6, 423-439, (2004)
[5]	T. Antczak, \(G\)-pre-invexity in mathematical programming, 2005 (in press)
[6]	Avriel, M.; Diewert, W.E.; Schaible, S.; Zang, I., Generalized concavity, (1987), Plenum Press New York, London
[7]	Avriel, M., \(r\)-convex functions, Mathematical programming, 2, 309-323, (1972) · Zbl 0249.90063
[8]	Bazaraa, M.S.; Sherali, H.D.; Shetty, C.M., Nonlinear programming: theory and algorithms, (1991), John Wiley and Sons New York
[9]	Ben-Israel, A.; Mond, B., What is invexity?, Journal of Australian mathematical society series B, 28, 1-9, (1986) · Zbl 0603.90119
[10]	Craven, B.D., Invex functions and constrained local minima, Bulletin of the Australian mathematical society, 24, 357-366, (1981) · Zbl 0452.90066
[11]	Hanson, M.A., On sufficiency of the kuhn – tucker conditions, Journal of mathematical analysis and applications, 80, 545-550, (1981) · Zbl 0463.90080
[12]	Hanson, M.A.; Mond, B., Necessary and sufficient conditions in constrained optimization, Mathematical programming, 37, 51-58, (1987) · Zbl 0622.49005
[13]	Mangasarian, O.L., Nonlinear programming, (1969), McGraw-Hill New York · Zbl 0194.20201
[14]	Martin, D.H., The essence of invexity, Journal of optimization theory and applications, 42, 65-76, (1985) · Zbl 0552.90077
[15]	Martos, B., Nonlinear programming. theory and methods, (1975), North Holland Amsterdam · Zbl 0357.90027
[16]	Mohan, S.R.; Neogy, S.K., On invex sets and preinvex functions, Journal of mathematical analysis and applications, 189, 901-908, (1995) · Zbl 0831.90097
[17]	Pini, R., Invexity and generalized convexity, Optimization, 22, 513-525, (1991) · Zbl 0731.26009
[18]	Rueda, N.G.; Hanson, M.A., Optimality criteria in mathematical programming involving generalized invexity, Journal of mathematical analysis and applications, 130, 375-385, (1988) · Zbl 0647.90076
[19]	Suneja, S.K.; Singh, C.; Bector, C.R., Generalizations of pre-invex functions and \(B\)-vex functions, Journal of optimization theory and applications, 76, 577-587, (1993) · Zbl 0802.49026
[20]	Weir, T.; Jeyakumar, V., A class of nonconvex functions and mathematical programming, Bulletin of the Australian mathematical society, 38, 177-189, (1988) · Zbl 0639.90082