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$(p,r)$-invex sets and functions. (English) Zbl 1051.90018
Summary: Notions of invexity of a function and of a set are generalized. The notion of an invex function with respect to $\eta$ can be further extended with the aid of $p$-invex sets. Slight generalization of the notion of $p$-invex sets with respect to $\eta$ leads to a new class of functions. A family of real functions called, in general, $(p, r)$-pre-invex functions with respect to $\eta$ (without differentiability) or $(p,r)$-invex functions with respect to $\eta$ (in the differentiable case) is introduced. Some (geometric) properties of these classes of functions are derived. Sufficient optimality conditions are obtained for a nonlinear programming problem involving $(p, r)$-invex functions with respect to $\eta$ .

90C26Nonconvex programming, global optimization
26B25Convexity and generalizations (several real variables)
90C29Multi-objective programming; goal programming
Full Text: DOI
[1] Antczak, T.: Invexity in optimization theory. (1998)
[2] T. Antczak, r-Invexity in Mathematical Programming, to be published. · Zbl 1097.90042
[3] Antczak, T.: (p,r)-invex sets and functions. (1998)
[4] Avriel, M.: R-convex functions. Math. programming 2, 309-323 (1972) · Zbl 0249.90063
[5] Bazaraa, M. S.; Sherali, H. D.; Shetty, C. M.: Nonlinear programming: theory and algorithms. (1991) · Zbl 1140.90040
[6] Ben-Israel, A.; Mond, B.: What is invexity?. J. austral. Math. soc. Ser. B 28, 1-9 (1986) · Zbl 0603.90119
[7] Hanson, M. A.: On sufficiency of the Kuhn--Tucker conditions. J. math. Anal. appl. 80, 545-550 (1981) · Zbl 0463.90080
[8] Hanson, M. A.; Mond, B.: Necessary and sufficient conditions in constrained optimization. Math. programming 37, 51-58 (1987) · Zbl 0622.49005
[9] Mangasarian, O. L.: Nonlinear programming. (1969) · Zbl 0194.20201
[10] Martin, D. H.: The essence of invexity. J. optim. Theory appl. 42, 65-76 (1985) · Zbl 0552.90077
[11] Mirtinovic, D. S.: Elementarne nierówności. (1972)
[12] Rockafellar, R. T.: Convex analysis. (1970) · Zbl 0193.18401
[13] Rueda, N. G.; Hanson, M. A.: Optimality criteria in mathematical programming involving generalized invexity. J. math. Anal. appl. 130, 375-385 (1988) · Zbl 0647.90076
[14] Weir, T.; Jeyakumar, V.: A class of nonconvex functions and mathematical programming. Bull. austral. Math. soc. 38, 177-189 (1988) · Zbl 0639.90082