# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
$(p,r)$-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 $\eta$ .

##### MSC:
 90C26 Nonconvex programming, global optimization 26B25 Convexity and generalizations (several real variables) 90C29 Multi-objective programming; goal programming
Full Text:
##### References:
 [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