Notions of invexity of a function and of a set are generalized. The notion of an invex function with respect to η can be further extended with the aid of p-invex sets. Slight generalization of the notion of p-invex sets with respect to η leads to a new class of functions. A family of real functions called, in general, p r -preinvex functions with respect to η (without differentiability) or p r -invex functions with respect to η (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 η.  2001 Academic Press Key Words: p r -invex set with respect to η; r-invex set with respect to η; p r -pre-invex function with respect to η p r -invex function with respect to η.

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In the present paper, we introduce the concept of G-pre-invex functions with respect to defined on an invex set with respect to . These function unify the concepts of nondifferentiable convexity, pre-invexity and r-pre-invexity. Furthermore, relationships of Gpre-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.

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