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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