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$\left(p,r\right)$-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, $\left(p,r\right)$-pre-invex functions with respect to $\eta$ (without differentiability) or $\left(p,r\right)$-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 $\left(p,r\right)$-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