\((p,r)\)-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 \(\eta\) .


90C26 Nonconvex programming, global optimization
26B25 Convexity of real functions of several variables, generalizations
90C29 Multi-objective and goal programming
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