\((p,r)\)-invex sets and functions. (English) Zbl 1051.90018

Antczak, Tadeusz

J. Math. Anal. Appl. 263, No. 2, 355-379 (2001).

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 of real functions of several variables, generalizations
90C29	Multi-objective and goal programming

Keywords:

\((p,r)\)-invex set with respect to \(\eta\); \(r\)-invex set with respect to \(\eta\); \((p,r)\)-pre-invex function with respect to \(\eta\); \((p,r)\)-invex function with respect to \(\eta\)

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