Some characterizations of strongly preinvex functions.(English)Zbl 1093.26006

Let $$K$$ be a subset of a Hilbert space $$H$$ and $$\eta:K\times K\rightarrow H$$ be a function. The set $$K$$ is called invex with respect to $$\eta$$ if for all $$u,v\in K$$ and $$t\in[0,1]$$, $$u+t\eta(v,u)\in K$$. Given such a set, a function $$F:K\rightarrow\mathbb{R}$$ is called preinvex with respect to $$\eta$$ if or all $$u,v\in K$$ and $$t\in[0,1]$$, $$F(u+t\eta(v,u))\leq (1-t)F(u)+tF(v)$$. In this paper, one considers another function $$\alpha :K\times K\rightarrow\mathbb{R}\backslash\{0\}$$ and defines the set $$K$$ to be $$\alpha$$-invex with respect to $$\eta$$ and $$\alpha$$ if for all $$u,v\in K$$ and $$t\in[0,1]$$, $$u+t\alpha(v,u)\eta(v,u)\in K$$. Likewise, the function $$F$$ is called $$\alpha$$-preinvex with respect to $$\alpha$$ and $$\eta$$ if for all $$u,v\in K$$ and $$t\in[0,1]$$, $$F(u+t\alpha(v,u)\eta(v,u))\leq (1-t)F(u)+tF(v)$$. Obviously, these notions are a transcription of the previous ones, if one considers the function $$\eta_{1}(\cdot,\cdot)=\alpha(\cdot ,\cdot)\eta(\cdot,\cdot)$$. Many other notions and properties introduced in this paper can be derived in the same way from the usual generalized invexity notions that can be found in other papers in the field. When this is not the case, mistakes occur frequently: for instance, in the proof of Theorem 3.2, relation (3.8) does not follow from (3.7), because $$\bar{\mu} =\mu/\alpha(v_{t},u)$$ in (3,7), but $$\bar{\mu}=\mu/\alpha(v,u)$$ is needed in (3.8) . In the proof of Theorem 3.5, relation (3.10) does not follow from (3.9), because (3.9) holds only for the particular choice of $$u,v$$ for which the relation in line 2 of the same page holds, thus one cannot take $$v=v_{t}$$.

MSC:

 26B25 Convexity of real functions of several variables, generalizations 90C25 Convex programming 90C46 Optimality conditions and duality in mathematical programming

Keywords:

invex functions; preinvex functions
Full Text:

References:

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