Some properties of semi-\(E\)-preinvex maps in Banach spaces (Q619751)

There are many generalizations of the definition of convex functions \(f: X\to Y\) (where \(X\) and \(Y\) are Banach spaces and \(Y\) is partially ordered) which are described by Jensen's inequality \[ f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y),\;\forall x,y,\;\forall\lambda\in [0,1], \] or, equivalently, \[ f(y+ \lambda(x- y))\leq f(y)+ \lambda(f(x)- f(y)),\;\forall x,y,\;\forall\lambda\in [0,1]. \] So the function \(f\) is called preinvex with respect to \(\eta: X \times X\to X\) if, in the second inequality, the difference \(x- y\) is replaced by the term \(\eta(x, y)\). The function \(f\) is called \(E\)-convex (or semi-\(E\)-convex) with respect to \(E: X\to X\) if in the inequalities \(x\) and \(y\) are (partly) replaced by \(E(x)\) and \(E(y)\). Combining both definitions, the authors introduce \(E\)-preinvex and semi-\(E\)-preinvex functions with respect to \(\eta\) and \(E\) according to \[ f(y +\lambda\eta(E(x), E(y)))\leq f(E(y))+ \lambda(E(f(x))- E(f(y)))\;\forall x,y,\;\forall\lambda\in[0,1], \] and \[ f(y+ \lambda\eta(E(x), E(y)))\leq f(y)+ \lambda(f(x)- f(y))\;\forall x,y,\;\forall\lambda\in[0, 1]. \] In the same manner, \(E\)-quasipreinvex and \(E\)-pseudopreinvex functions are introduced. The authors provide characterizations for these notions and discuss their applicability in mathematical optimization.