Duality for fractional complex programming with generalized convexity (Q2764702)

The authors consider four kinds of parameter free dual models for the following fractional complex programming problem involving nonlinear analytic functions with generalized convexity: NEWLINE\[NEWLINE\text{Minimize} \frac{\text{Re} [f(z, \overline z)+(z^HAz)^{1/2}]} {\text{Re}[g(z,\overline z)-(z^HBz)^{1/2}]}= \frac {\psi(z)}{\varphi(z)}\tag{P}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\text{subject to }z\in\mathbb{C}^n\text{ and }h(z, \overline z)\in S \subset\mathbb{C}^mNEWLINE\]NEWLINE where \(f,g:\mathbb{C}^{2n}\to\mathbb{C}\) and \(h: \mathbb{C}^{2n} \to\mathbb{C}^m\) are analytic functions on a specific set \(Q=\{(z,z')\in \mathbb{C}^n \times \mathbb{C}^n\mid z'=\overline z\}\subset \mathbb{C}^{2n}\); \(z^H\) means the transpose of \(\overline z\); \(S\) is a polyhedral cone; \(A\) and \(B\) are positive semidefinite Hermitian matrices. It is assumed that \(\psi(z)\geq 0\) and \(\varphi (z)>0\).NEWLINENEWLINE The weak, strong and strict converse duality theorems are proved.