Nonlinear Programming in Complex Space: Necessary Conditions

Abstract

Necessary conditions of the Kuhn–Tucker type are given for two classes of nonlinear programming problems over polyhedral cones in finite-dimensional complex space. The first class consists of problems of the form \[{\text{Minimize }} \operatorname{Re} \quad f(z)\quad {\text{subject to}}\, g(z) \in S,\] where S is a polyhedral cone in $C^m $ and $f:C^n \to C$,$g:C^n \to C^m $ are analytic functions. A necessary condition for a feasible point $z^0 $ to be optimal is that there exist a vector $u \in S^ * $ such that $\overline {\nabla f(z^0 )} = [D_z^H g(z^0 )]u$ and $\operatorname{Re} (g(z^0 ),u) = 0$. The second class consists of problems of the form \[{\text{Minimize }} \operatorname{Re} \quad f(z,\bar z)\quad {\text{subject to }} g(z,\bar z) \in S,\] where $f:C^{2n} \to C$, $g:C^{2n} \to C^m $ are analytic. A necessary condition for a feasible point $z^0 $ to be optimal is that there exist $u \in S^ * $ such that $\overline {\nabla _z f(z^0 ,\overline {z^0 } )} + \nabla _{\overline z } f(z^0 ,\overline {...