Nonlinear Programming in Complex Space: Necessary Conditions
Robert A. Abrams, Adi Ben-Israel · SIAM Journal on Control · 1971
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 $\overline { abla f(z^0 )} = [D_z^H g(z^0 )]u$ 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 { abla _z f(z^0 ,\overline {z^0 } )} + abla _{\overline z } f(z^0 ,\overline {z^0 } ) = [D_z^H g(z^0 ,\overline {z^0 } )]u + [D_{\overline z }^T g(z^0 ,\overline {z^0 } )]\overline u $ and $\operatorname{Re} (g(z^0 ,\overline {z^0 } ),u) = 0$. The derivation of necessary conditions for problems of the first class is analogous to that used in the real case. For problems of the second class, necessary conditions are obtained by considering an equivalent problem in the form of the first class.