Convex Optimization · Problem 4.42

Duality — Boyd & Vandenberghe

Mathematics — Optimization · complex-lmis-sdps

Problem

Complex LMIs and SDPs. A complex LMI has the form

x_1 F_1 + \cdots + x_n F_n + G \preceq 0

where

F_1, \ldots, F_n, G

are complex

n \times n

Hermitian matrices, i.e.,

F_i^H = F_i

G^H = G

, and

x \in \mathbf{R}^n

is a real variable. A complex SDP is the problem of minimizing a (real) linear function of

x

subject to a complex LMI constraint. Complex LMIs and SDPs can be transformed to real LMIs and SDPs, using the fact that

X \succeq 0 \iff \left[ \begin{array}{cc} \Re X & -\Im X \\ \Im X & \Re X \end{array} \right] \succeq 0,

where

\Re X \in \mathbf{R}^{n \times n}

is the real part of the complex Hermitian matrix

X

, and

\Im X \in \mathbf{R}^{n \times n}

is the imaginary part of

X

. Verify this result, and show how to pose a complex SDP as a real SDP.