# Objective function of complex variables, with real constraints

I have an objective function $$L:\mathbb{C}^M \to \mathbb{R}$$ that can be written:

$$L(x) = x^H A x - b^H x - x^H b + d$$

where superscript $$H$$ denotes the conjugate transpose. $$A \in \mathbb{C}^M$$ is a hermitian matrix of size $$M$$ (so $$A^H = A$$), $$b$$ is a complex vector $$b \in \mathbb{C}^M$$, and $$d$$ is a real constant. I believe $$L(x)$$ is known to be convex, as it can be equivalently written as the squared euclidean norm of an affine function of $$x$$ (i.e. $$L(x) = (Fx - g)^H (Fx - g)$$).

I'm looking to solve the following minimization problem, with or without the added constraint that the vector $$x$$ is real. Formally:

$$\text{minimize} \; L(x)$$ $$\text{subject to} \; \text{Im}(x) = 0$$

In other words, I want to find the real vector $$x \in \mathbb{R}^M$$ that minimizes the convex loss function, which is formally a function of complex input vector $$x$$. I have a feeling this should be straightforward but I'm not sure exactly how to go about doing so.

In the case that the "reality" constraint is quite non-trivial, it can be discarded. Discarding the constraint, I believe Wirtinger calculus can be used to solve this. Unfortunately, I'm not 100% clear on how to apply Wirtinger derivatives to actually obtain first-order conditions for a minima.

For example, to solve the unconstrained problem, is it valid to write: $$L(x) = x^H A x - 2 \text{Re}(b^H x) + d$$ $$\nabla{L}(\hat{x}) = A\hat{x} - 2\text{Re}(b) = 0$$ $$A \hat{x} = 2\text{Re}(b)$$

Thus obtaining the solution $$\hat{x} = A^{-1} 2\text{Re}(b)$$ ?

First, write $$A = H + i K$$, where both $$H,K$$ are matrices with real entries. The fact that $$A$$ is Hermitian means that $$A = A^H \implies H + iK = H^T - i K^T \implies \begin{cases} H = H^T,\\ K = -K^T. \end{cases}$$ From the fact that $$K^T = -K$$, we find that for any real vector $$x$$, $$x^TKx = (x^TKx)^T = x^TK^Tx = -x^TKx \implies x^TKx = 0.$$ Thus, for any such $$x$$, $$x^TAx = x^T(H + iK)x = x^THx + ix^TKx = x^THx.$$ Similarly, we note that for any real $$x$$, we have $$b^Hx + x^Hb = b^Hx + \overline{b^Hx} = b^Hx + \overline{b^H} x = (b + \bar b)^T x,$$ where $$\bar x$$ denotes the complex-conjugate of $$x$$. Note that $$v := b + \bar b = 2 \operatorname{Re}(b)$$.
Putting everything together, we see that it suffices to minimize the unconstrained objective function $$f:\Bbb R^M \to \Bbb R$$ given by $$f(x) = x^THx - v^Tx + d,$$ where $$H$$ is a real, symmetric, positive definite matrix and $$v \in \Bbb R^n$$.
• Thank you so much, this makes a lot of sense! :) I had previously been solving the problem by simply "discarding" the complex components of $A$ and $b$, but was not sure at all if that was correct — but now it is clear that this does in fact solve the same optimization problem. Thanks again! Jul 20, 2021 at 15:17