Maximizing the expectation of the quadratic product $\mathbf{x}^{\mathrm{T}} \mathbf{A} \mathbf{x}$ w.r.t. $\mathbf{A}$

Question

Let $\mathbf{A}$ be a $N$-dimensional positive semidefinite matrix, and let $\mathbf{x} \sim \mathcal{N}(\mathbf{m},\mathbf{C})$. I'm looking for a solution to the following problem:

$$\mathrm{argmax}_{\mathbf{A}} \mathbb{E}_{\mathbf{x}} \big[ \mathbf{x}^{\mathrm{T}} \mathbf{A} \mathbf{x} \big], \quad \mathrm{s.t} \quad \mathrm{Tr}(\mathbf{A}) \leq 1.$$

I know that, for a given $\mathbf{A}$, we have that $\mathbb{E}_{\mathbf{x}} \big[ \mathbf{x}^{\mathrm{T}} \mathbf{A} \mathbf{x}\big] = \mathrm{Tr}(\mathbf{A} \mathbf{C}) + \mathbf{m}^{\mathrm{T}} \mathbf{A} \mathbf{m}$ and therefore I can rewrite the above problem without the expectation as

$$\mathrm{argmax}_{\mathbf{A}} \mathrm{Tr} \big( \mathbf{A} (\mathbf{C} + \mathbf{m} \mathbf{m}^{\mathrm{T}}) \big), \quad \mathrm{s.t} \quad \mathrm{Tr}(\mathbf{A}) \leq 1$$

where $(\mathbf{C} + \mathbf{m} \mathbf{m}^{\mathrm{T}})$ is positive semidefinite. But how can I solve the latter problem?

Answer 1

This is an LMI (Linear Matrix Inequality) constrained Linear Program. It would be a Linear Programming problem but for the constraint that A be positive semidefinite.

Here is an example with made up data, solved using CVX under MATLAB.

C = rand(4,4); C = C*C'; % generate a covariance matrix
m = randn(4,1); % generate an m vector
cvx_begin
variable A(4,4)
maximize(trace(A*(C+m*m')))
trace(A) <= 1
A == semidefinite(4)
cvx_end

The resulting optimal A with these values of C and m is

   0.464556277270533   0.231014404408925   0.440445162363771  -0.037204116358224
   0.231014404408925   0.114878772904881   0.219024436539946  -0.018500851686589
   0.440445162363771   0.219024436539946   0.417585447860668  -0.035273171137730
  -0.037204116358224  -0.018500851686589  -0.035273171137730   0.002979501907052

which has 3 eigenvalues equal to zero, and one eigenvalue equal to 1.