I have the following quadratic maximization problem

$\max_{\mathbf X} \quad tr(\mathbf A\mathbf X\mathbf B\mathbf X^H)+tr(\mathbf C\mathbf X)+tr(\mathbf C^H\mathbf X^H)$

subject to the quadratic constraint $tr(\mathbf X \mathbf X^H)\leq r$ where $\mathbf A, \mathbf B \succeq \mathbf 0$ (positive semi-definite) and $\mathbf C$ is any matrix (not necessarily hermitian).

I am looking for a closed-form solution/structure for $\mathbf X$ or even a nice upper bound. A simple upper bound of it can be found using inequality results, so we know that the problem has a finite maximum.

I would really appreciate any help/comment/solution :)

Saeed

• I don't think that there will be a closed-form solution; this problem will demand a numerical solution. – Suvrit Jun 22 '11 at 22:58

$\max_x f(x)$ subject to $g(x) < 0$, where $x$ is the set of components $x_i^j$ of the matrix $X$.
Use one Lagrange multiplier to write $f(x) + \lambda g(x)$. Both nonlinear terms are quadratic, so this is the most favorable case for the purpose of obtaining solutions. To take the derivative of the expression above with respect to the set of unknowns $x$, index notation as in tensor calculus could be helpful. Then you are in the standard form of constrained optimization.