I have the following problem $$ \max_x ~~ \sum_k | a^H_k B x |^2 \\ \text{s. t.} ~~ x^H B^H B x \leq c \\ $$ where $ x\in \mathbb{C}^N $, $ B \in \mathbb{C}^{M \times N} $, $ M > N $.
I know the problem is non-convex since the objective is a convex function. However, I would like to find a close-form solution for this problem. Is it possible? One idea is to use the Lagrangian although I may not attain a global optimum. However, even for this case, I am unable to go further. This is what I have thus far.
The Lagrangian is $$ L(x,\lambda) = -\sum_k x^H B^H a_k a^H_k B x + \lambda (x^H B^H B x - c) $$
The KKT conditions are:
(1): $ (\sum_k B^H a_k a^H_k B ) x = \lambda B^H B x $
(2): $ \lambda (x^H B^H B x - c) = 0 $
(3): $ \lambda \geq 0 $
(4): $ x^H B^H B x \leq c $
Case I: When $ x^H B^H B x \leq c $, then $ \lambda = 0 $ and $ ( \sum_k B^H a_k a^H_k B ) x = 0 $
Case II: When $ x^H B^H B x = c $, then $ \lambda > 0 $. If we multiply by $ x^H $ at both sides of (1) then it yields $ \lambda \cdot c - x^H ( \sum_k B^H a_k a^H_k B ) x = 0 $.
Any hints how to proceed? Or do you see any alternative ways to approach the problem? Thanks in advance.
Note that $\sum_k |a_k^* B x|^2 = \sum_k x^*B^* a_k a_k^* B x = x^* B^* (\sum_k a_ka_k^*)B x$, so the problem becomes $\max \{ x^*B^* A B x | \ \|Bx\| \le \sqrt{c} \} $ with $A=\sum_k a_ka_k^*$.
Let $Q$ be such that $Q^*Q = I$ and ${\cal R} Q = {\cal R}B$, then we have \begin{eqnarray} \max \{ x^*B^* A B x | \ \|Bx\| \le \sqrt{c} \} &=& \max \{ w^* A w | \ \|w\| \le \sqrt{c} , w \in {\cal R B}\} \\ &=& \max \{ w^* Q^*AQ w | \ \|Qw\| \le \sqrt{c}\} \\ &=& \max \{ w^* Q^*AQ w | \ \|w\| \le \sqrt{c}\} \\ &=& c\max \{ w^* Q^*AQ w | \ \|w\| \le 1\} \\ &=& c \lambda_\max (Q^*AQ) \end{eqnarray}