# Analytical algorithm to obtain solution to convex optimization problem.

Assume a vector $\vec{P}$ with N elements $\in \mathbb{R}^+$ and constants $T_P$ and $\epsilon$. The vector $\vec{P}$ is arranged in a column $(N\times 1)$.

Consider the problem: \begin{aligned} &{\text{minimize}} & & \sum_{k \in \mathcal{K}} \frac{1}{P_{k}^2} \\ & \text{subject to} & & \vec{P}^H\vec{P} \le T_P \\ &&& \vec{Z}^H\vec{Z} \le \epsilon \end{aligned} where $\mathcal{K}$ is a subset of the set $\{0, 1, ... N-1\}$, and $\vec{Z} = W\vec{P}$ where $W$ is a $L \times N$ matrix (L > N).

I have converted it into a convex problem (SDP) given below:

\begin{aligned} &{\text{minimize}} & & ||\vec{t}|| \\ & \text{subject to} & & \begin{bmatrix} diag(\vec{P}_{\mathcal{K}}) & I \\ I & diag(\vec{t}) \end{bmatrix} \succ 0\\ &&& \vec{P}^H\vec{P} \le T_P \\ &&& \vec{Z}^H\vec{Z} \le \epsilon \end{aligned}

I have solved the above in cvx and obtained optimal values of P. Now I need to obtain analytical solutions of this or of the original problem. Is there any iterative algorithm to do the same? If so, how does it converge?

• I don't know what you mean here. It's almost certain that there is no analytical solution to this problem. And CVX uses a standard iterative algorithm to solve it. So you already have what you've asked for. – Michael Grant Jan 5 '16 at 16:22
• I see. Thank you for clearing it. – Gourab Ghatak Jan 7 '16 at 15:36