I have a problem which needs to be expressed as either LP, QP, QCQP or SOCP. \begin{array}{ll} \text{minimise}_{x,y,t} & a||\mathbf{x}||_2^2 + b||\mathbf{y}||_2^2 \\ \text{subject to}& ||\mathbf{x}+\mathbf{y}||_2^2 \le t. \end{array} Its given that $a,b \in \mathbb{R}_+$ and $\mathbf{x},\mathbf{y}\in \mathbb{R}^n$.
I tried to express it as SOCP by opening up the equations.
\begin{array}{ll} \text{minimise}_{x,y,t} & \sum_i^n ax_i^2+by_i^2 \\ \text{subject to}& \sum_1^n(x_i+y_i)^2 \le t \iff \sum_i^n(x^2_i+ y_i^2) \le t. \end{array}
I am not sure how to proceed further, as the objective is still not affine/linear.
Or is it better to try for QCQP as, \begin{array}{ll} \text{minimise}_{x,y,t} & a\mathbf{x}^T\mathbf{x} + b\mathbf{y}^T\mathbf{y} \\ \text{subject to}& (\mathbf{x+y})^T (\mathbf{x+y}) \le t \iff \mathbf{x}^T\mathbf{x} + \mathbf{y}^T\mathbf{y} + 2\mathbf{x}^T\mathbf{y} \le t. \end{array}.
I am right in my formulations ? How to I proceed further ? I would appreciate the help.
Here's an SOCP reformulation. Introduce variable $z \ge 0$ to represent $a ||x||_2^2 + b ||y||_2^2$. Introduce variable $u \ge 0$ to represent $||x+y||_2^2$. Introduce variable $v$ to represent the constant $1/2$. Introduce variable $w_i$ to represent $x_i + y_i$. The problem is to minimize $z$ subject to \begin{align} u &\le t \\ v &= 1/2 \\ w_i &= x_i + y_i &&\text{for all $i$} \\ 2 u v &\ge \sum_i w_i^2 \\ 2 v z &\ge \sum_i \left((\sqrt{a} x_i)^2 + (\sqrt{b} y_i)^2\right) \end{align} The last two constraints are rotated SOC constraints, and everything else is linear.
Note that $x_i=y_i=t=0$ is optimal.