Change of variables in QCLP

Question

Is there any change of variables that makes the following optimization problem easier to solve?

\begin{align} \max_{x\in\mathbb{R}^n,t\in\mathbb{R}}\quad & c^\top x,\\ \mbox{s.t.}\quad\quad & ax+b\geq t,\\ & x^\top P x\leq t^2,\\ & x^\top Q x\leq 1,\\ & t\geq 0. \end{align}

where matrices $Q,P\in\mathbb{R}^{n\times n}$ are positive definite.

Answer 1

The constraints are convex. The problem is convex, and can be formulated and solved as a Second Order Cone Problem (SOCP), for which there are many high quality solvers.

In CVX under MATLAB, this can be formulated as

cvx_begin
variable x(n) t
maximize(c''*x)
a*x + b >= t
norm(chol(P)*x) <= t
norm(chol(Q)*x) <= 1
t >= 0
cvx_end