# Is there any way to transform a non-convex optimization problem into a convex one?

I have an optimization problem which is described as $$\begin{array}{ll} \text{minimize}_x & c^{T}x\\ \text{subject to} & Gx \preceq h\\ & -x^{T}Px - qx - r \leq 0 \end{array}$$ where $P$ is positive semi-definite.

Is there any way to convert this problem into a convex problem?

• No, there is not. See this PDF for some heuristic alternatives. – Michael Grant Nov 3 '14 at 16:52