# How to prove that this is a Quadratic Optimization Problem?

Show that the following problem is a Quadratic Optimization Problem. $$\begin{array}{ll} \max & \sum\limits_{i=1}^n \left( \mu_i(z_i + x_i) - a_i \lvert x_i \rvert - b_i x_i^2 \right) \\ \text{s.t.} & (z+x)' \sum(z+x) \le s \\ & z_i + x_i \le \gamma_i z_i^{\text{total}} & i = 1, \ldots, n \\ & -\delta_i \le x_i \le \delta_i & i = 1, \ldots, n \\ & -L \le \sum\limits_{i=1}^n p_i x_i \le L \\ & \sum\limits_{i=1}^n p_i \lvert x_i \rvert \le t \\ & z_i + x_i \ge 0 & i = 1, \ldots, n \end{array}$$

• You should write out the problem in LaTeX here. I understand that you are new to this site, and the problem statement is quite complicated. So this is a suggestion for next time – Shailesh Oct 27 '15 at 6:44
• Also include your efforts in solving the problem – Shailesh Oct 27 '15 at 6:44
• As far as i can tell, this can't be considered as a quadratic optimization problem, due to the first of constraints not being linear. – Narek Margaryan Oct 27 '15 at 6:51
• It's a quadratically constrained quadratic program. Convex if $b_i$ is positive and $\Sigma$ positive semidefinite. Depending on the signs of $a_i$ and $p_i$, the absolute values might be LP representable, but in the worst-case you might have to introduce binary variables to model those (if $a_i$ or $pi$ are negative) – Johan Löfberg Oct 27 '15 at 7:20
• Should have explicitly said that it is convex if $a_i, b_i,p_i,\Sigma$ are positive...Nonconvexity of the objective due to $b_i$ being negative is sort of different than nonconvexity due to negative $a_i$, as the latter can be saved by binary variables to model the absolute values. – Johan Löfberg Oct 27 '15 at 12:37