I want to solve the following quadratic program in $x \in \mathbb R^{n}$

$$\begin{array}{ll} \text{minimize} & \frac 12 x^\top Q \, x + c^\top x\\ \text{subject to} & x \geq 0_n\end{array}$$

where $Q = q I_n$ and $q>0$. I know we can solve a general QP using Lagrangian multipliers and similar methods. However, I was wondering if $Q$ being a multiple of the identity matrix does simplify the problem. Is there a closed-form solution?

  • $\begingroup$ What do you know about vector $c$? And $q$? Note that the QP is non-convex if $q < 0$. $\endgroup$ – Rodrigo de Azevedo Sep 27 '17 at 17:01
  • $\begingroup$ You might find this paper on non negative quadratic programming useful (looks at a general positive definite $Q$) $\endgroup$ – nemo Sep 27 '17 at 18:37

Maybe write the formula as this form?\begin{align} \frac{1}{2}x^\top Q x+c^\top x= & \frac{q}{2}\left( \sum_{i=1}^{n}\left(x_i^2+\frac{c_i}{q}x_i\right)\right)\\ {}= & \frac{q}{2}\left( \sum_{i=1}^{n}\left(x_i+\frac{c_i}{2q}\right)^2-\frac{c_i^2}{4q^2}\right) \end{align} So it depends on $c=(c_1,\dots,c_n)$. Assume $q>0$, if ${c_i}<0$, then $x_i=-\frac{c_i}{q}$; if $c_i>0$, then $x_i=0$.

  • $\begingroup$ can you please help me with the following problem? math.stackexchange.com/questions/2550644/… I like what you did here, and it seems that you know a lot about these. If you could take a look and help out, I would be forever in your debt! Thank you. $\endgroup$ – ALannister Dec 4 '17 at 15:40

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