1/x^2 (convex using DCP)

(Thomas) #1

How can I express 1/x^2 as a convex function using DCP rules?

(Let’s say I want to solve the following problem:
variables x
minimize 1/x
subject to

(Mark L. Stone) #2

You can use inv_pos(x) or pow_p(x,-1) for 1/x,
or use pow_p(x,-2) or square(inv_pos(x)) for 1/x^2.

This is addressed in the CVX Users Guide http://cvxr.com/cvx/doc/funcref.html#new-functions .

Edit: As discussed below, square(inv_pos(x)) is valid under CVX 3.0beta, but not under CVX 2.1.

(Thomas) #3

Thanks a lot.
pow_p(x,-2) works fine!
(However square(inv_pos(x)) produces the error Illegal operation: square( {convex} )).

(Mark L. Stone) #4

Sorry about that. square(inv_pos(x)) works under CVX 3.0beta which is more “lenient” (sophisticated) on sign-dependent matters, but not under CVX 2.1. square_pos(inv_pos(x)) should work under both CVX 2.1 and 3.0 beta. As well, of course, as pow_p(x,-2) .

(Thomas) #5

I have another question and I would be grateful if you could help:

Even though x/(x+1)^2 is not convex on the whole real line, it can be shown that it is convex for x>=2.

How can i formulate let’s say the following problem in CVX?

‘minimize x/(x+1)^2
subject to 2<=x<=3’

(Mark L. Stone) #6

I don;t see how to do it. Convexity (or concavity) only over a restricted range is often not possible to deal with in CVX. But I’m not saying it definitely can’t be done. So I’ll leave your question at X/(x+1)^2 convex for x>=2 open, so that someone can weigh in more definitively.

Edit: My suspicion is confirmed by mcg’s answer at the linked question.