Minimizing perspective of a function


(Arjun Anand) #1

Hello,
I would like to minimize the function h(x, y)=\frac{x^n}{y^{n-1}} where x >0, y >0 and n is a positive integer. h(x, y) is the perspective of function f(x)=x^n, and hence, it is convex. For n=2, I have used the built-in function quad_over_lin. Is there any standard procedure for a general value of n?.


(Mark L. Stone) #3

One way is to use gp mode via cvx_begin gp

minimize(x^n/y^(n-1))


(Erling D.Andersen) #4

Assume you have the power cone

t^{1/n}y^{1-1/n} \geq |x|

constraint. That implies

t \geq \frac{|x|^n}{y^{n-1}}

so you could minimize x subject to a power cone constraint.

In Mosek v9 we will support the 3 dimensional power cone. For n=3 you can do it with conic quadratic optimization (warning of changed notation):

\begin{array}{c} \left \{(t,x,y)\mid t \geq \frac{|x|^3}{y^2}, y\geq 0 \right \} \\ \Leftrightarrow \\ \left \{(t,x,y)\mid (z,x) \in Kq, (\frac{y}{2},s,z), (\frac{t}{2},z,s) \in Kr \right \}. \end{array}

%
where Kq is a quadratic cone and Kr is a rotated quadratic cone.

I think it should also be possible for other n but has not worked out the details.