Hello all,
I am trying to solve the problem minimize sum_{n} 1/f_n(x), subject to f_n(x) > 0, where f_n(x) is concave in x.
I am aware that in general, the objective function is not always convex. However, under the constraint f_n(x) > 0, 1/f_n(x) is guaranteed to be convex.
Nevertheless, when I solve the problem directly with cvx, I encountered this error:
You would have to write what exactly f_n(x) is. Then we can think if the inverse has a conic model. There is no general answer or any algorithm to derive a model for a general f_n.
Thanks for you reply.
Here is the cvx code to solve the problem:
cvx_begin
variable rho(L+K)
obj = 0;
for l = 1:L
S = (Nt-K)/Ntrho(l) + sum(rho(L+1:L+K))/Nt;
CN = sigma_s + sigma_err(rho(l)+sum(rho(L+1:L+K)));
for i = 1:4
denum((l-1)4+i) = 2y((l-1)4+i)sqrt(SLambda(l,i,i)+CN) - y((l-1)4+i)^2(abs(B_tilda(l,i,i))^2CN);
obj = obj + 1/denum((l-1)*4+i);
end
end
minimize obj
subject to
for l = 1:L
for i = 1:4
denum((l-1)*4+i) > 0;
end
end
sum(rho) <= Pmax;
rho >= 0;
cvx_end
In the above code, denum denotes f_n(x). Specifically, f_n(x) has a form of 2y_n*sqrt(A(x)) - y_n^2 * B(x), where A(x) and B(x) are affine function of x.
I would really appreciate if you can take a look and help me with the issue.
I couldn’t parse your code so let me just address the last paragraph. I understand that up to rescaling you need to be able to express something like
\frac{1}{\sqrt{x}-y}
on the set where \sqrt{x}>y . I see no reason why CVX would not allow you to enter it directly
inv_pos(sqrt(x)-y)
but if not then you can always write the inequality
t\geq\frac{1}{\sqrt{x}-y}
as
\sqrt{x}\geq \frac{1}{t}+y
which should be straightforward to enter as it stands.
Edit: Sorry, apparently I missed the sentence in your post where you write that f_n are concave. Then of course inv_pos is the only possible solution - it works if f_n has a conic model and otherwise one is probably doomed anyway.