Norm in equality constraint

[Yosia Hadisusanto]

Hi,

I would like to solve this optimization problem in CVX:

min: w'Sw
s.t.: ||S||=1

The optimization variable is matrix S.

w = rand(4,1);
w = w/norm(w);
cvx_begin
variable Sigma(4,4) symmetric
minimize(w'*Sigma*w)
subject to
norm(Sigma,2) == 1
Sigma == semidefinite(4)
cvx_end

I have the problem with the equality constraint since
norm(Sigma,2) is convex. Do you have any idea how to
reformulate the equality constraint?

Thanks,
Yosia

[Joachim Dahl]

This problem can be solved in closed form. One solution is

X = I - w*w'/||w||^2.

You cannot solve this problem in CVX, but you can lookup
semidefinite formulations/relaxations of trust-region problems
for relayionships with convex opt.

Also (as you might know) CVXOPT is not the same software package as CVX.

Joachim