Hi everyone,
I have reformulated the following optimization problem to eliminate matrix inversion, as it is not convex. Following previous posts, I used the Schur complement to enforce B_k as the inverse of A_k. However, my results in the CVX did not show that B_k = A_k^(-1). When I impose A_k as a positive definite (PD) matrix, B_k turns into a zero matrix.
min || A_k^(-1)(n, : ) || ^2, for all k and n.
where A is a binary matrix, and A_k is a squared submatrix of A.
The reformulated problem in SDP form is:
min t
s.t [B_k, I; I, A_k] >=0;
[sqrt(t)*I, B_k^(T)(n, : ) ; B_k(n,:), sqrt(t)] >=0;
How can I enforce CVX to ensure B_k = A_k^(-1) ? Any insights or alternative approaches would be greatly appreciated!
i have no idea what you ran. You haven’t shown any code. You haven’t tols us which norm you are using. I don’t even see how the last constraint is a linear SDP which would be accepted by CVX.
Moreover, there is no solver available under CVX for solving an SDP problem having binary or integer variables (Mosek does not allow that combination). CVX would submit such a problem to Mosek, but I think Mosek would generate an error message.
Perhaps you didn’t even submit an SDP; for instance, due to using >= without cvx_begin sdp, in which case >= is applied elementwise.
Perhaps you can show us your careful mathematical formulation, and your proof that the solution to your reformulated problem solves your original problem. In any event, you will nit be able to use CVX for a binary SDP, but you might be able to use YALMIP (BNB + Mosek) if you come up with a suitable binary SDP.
I’m sorry!. The code is below, and the norm-2 has been considered. I have used MOSEK with a binary variable A.
epsilon =1e-2;
cvx_begin
cvx_solver(‘mosek’);
variable A(MAC_size, all_cw_count) binary
variable B_k(MAC_size, MAC_size,DoF)
variable t nonnegative
minimize(t)
subject to
for k = 1:DoF
temp = find(any(all_cw == k, 2))';
A_k = A(:, temp);
[B_k(:,:,k), eye(MAC_size); eye(MAC_size), A_k] >= 0;
for n=1:MAC_size
[sqrt(t)*eye(MAC_size), B_k(n,:,k)'; B_k(n,:,k), sqrt(t)] >= 0;
end
A_k >= epsilon * eye(MAC_size);
end
cvx_end
As I wrote previously, there are no SDP constraints in your code. You did not use cvx_begin sdp, therefore, all the constraints are elementwise, and all my previous comments apply.
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