I am calculating the solution of a constrained linear least-squares problem as follows:
lb = zeros(7,1);
ub = ones(7,1);
for i = 1:size(b,2)
x(:,i) = lsqlin(C,b(:,i),[],[],[],[],lb,ub);
end
where C is m x 7 and b is m x n. n is quite large leading to a slow computation time. Is there any way to speed up this procedure and get rid of the slow for loop. I am using lsqlin instead of pinv or \ because I need to constrain my solution to the boundaries of 0–1 (lb and ub).
The for loop is not necessarily the reason for any slowness – you're not pre-allocating and lsqlin is probably printing out a lot of stuff on each iteration. However, you may be able to speed this up by turning your C matrix into a sparse block diagonal matrix, C2, with n identical blocks (see here). This solves all n problems in one go. If the new C2 is not sparse you may use a lot more memory and the computation may take much longer than with the for loop.
n = size(b,2);
C2 = kron(speye(n),C);
b2 = b(:);
lb2 = repmat(lb,n,1); % or zeros(7*n,1);
ub2 = repmat(ub,n,1); % or ones(7*n,1);
opts = optimoptions(@lsqlin,'Algorithm','interior-point','Display','off');
x = lsqlin(C2,b2,[],[],[],[],lb2,ub2,[],opts);
Using optimoptions, I've specified the algorithm and set 'Display' to 'off' to make sure any outputs and warnings don't slow down the calculations.
On my machine this is 6–10 times faster than using a for loop (with proper pre-allocation and setting options). This approach assumes that the sparse C2 matrix with m*n*7 elements can fit in memory. If not, a for loop based approach will be the only option (other than writing your own specialized version of lsqlin or taking advantage any other spareness in the problem).