Anyone knows the criterion for CVX to choose a solution from multiple optima? How can I change the criterion? For example when doing regression y=X*Beta + noise, X is row rank insufficient?
cvx_begin quiet
variable Beta(d)
objfun=(X*Beta-y)'*(X*Beta-y);
minimize( objfun )
cvx_endThe criteria depends upon the particular solver, and in general there is no way for you to change it. From a strict mathematical standpoint, a modeler should have no expectation that a solver should prefer one optimum over another.
If you do have a preference of one solution over another, then you should modify your objective function to reflect that. For instance, you could select a relatively small value of \lambda and specify
cvx_begin quiet
variable Beta(d)
minimize(sum_square(X*beta-y)+lambda*norm(Beta))
cvx_end
which would prefer smaller values of \beta to larger ones. Alternatively, once you have determined the optimal value of the original problem, you could do something like this:
cvx_begin quiet
variable Beta(d)
minimize(norm(Beta))
sum_square(X*beta-y)<=1.01*optval
cvx_end
where optval is the optimal value of the original problem. Of course, you can adjust the 1.01 factor there as well, but it can’t be less than 1.
The point is: if you have a preference, your model must reflect it.