Which is more efficient? Does fmincon "feel" along a direction if lb=ub for that dimension?
I don't think efficiency will be impacted so much by what fmincon is doing as by what the objective function and constraints that you supply are doing. When you have parameters that are apriori known, you can usually use that to simplify the objective function calculations, a benefit you don't get if you use lb=ub to convey the known value artificially. As an example, consider the simple objective function,
f(x) = sum(x.^2)
Suppose that x is a vector of length N and x(2:N) are in truth, apriori known. Then one option is to use that to reduce the above analytically to the minimization of
f(x)=x(1)^2
which is an N-fold cheaper computation. Same thing goes for the constraints. An MxN matrix inequality system
A*x<=b
reduces to an Mx1 system
A(:,1)*x(1) <=b - A(:,2:N)*x(2:N)
Notice that everything on the RHS is known, fixed, and pre-computable.
Conversely, if you use lb,ub to constrain the values of x(2:N) then, even if fmincon doesn't vary x(2:end) in the course of its iterations, it will still be repeatedly calling/evaluating the full function f(x)=sum(x.^2) and doing the complete matrix multiplications A*x with all of their O(N) computational demands.
