Iterative solutions of the linear least squares subject to linear constraints

This paper presents the iterative solutions and algorithms of the linear Least Squares (LS) subject to linear constraints. First, the case with linear equality constraints is studied. The general LS solution without any assumption on the data matrix for this case is given by the matrix pseudoinverse technique and the form of the solution is quite similar to the unconstrained one. Then, under the same assumption as that for the unconstrained case, the iterative solution of linearly constrained LS still holds, and they both include the solutions of the unconstrained LS as the special cases. Then, we apply the above results to derive the LS iterative solution subject to linear inequality constraints by the quadratic programming theory. Finally, we demonstrate that under the regularity conditions similar to those for unconstrained iterative LS problems the linearly constrained iterative LS solutions as well as the iterative algorithms (not strict LS solutions) still converge to the estimated true value, and do not depend on initial conditions in some sense