| # Linearly Constrained Least-Squares Problems Least-Squares > Definitions >> [ LS | Regularized LS | Constrained LS ] | Solution | Sensitivity | Limitations Linearly-constrained least-squares Minimum-norm solutions to linear equations # Linearly constrained least-squares ## Definition An interesting variant of the ordinary least-squares problem involves equality constraints on the decision variable : where , and are given. Examples: Minimum-variance portfolio. ## Solution We can express the solution by first computing the nullspace of . Assuming that the feasible set of the constrained LS problem is not empty, it can be expressed as where is the dimension of the nullspace of , and is a particular solution to the equation . Expressing in terms of the free variable , we can write the constrained problem as an unconstrained one: where , and . # Minimum-norm solution to linear equations A special case of linearly constrained LS is in which we implicitly assume that the linear equation in : , has a solution, that is, is in the range of . The above problem allows to select a particular solution to a linear equation, in the case when there are possibly many, that is, the linear system is under-determined. As seen here, when is full row rank, the matrix is invertible, and the above has the closed-form solution Examples: Control positioning of a mass.