| # Variants of the Least-Squares Problem Least-Squares > Ordinary LS | Variants | Kernels | Applications Linearly-constrained least-squares Minimum-norm solutions to linear equations Regularized least-squares ## 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, that is, is in the range of , this set can be expressed as where is the dimension of the nullspace of , is a matrix whose columns span 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, that is, the matrix is invertible, the above has the closed-form solution Examples: Control positioning of a mass. ## Regularized least-squares In the case when the matrix in the OLS problem is not full column rank, the closed-form solution cannot be applied. A remedy often used in practice is to transform the original problem into one where the full column rank property holds. The regularized least-squares problem has the form where is a (usually small) parameter. The regularized problem can be expressed as an ordinary least-squares problem, where the data matrix is full column rank. Indeed, the above problem can be written as the ordinary LS problem where The presence of the identity matrix in the matrix ensures that it is full (column) rank. ## Solution Since the data matrix in the regularized LS problem has full column rank, the formula seen here applies. The solution is unique, and given by For , we recover the ordinary LS expression that is valid when the original data matrix is full rank. The above formula explains one of the motivations for using regularized least-squares in the case of a rank-deficient matrix : if , but is small, the above expression is still defined, even if is rank-deficient. ## Weighted regularized least-squares Sometimes, as in kernel methods, we are led to problems of the form where is positive definite (that is, for every non-zero ). The solution is again unique and given by