Constrained least squares


In constrained least squares one solves a linear least squares problem with an additional constraint on the solution.
I.e., the unconstrained equation must be fit as closely as possible while ensuring that some other property of is maintained.
There are often special-purpose algorithms for solving such problems efficiently. Some examples of constraints are given below:
When the constraint only applies to some of the variables, the mixed problem may be solved using separable least squares by letting and represent the unconstrained and constrained components. Then substituting the least-squares solution for, i.e.
back into the original expression gives an equation that can be solved as a purely constrained problem in.
where is a projection matrix. Following the constrained estimation of the vector is obtained from the expression above.