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In constrained least squares one solves a linear least squares problem with an additional constraint on the solution.[1][2] This means, the unconstrained equation must be fit as closely as possible (in the least squares sense) while ensuring that some other property of is maintained.
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There are often special-purpose algorithms for solving such problems efficiently. Some examples of constraints are given below:
If the constraint only applies to some of the variables, the mixed problem may be solved using separable least squares[4] by letting and represent the unconstrained (1) and constrained (2) components. Then substituting the least-squares solution for , i.e.
(where + indicates the Moore–Penrose pseudoinverse) back into the original expression gives (following some rearrangement) 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.
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