There is a theory for general constraints in quadratic form minimization. I haven't found the theory to be useful in any application I've run into so far, but it should be useful for writing erudite theoretical articles.
Constraint equations are an underdetermined set of equations, say (the number of components in exceeds that in ), that must be solved exactly while some other set is solved in the least-squares sense, say .This is formalized as
The great mathematician Lagrange apparently looked at the result, equation (2), and realized that he could have it far more simply by extremalizing the following quadratic form
When I sat down to write this book I promised myself to include no equations without a practical use, so now I'll explain how you can use these equations to prepare pedantic articles on geophysical inversion. You can derive marvelously complicated equations by formally solving equation (2) as you would any set. The solution balloons up in size, particularly since the elements of equation (2) are submatrices and they cannot be commuted. You justify these equations by stating that geophysical measurements are finite in number and thus all equations with measurements are constraints to an infinite-dimensional optimization problem for finding the earth model which is a continuous function. You can obfuscate further by introducing weighting functions that are inverse covariance matrices of the unknowns, and thus fully as unknown as the unknowns themselves.