Gradient projection and the total inverse

Let us start with the illustrative case where we seek to minimize the distance (jc—jc0)t(jc—jc0) subject to one constraint which we write g(x)=0. We want to minimize the sum c2 such as 

On the right-hand side, we recognize the projector P onto the gradient of g 

More generally, if there are n data points, n Lagrange multipliers Aj will be needed. Let m be the sum of the number of observations and the number of parameters. The general function c2 to minimize will be 

Let us rewrite the term under the summation sign in a matrix form 

We now lump the n Lagrange multipliers into the vector X, and the g3(x) into a vector g(x). We further define the m x n matrix F of partial derivatives by its current term fp which is the derivative of the y th constraint with respect to the ith parameter, as 

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