An efficient Newton's method for optimization under equality constraints

An efficient procedure for optimizing a nonlinear objective functional ?(x) under linear and/or nonlinear equality constraints is given. The linearly constrained, quadratic ?(x) case is shown to have a solution given by the explicit formula x = x_p - N(N′AN)^-1N′(Ax_p + b/2), where ?(x) = a+b′x+x′Ax(x?Rⁿ) is convex, and both x_p?R_n and N an n×(n-r) matrix]; can be obtained simultaneously from the constraint set, Kx=c (K of rank r<n), by a single Gaussian elimination. The nonlinearly constrained, arbitrary ?(x) case is treated by an interative scheme in which the above formula is used to “project” onto approximate solutions satisfying linear approximations of the constraints. This method does not require the initial guess or the iterated values to be in the feasible region. The resulting algorithm does appear to be efficient.