An efficient Newton's method for optimization under equality constraints

摘要

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 = xp - N(N′AN)-1N′(Axp + b/2), where ⨍(x) = a+b′x+x′Ax(xϵRn) is convex, and both xpϵRn and N [an n×(n-r) matrix]; can be obtained simultaneously from the constraint set, Kx=c (K of rank r