A class of quadratically convergent algorithms for constrained function minimization

The problem of minimizing a functionf(x) subject to the constraint ?(x)=0 is considered. Here,f is a scalar,x is ann-vector, and ? is anm-vector, wherem <n. A general quadratically convergent algorithm is presented. The conjugate-gradient algorithm and the variable-metric algorithms for constrained function minimization can be obtained as particular cases of the general algorithm. It is shown that, for a quadratic function subject to a linear constraint, all the particular algorithms behave identically if the one-dimensional search for the stepsize is exact. Specifically, they all produce the same sequence of points and lead to the constrained minimal point in no more thann ?r descent steps, wherer is the number of linearly independent constraints. The algorithms are then modified so that they can also be employed for a nonquadratic function subject to a nonlinear constraint. Some particular algorithms are tested through several numerical examples.