A class of differential descent methods for constrained optimization

In this paper a class of algorithms is presented for minimizing a nonlinear function subject to nonlinear equality constraints along curvilinear search paths obtained by solving a linear approximation to an initial-value system of differential equations. The system of differential equations is derived by introducing a continuously differentiable matrix whose columns span the subspace tangent to the feasible region. The new approach provides a convenient way for working with the constraint set itself, rather than with the subspace tangent to it. The algorithms obtained in this paper may be viewed as curvilinear extensions of two known and successful minimization techniques. Under certain conditions, the algorithms converge to a point satisfying the first-order Kuhn-Tucker optimality conditions at a rate that is asymptotically at least quadratic.