Iterative methods using lagrange multipliers for solving extremal problems with constraints of the equation type

Abstract

Some methods of minimization under constraints of the equation type will be discussed, in which the iterative process is based both on the initial variables and on dual variables (Lagrange multipliers). In the classification given in [1], these methods are of the duality type. Alternatively, they may be interpreted as iterative methods for finding the stationary (in particular, saddle) points of the Lagrange function. We assume that both the functional to be minimized, and the constraints, are reasonably smooth. In Section 1 we describe the methods, prove their local convergence, and estimate the rate of convergence. In Section 2 we consider the merits and drawbacks of the methods from the computational point of view, and outline a means for selecting the initial approximation.