The Method of Multipliers for Equality Constrained Problems

This chapter discusses the method of multipliers for equality constrained problems. By solving an approximate problem, an approximate solution of the original problem can be obtained. However, if a sequence of approximate problems can be constructed that converges in a well-defined sense to the original problem, then the corresponding sequence of approximate solutions would yield in the limit a solution of the original problem. The basic idea in penalty methods is to eliminate some or all of the constraints and add to the objective function a penalty term that prescribes a high cost to infeasible points. A parameter that determines the severity of the penalty and as a consequence the extent to which the resulting unconstrained problem approximates the original constrained problem is associated with the penalty methods.