On minimization under linear equality constraints

Abstract

The problem of minimizing a general function subject to linear equalit, constraints is transformed into an equivalent unconstrained problem. Ir, order to solve the first one may apply any unconstrained minimization method to the second problem. By translating this into the language of the first problem one obtains equivalent methods, together with the equivalent convergence properties. An application is given to Goldfarb’s extension of the variable metric method to linearly constrained problems.

The problem of minimizing a function subject to linear inequality constraints is often solved by a so-called “active set strategy”. During several iterations some of the inequality constraints are regarded as equalities (the “active” constraints); therefore there are subproblems with linear equality constraints to solve, which will be treated in the following sections.