Linearly constrained minimization

Abstract

After chapter 5 it should come as no surprise that there are specific methods for linear, as opposed to non-linear, constraints. Broadly speaking these methods, which may be called projection methods, take the search vectors generated by a standard unconstrained minimization technique and project them so that they lie in the intersection of a set of constraint hyperplanes. This is necessary to maintain feasibility when the constraints are equalities, but it is also the approach for inequalities in conjunction with a technique, called active set strategy, for deciding which subset of the constraints should temporarily be considered as equalities at any point. This kind of idea is not applicable to non-linear constraints without substantial modifications because the concept of feasible directions is less useful there (compare section 5.3.1).