Local Methods for Problems with Equality Constraints

Abstract

In this chapter, we present and study several local methods for minimizing a nonlinear function subject only to nonlinear equality constraints. This is the problem (P _E ) represented in Figure 12.1: Ω is an open set of ℝⁿ, while f : Ω → ℝ and c : Ω → ℝ^m are differentiable functions. Since we always assume that c is a submersion, which means that c′(x) is surjective (or onto) for all x ∈ Ω, the inequality m < n is natural. Indeed, for the Jacobian of the constraints to be surjective, we must have m ≤ n; but if m = n, any feasible point is isolated, which results in a completely different problem, for which the algorithms presented here are hardly appropriate. Therefore, a good geometrical representation of the feasible set of problem (P _E ) is that of a submanifold M _* of ℝⁿ, like the one depicted in Figure 12.1.