Numerical Optimization pp 169-202 | Cite as

# Local Methods for Problems with Equality Constraints

Chapter

## 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 ℝ^{ n }, 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 ℝ^{ n }, like the one depicted in Figure 12.1.## Keywords

Stationary Point Equality Constraint Null Space Local Method Quadratic Convergence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2003