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 → ℝm 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.
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© 2003 Springer-Verlag Berlin Heidelberg
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Bonnans, J.F., Gilbert, J.C., Lemaréchal, C., Sagastizábal, C.A. (2003). Local Methods for Problems with Equality Constraints. In: Numerical Optimization. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05078-1_12
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DOI: https://doi.org/10.1007/978-3-662-05078-1_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00191-1
Online ISBN: 978-3-662-05078-1
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