Local Methods for Problems with Equality Constraints

  • J. Frédéric Bonnans
  • J. Charles Gilbert
  • Claude Lemaréchal
  • Claudia A. Sagastizábal
Part of the Universitext book series (UTX)

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 xΩ, the inequality m < n is natural. Indeed, for the Jacobian of the constraints to be surjective, we must have mn; 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.
Fig. 12.1.

Problem (P E ) and its feasible set

Keywords

Manifold 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • J. Frédéric Bonnans
    • 1
  • J. Charles Gilbert
    • 1
  • Claude Lemaréchal
    • 2
  • Claudia A. Sagastizábal
    • 3
  1. 1.INRIA RocquencourtLe ChesnayFrance
  2. 2.INRIA Rhône-AlpesMontbonnot, Saint IsmierFrance
  3. 3.IMPAJardim Botânico, Rio de Janeiro-RJBrazil

Personalised recommendations