# Résolution de problèmes non linéaires par les méthodes de points intérieurs. Théorie et algorithmes.

Abstract : The barrier methods solve the nonlinear problem by solving a sequence of penalized problems. The relation between the sequence, known as external, of the solutions of the penalized functions and the solution of the initial problem was established in the Sixties.

In this thesis, we used a logarithmic barrier function. At each external iteration, SQP techniques produce a series of quadratic subproblems whose solutions form a sequence, known as internal, of descent directions, to solve the penalized nonlinear problem.

We introduced a change of variable on the step what allow us to obtain optimality conditions more stable numerically. We gave simulations to compare the performances of the G.C. method with that of D.C. method, applied to solve trust-region quadratic problems.

We adapted D.C. method to solve the vertical subproblems, which allowed us to reduce their dimensions from $n+m$ to $m+p$ ($p < n$).

The evolution of the algorithm is controlled by the merit function. Numerical tests make it possible to compare the advantages of various forms of the them.

We introduced new rules to improve this evolution. The numerical experiments show a profit concerning the number of solved problems. The study of the convergence of our method SDC, closes this work.
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https://tel.archives-ouvertes.fr/tel-00011376
Contributor : Mohammed Ouriemchi <>
Submitted on : Friday, January 13, 2006 - 2:27:33 PM
Last modification on : Tuesday, February 5, 2019 - 11:41:16 AM
Long-term archiving on: : Saturday, April 3, 2010 - 9:22:37 PM

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• HAL Id : tel-00011376, version 1

### Citation

Mohammed Ouriemchi. Résolution de problèmes non linéaires par les méthodes de points intérieurs. Théorie et algorithmes.. Mathématiques [math]. Université du Havre, 2005. Français. ⟨tel-00011376⟩

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