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Combinatorial Structures in Nonlinear Programming

This article by Stefan Scholtes from April 2002 explores the generalization of Combinatorial Structures in Nonlinear Programming. It discusses the evolution from Nonlinear Programs (PNL) to Nonlinear Combinatorial Programs (PCNL) and the implications for traditional concepts such as active/inactive constraints, regular points, stationary points, and Lagrange multipliers. The article introduces additional conditions on g and h, leading to conditions on Z for a more holistic approach. Various examples, definitions, and algorithms like Sequential Quadratic Programming (SQP) are presented to illustrate the applicability and extension of classical notions in this context.

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Combinatorial Structures in Nonlinear Programming

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  1. Combinatorial Structures in Nonlinear Programming Présentation d’un article de Stefan Scholtes April 2002

  2. Idée: Généralisation • Programme Non Linéaire (PNL) • Programme Combinatoire Non Linéaire (PCNL)

  3. Idée: Généralisation (2) Conditions sur g, h  Conditions sur Z

  4. Idée: Généralisation (3) • Biniveau vs PCNL

  5. Que deviennent les notions classiques? • Contraintes actives, inactives • Points réguliers • Points stationnaires • Multiplicateurs de Lagrange

  6. Activité des contraintes/composantes • Définition

  7. x: z1 (i), z2 (a) y: z1 (a), z2 (i) p: z1 (a), z2 (a) z2 z2 x x y z2 y z1 z1 y p x z1 Activité - Illustration • x: z1 (a), z2 (a) • y: z1 (a), z2 (a) • x: z1 (i), z2 (a) • y: z1 (a), z2 (a)

  8. Régularité et Stationnarité • Régularité • Stationnarité • Dans le cas classique

  9. Stationnarité • PNL • PCNL • Il faut introduire une condition supplémentaire

  10. Condition supplémentaire: Exemple f Z

  11. Ensemble localement étoilé • Définition • PCNL

  12. Ensemble localement étoilé Exemples z2 z2 z1 z1 z2 z1

  13. Multiplicateurs de Lagrange • Définition

  14. Multiplicateurs de Lagrange (2) • Proposition • PNL • PCNL

  15. Z Multiplicateurs de Lagrange (3) Exemple (Utilisations des nouvelles notions) l

  16. Multiplicateurs de Lagrange (4) • Complémentarité stricte

  17. PNL - Sequential Quadratic Programming (SQP)

  18. PNL - Sequential Quadratic Programming (SQP)

  19. PNL - Sequential Quadratic Programming (SQP) (2) • Algorithme SQP

  20. Les notions classiques sont généralisables! • Contraintes actives, inactives • Composantes • Points réguliers • Points stationnaires • Multiplicateurs de Lagrange • Points Critiques

  21. PCNL - Sequential Quadratic Programming (SQP généralisé) • Algorithme SQP généralisé

  22. PCNL - SQP généralisé (2) • Convergence en

  23. PCNL - SQP généralisé (3) CP CQP

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