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# Cours scientifiques - MAA307 : Convex Optimization and Optimal Control

## Descriptif

Prerequisite: MAA202

MAA307 is composed of three connected parts. The first one lays the foundation of convex analysis in Hilbert spaces, and covers topics such as: convex sets, projection, separation, convex cones, convex functions, Legendre-Fenchel transform, subdifferential. The second part deals with optimality conditions in convex or
differentiable optimization with equality and inequality constraints, and opens the way to duality theory and related algorithms (Uzawa, augmented Lagrangian, decomposition and coordination). The last part is an introduction to the optimal control of ordinary differential equations.

MAA307 complements MAA209 on the theoretical side, but MAA209 is not mandatory.

## Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Vous devez avoir validé l'équation suivante : UE MAA202

Numérique sur 20

## Pour les étudiants du diplôme Echanges PEI

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 10)
L'UE est acquise si note finale transposée >= D
• Crédits ECTS acquis : 4 ECTS

La note obtenue rentre dans le calcul de votre GPA.

## Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 10)
L'UE est acquise si Note finale >= 9
• Crédits ECTS acquis : 4 ECTS

La note obtenue rentre dans le calcul de votre GPA.

### Programme détaillé

1. Convex sets (convex combinations, convex hull, projection and separation, cones)
2. Convex functions (including indicator and support functions, lower semicontinuity, closed convex hull, Legendre-Fenchel transform)
3. Optimization without explicit constraint (existence issues, elements of subdifferential calculus, parametric duality)
4. Optimality conditions with equality and inequality constraints (KKT conditions in convex or differentiable optimization)
5. Lagrangian duality and algorithmic notions (proximal and projection methods, Lagrangian duality, Uzawa's algorithm, augmented Lagrangian, decomposition and coordination)
6. Introduction to the optimal control of ordinary differential equations (adjoint method, Lagrangian, Hamiltonian, Pontryagin's principle, Riccati's equation and feedback law)

Support pédagogique multimédia