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# Cours scientifiques - MAA307 : Convex Optimization and Optimal Control

## Descriptif

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.

Convex Optimization and Optimal Control 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. The last part is an introduction to the optimal control of ordinary differential equations and discusses, in particular, the concepts of adjoint state, Hamiltonian and feedback law.

## Objectifs pédagogiques

At the end of the course the students should be able to

• Handle the main concepts of convex analysis in finite dimensional and Hilbert spaces,
• Write down and exploit first order optimality conditions for convex and non-convex minimization problems,
• Construct dual problems using the concept of Lagrangian,
• Develop and implement in Python first order algorithmic strategies in (mainly) convex optimization,
• Apply differential calculus techniques and write optimality conditions for the control of ordinary differential equations.

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

Vous devez avoir validé l'équation suivante : UE MAA202 Ou UE MAA301

Numérique sur 20

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

Le rattrapage est autorisé (Max entre les deux notes écrêté à une note seuil)
L'UE est acquise si Note finale >= 9
• Crédits ECTS acquis : 5 ECTS

La note obtenue rentre dans le calcul de votre GPA.

## Pour les étudiants du diplôme Programmes d'échange internationaux

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.

### 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