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.
Diplôme(s) concerné(s)
Parcours de rattachement
- Bachelor en sciences - S5 - Double spécialité Mathématiques et Physique
- Bachelor en Sciences-S5-Double specialite Mathematiques et Informatique
- Bachelor en sciences - S5 - Double spécialité Mathématiques et Économie
- Bachelor en sciences - S5 - Double spécialité Mathématiques et Économie - Mineure en Biologie
- Bachelor en sciences - S5 - Double spécialité Mathématiques et Économie - Mineure en Chimie
- Bachelor en sciences - S5 - Double spécialité Mathématiques et Physique - Mineure en Biologie
- Bachelor en sciences - S5 - Double spécialité Mathématiques et Physique - Mineure en Chimie
- Bachelor en Sciences-S5-Double specialite Mathematiques et Informatique - Mineure en Biologie
- Bachelor en Sciences-S5-Double specialite Mathematiques et Informatique - Mineure en Chimie
Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique
Vous devez avoir validé l'équation suivante : UE MAA202 Ou UE MAA301
Format des notes
Numérique sur 20Littérale/grade américainPour 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)- 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)- Crédits ECTS acquis : 4 ECTS
La note obtenue rentre dans le calcul de votre GPA.
Programme détaillé
- Convex sets (convex combinations, convex hull, projection and separation, cones)
- Convex functions (including indicator and support functions, lower semicontinuity, closed convex hull, Legendre-Fenchel transform)
- Optimization without explicit constraint (existence issues, elements of subdifferential calculus, parametric duality)
- Optimality conditions with equality and inequality constraints (KKT conditions in convex or differentiable optimization)
- Lagrangian duality and algorithmic notions (proximal and projection methods, Lagrangian duality, Uzawa's algorithm, augmented Lagrangian, decomposition and coordination)
- 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
