This information is indicative and can be subject to change.
Convex analysis and optimization
Teacher: Bruno Nazaret
E-mail: [email protected]
ECTS: 2.5
Evaluation: Final Exam
Previsional Place and time: 3h lectures (x6) in the first period (Sept.-Oct)
Prerequisites: Introduction to functional analysis (Hilbert spaces), Analysis in vector spaces, smooth optimization.
Aim of the course: This course aims to give the main ingredients of convex analysis in the Hilbert setting with applications to optimization an duality, with some basic numerical gradient descent algorithms
Syllabus:
Ch 1. Hilbert spaces - orthogonal projection - duality - separation theorems
Ch 2. Geometric and topological properties of convex sets
Ch 3. Convex functions - Subdifferentiability - Fenchel transform
Ch 4. Optimization of convex functionals - Fenchel Rockafellar duality théorem and applications
Ch 5. Gradient descent algorithms
References:
Functional analysis:
- W. Rudin : "Real and complex analysis". McGraw-Hill Higher Education, 1986.
- H. Brézis : "Functional Analysis, Sobolev Spaces and Partial Differential Equations". Springer Verlag New York, 2011.
Convex analysis and optimization:
- I. Ekeland and R.Temam : "Convex Analysis and Variational Problems". Amsterdam‐Oxford. North‐Holland Publ. Company, 1976.
- H. H. Bauschke and P.L. Combettes : "Convex Analysis and Monotone Operator Theory in Hilbert Spaces". Springer International Publishing, 2017.
- G. Carlier : "Classical and modern optimization". Advanced Textbooks in Mathematics, World Scientific, 2022.
Convex analysis and optimization
Teacher: Bruno Nazaret
E-mail: [email protected]
ECTS: 2.5
Evaluation: Final Exam
Previsional Place and time: 3h lectures (x6) in the first period (Sept.-Oct)
Prerequisites: Introduction to functional analysis (Hilbert spaces), Analysis in vector spaces, smooth optimization.
Aim of the course: This course aims to give the main ingredients of convex analysis in the Hilbert setting with applications to optimization an duality, with some basic numerical gradient descent algorithms
Syllabus:
Ch 1. Hilbert spaces - orthogonal projection - duality - separation theorems
Ch 2. Geometric and topological properties of convex sets
Ch 3. Convex functions - Subdifferentiability - Fenchel transform
Ch 4. Optimization of convex functionals - Fenchel Rockafellar duality théorem and applications
Ch 5. Gradient descent algorithms
References:
Functional analysis:
- W. Rudin : "Real and complex analysis". McGraw-Hill Higher Education, 1986.
- H. Brézis : "Functional Analysis, Sobolev Spaces and Partial Differential Equations". Springer Verlag New York, 2011.
Convex analysis and optimization:
- I. Ekeland and R.Temam : "Convex Analysis and Variational Problems". Amsterdam‐Oxford. North‐Holland Publ. Company, 1976.
- H. H. Bauschke and P.L. Combettes : "Convex Analysis and Monotone Operator Theory in Hilbert Spaces". Springer International Publishing, 2017.
- G. Carlier : "Classical and modern optimization". Advanced Textbooks in Mathematics, World Scientific, 2022.
