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Degrees incorporating this pedagocial element :
Description
Theoretical foundations of convex optimization.
This course deals with:
Topic 1: convex analysis
Topic 2: convex programming
Basic notions: vector space, affine space, metric, topology, symmetry groups, linear and affine hulls, interior and closure, boundary, relative interior
Convex sets: definition, invariance properties, polyhedral sets and polytopes, simplices, convex hull, inner and outer description, algebraic properties, separation, supporting hyperplanes, extreme and exposed points, recession cone, Carathéodory number, convex cones, conic hull
Convex functions: level sets, support functions, sub-gradients, quasi-convex functions, self-concordant functions
Duality: dual vector space, conic duality, polar set, Legendre transform
Optimization problems: classification, convex programs, constraints, objective, feasibility, optimality, boundedness, duality
Linear programming: Farkas lemma, alternative, duality, simplex method
Algorithms: 1-dimensional minimization, Ellipsoid method, gradient descent methods, 2nd order methods
Conic programming: barriers, Hessian metric, duality, interior-point methods, universal barriers, homogeneous cones, symmetric cones, semi-definite programming
Relaxations: rank 1 relaxations for quadratically constrained quadratic programs, Nesterovs π/2 theorem, S-lemma, Dines theorem
Polynomial optimization: matrix-valued polynomials in one variable, Toeplitz and Hankel matrices, moments, SOS relaxations
The course is composed of 18 hours lectures.
Evaluation : A two-hours written exam (E1) in December. For those who do not pass there will be another two-hours exam (E2) in session 2 in spring.
Prerequisites
Linear algebra: matrices, vector spaces, linear functions
Analysis: differentiability, gradients, convergence, continuity
Targeted skills
At the end of the course, the student will be able to convert optimization problems into a standard form amenable to a solution by a solver.
Bibliography
S. Boyd, L. Vandenberghe. Convex Optimization. Cambridge University Press. 2004. http://stanford.edu/~boyd/cvxbook/
A. Ben-Tal, A. Nemirovski. Lectures on modern convex optimization. SIAM, Philadelphia, 2001.
R.T. Rockafellar. Convex analysis. Princeton University Press, Princeton, 1970
J.-B. Lasserre. An Introduction to Polynomial Optimization. Cambridge University Press, Cambridge, 2015
In brief
Period : Semester 9Credits : 3
Number of hours Lectures (CM) : 18h
Culmination Code (APOGEE) : GBX9AM16
Hing methods : In person
Location(s) : Grenoble - University campus
Language(s) : English
Contacts
Roland Hildebrand
