Mathematical optim

Mathematical 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

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.