The student will have an overview over modern methods of convex programming and their problem solving capabilities. He will be able to recognize a convex problem and to formulate it in standard form.
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 ?/2 theorem, S-lemma, Dines theorem
Polynomial optimization: matrix-valued polynomials in one variable, Toeplitz and Hankel matrices, moments, SOS relaxationsPrerequisites
Linear algebra: matrices, vector spaces, linear functions
Analysis: differentiability, gradients, convergence, continuity
The course is composed of 18 hours lectures.
Evaluation : A two-hours written exam (E1) in session 1. For those who do not pass there will be another two-hours exam (E2) in session 2.
N1 = E1
N2 = E2
The exam is given in english only
The course exists in the following branches:
Course ID : WMM9AM16
You can find this course among all other courses.
Aharon Ben-Tal, Laurent El Ghaoui, Arkadi Nemirovski. Robust Optimization. Princeton University Press, 2009.
Aharon Ben-Tal, Arkadi Nemirovski. Lectures on Modern Convex Optimization - Analysis, Algorithms, and Engineering Applications. Vol. 2 of MPS/SIAM Series on Optimization. SIAM, 2001.
Steven Boyd, Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004.
Jean-Bernard Lasserre. Moments, Positive Polynomials and their Applications. Vol. 1 of Imperial College Press Optimization Series. Imperial College Press, 2009.
Yurii Nesterov, Arkadi Nemirovski. Interior-Point Algorithms in Convex Programming. Vol. 13 of SIAM Stud. Appl. Math. SIAM, Philadelphia, 1994.
Date of update January 15, 2017