Mathematical optimization - WMM9AM90

Goal(s)

The course is meant as sequel to the course "Non-smooth convex optimization methods". It treats more advanced methods and applications of convex optimization, notably structured conic programs, their solution by interior-point methods, and reformulations or approximations of common optimization problems as conic symmetric programs.

Content(s)

The subject of this half-semester course are more advanced methods in convex optimization. It consists of 6 lectures, 2 x 1,5 hours each, and can be seen as continuation of the course "Non-smooth convex optimization methods". Approximate content of each lecture:

• Linear programs / Conic programs
• Representations of linear programs
• Duality
• Symmetric cones
• Second order conic / semi-definite programming
• Liftings / Complexity

• Robust optimization
• Robust counterparts of conic programs
• Robust Linear / Second order conic / Semi-definite programs

• Interior-point methods
• Self-concordant barriers
• Short / Long step path-following methods
• Barriers for non-symmetric cones

• Applications
• Control problems
• Analysis of dynamical systems
• Topology / Shape optimization

• Relaxations
• MaxCut (Goemans / Williamson)
• Stable set (Lovasz / Schrijver)
• Copositive programming relaxations

• Polynomial optimization
• Sums of squares relaxations
• Moment relaxations

Linear algebra: matrices, vector spaces, linear functions

Analysis: differentiability, gradients, convergence, continuity