OPTEC - K.U.Leuven Center of Excellence: Optimization in Engineering

"Solving a large class of structured convex optimization
problems with a unified conic formulation"

*** The talk will start at 16:00. Before (starting at 15:30) there is a coffee break and the opportunity to meet and have discussions. ***

(joint work with Robert Chares)

The first part of this talk will briefly survey the field of
convex optimization (a subset of nonlinear optimization), whose
main paradigm is the following: in exchange for voluntarily
restricting your optimization model to specific (convex)
objective functions and (convex) feasible sets, you get a
better understanding of your problem (using the notion of
duality) and the ability to use efficient algorithms, both from
the theoretical (algorithmic complexity) and practical (ability
to solve large-scale problems efficiently) points of view. In
addition, we will focus on the necessity to consider structured
problems, and in particular on the conic formulation of convex
problems.

The second part of this talk will describe a common framework
recently introduced to unify several classes of structured
convex optimization problems, including linear programming,
second-order cone programming, quadratically constrained convex
quadratic programming, l_p programming, minimization of sums of
Euclidean or p-norms, geometric programming, entropy
programming, etc.

Each of these problems can be modelled as a conic optimization
problem, where all the cones used in the formulation belong to
a single family of three-dimensional self-dual convex cones,
defined as { (x,y,z) | x^a y^(1-a) >= |z|, x>=0, y>=0 } where a
is a real parameter between 0 and 1.

This unified formulation features several advantages, including
the possibility to effortlessly derive the dual of any of these
problems and the ability to apply the same algorithmic
framework to any of those classes of structured convex
problems, more specifically interior-point methods based on the
theory of self-concordant barriers.

The presentation of a solver currently under development will
conclude the talk.

Slides