Towards optimal barriers for convex cones
Résumé
Self-concordant barriers are essential for interior-point algorithms in conic programming. To speed up the convergence it is of interest to find a barrier with the lowest possible parameter for a given cone. The barrier parameter is a non-convex function on the set of self-concordant barriers, and finding an optimal barrier amounts to a non-convex infinite-dimensional optimization problem. We describe this problem and show how it can be convexified at the cost of obtaining sub-optimal solutions with guaranteed performance. Finite-dimensional approximations yield lower bounds on the parameter.