We take a closer look at copositive matrices satisfying a known condition for the copositivity of the (Moore-Penrose generalized) inverse of the matrix.
joint work with Bruno Figueira Lourenço, Masakazu Muramatsu
We analyze the duality of SDP to understand how positive duality gaps arise. We show that there exists a relaxation of the dual whose optimal value coincides with the primal, whenever the primal and dual are weakly feasible. The relaxed dual is constructed by removing several linear constraints from the dual, and therefore is likely to have a larger optimal value generically, leading to a positive duality gap. The analysis suggests that positive duality gap is a rather generic phenomenon when both the primal and dual are weakly feasible.
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.