An inexact proximal bundle method with applications to convex conic programming

We present an inexact bundle method for minimizing an unconstrained convex sup-function with an open domain. Under some mild assumptions, we reformulate a convex conic programming problem as such problem in terms of the support function. This method is a first-order method, hence it requires much less computational cost in each iteration than second-order approaches such as interior-point methods. While sometimes providing solutions of low accuracy, such method can attack large scale problems. We show the global convergence of this method. Finally, we give an explicit model for the objective function and a concrete routine to compute the largest eigenvalue inexactly in the case of convex quadratic symmetric cone programming.