A strongly convergent proximal bundle method for convex minimization in Hilbert spaces

Akey procedure in proximal bundle methods for convex minimization problems is the definition of stability centres, which are points generated by the iterative process that successfully decrease the objective function. In this paper we study a different stability-centre classification rule for proximal bundle methods. We show that the proposed bundle variant has at least two particularly interesting features: (i) the sequence of stability centres generated by the method converges strongly to the solution that lies closest to the initial point; (ii) if the sequence of stability centres is finite, x being its last element, then the sequence of nonstability centres (null steps) converges strongly to x. Property (i) is useful in some practical applications in which a minimal norm solution is required. We show the interest of this property on several instances of a full sized unit-commitment problem.