In this paper, we consider a first-order block-decomposition method for minimizing the sum of a convex differentiable function with Lipschitz continuous gradient, and two other proper closed convex (possibly, nonsmooth) functions with easily computable resolvents. The method presented contains two important ingredi- ents from a computational point of view, namely: an adaptive choice of stepsize for performing an extragradient step; and the use of a scaling factor to balance the blocks. We then specialize the method to the context of conic semidefinite programming (SDP) problems consisting of two easy blocks of constraints. Without putting them in standard form, we show that four important classes of graph-related conic SDP problems auto- matically possess the above two-easy-block structure, namely: SDPs for O-functions and O+-functions of graph stable set problems, and SDP relaxations of binary inte- ger quadratic and frequency assignment problems. Finally, we present computational results on the aforementioned classes of SDPs showing that our method outperforms the three most competitive codes for large-scale conic semidefinite programs, namely: the boundary point (BP) method introduced by Povh et al., a Newton-CG augmented Lagrangian method, called SDPNAL, by Zhao et al., and a variant of the BP method, called the SPDAD method, by Wen et al.

We introduce a new efficient method to solve the continuous quadratic knapsack problem. This is a highly structured quadratic program that appears in dif- ferent contexts. The method converges after O(n) iterations with overall arithmetic complexity O(n2). Numerical experiments show that in practice the method con- verges in a small number of iterations with overall linear complexity, and is faster than the state-of-the-art algorithms based on median finding, variable fixing, and secant techniques.

The growing cost of transportation and distribution pushes companies, especially small and medium transportation enterprises, to form partnership and to exploit economies of scale. On the other hand, to increase their competitiveness on the market, companies are asked to consider preferences of the customers as well. Therefore, tools for logistics management need to manage collective resources, as many depots and heterogeneous fleets, providing flexible preference handling at the same time. In this paper we tackle a pickup and delivery vehicle routing problem involving such aspects; customers place preferences on visiting time (represented as soft time windows), and their violation is allowed at a price. Our interest in this problem stems from an ongo- ing industrial project. First we propose an exact branch-and-price algorithm, having as a core advanced dynamic programming techniques. Then we analyze through a computational campaign the impact of soft time windows management on the optimal solution in terms of both routing and overall distribution costs. Our experiments show that our approach can solve instances of real size, and clarify the practical usefulness of soft time windows management.