We propose a version of the bundle scheme for convex nondifferentiable optimization suitable for the case of a sum-function where some of the components are "easy", that is, they are Lagrangian functions of explicitly known compact convex programs. This corresponds to a stabilized partial Dantzig-Wolfe decomposition, where suitably modified representations of the "easy" convex subproblems are inserted in the master problem as an alternative to iteratively inner-approximating them by extreme points, thus providing the algorithm with exact information about a part of the dual objective function. The resulting master problems are potentially larger and less well-structured than the standard ones, ruling out the available specialized techniques and requiring the use of general-purpose solvers for their solution; this strongly favors piecewise-linear stabilizing terms, as opposed to the more usual quadratic ones, which in turn may have an adverse effect on the convergence speed of the algorithm, so that the overall performance may depend on appropriate tuning of all these aspects. Yet, very good computational results are obtained in at least one relevant application: the computation of tight lower bounds for Fixed-Charge Multicommodity Min-Cost Flow problems.

Abstract. We present a bundle method for convex nondifferentiable minimization where the model is a piecewise-quadratic convex approximation of the objective function. Unlike standard bundle approaches, the model only needs to support the objective function from below at a properly chosen (small) subset of points, as opposed to everywhere. We provide the convergence analysis for the algorithm, with a general form of master problem which combines features of trust region stabilization and proximal stabilization, taking care of all the important practical aspects such as proper handling of the proximity parameters and the bundle of information. Numerical results are also reported.

Abstract. We present a convex nondifferentiable minimization algorithm of proximal bundle type that does not rely on measuring descent of the objective function to declare the so-called serious steps; rather, a merit function is defined which is decreased at each iteration, leading to a (potentially) continuous choice of the stepsize between zero (the null step) and one (the serious step). By avoiding the discrete choice the convergence analysis is simplified, and we can more easily obtain efficiency estimates for the method. Some choices for the step selection actually reproduce the dichotomic behavior of standard proximal bundle methods but shed new light on the rationale behind the process, and ultimately with different rules; furthermore, using nonlinear upper models of the function in the step selection process can lead to actual fractional steps.

We discuss a Lagrangian-relaxation-based heuristics for dealing with feature selection in the Support Vector Machine (SVM) framework for binary classification. In particular we embed into our objective function a weighted combination of the L1 and L0 norm of the normal to the separating hyperplane. We come out with a Mixed Binary Linear Programming problem which is suitable for a Lagrangian relaxation approach. Based on a property of the optimal multiplier setting, we apply a consolidated nonsmooth optimization ascent algorithm to solve the resulting Lagrangian dual. In the proposed approach we get, at every ascent step, both a lower bound on the optimal solution as well as a feasible solution at low computational cost. We present the results of our numerical experiments on some benchmark datasets

We treat the Feature Selection problem in the Support Vector Machine (SVM) framework by adopting an optimization model based on use of the 0 pseudo-norm. The objective is to control the number of non-zero components of normal vector to the separating hyperplane, while maintaining satisfactory classification accuracy. In our model the polyhedral norm. [k] , intermediate between. 1 and. ∞, plays a significant role, allowing us to come out with a DC (Difference of Convex) optimization problem that is tackled by means of DCA algorithm. The results of several numerical experiments on benchmark classification datasets are reported.

Subgradient methods (SM) have long been the preferred way to solve the large-scale Nondifferentiable Optimization problems arising from the solution of Lagrangian Duals (LD) of Integer Programs (IP). Although other methods can have better convergence rate in practice, SM have certain advantages that may make them competitive under the right conditions. Furthermore, SM have significantly progressed in recent years, and new versions have been proposed with better theoretical and practical performances in some applications. We computationally evaluate a large class of SM in order to assess if these improvements carry over to the IP setting. For this we build a unified scheme that covers many of the SM proposed in the literature, comprised some often overlooked features like projection and dynamic generation of variables.We fine-tune the many algorithmic parameters of the resulting large class of SM, and we test them on two different Lagrangian duals of the Fixed-Charge Multicommodity Capacitated Network Design problem, in order to assess the impact of the characteristics of the problem on the optimal algorithmic choices.Our results show that, if extensive tuning is performed, SM can be competitive with more sophisticated approaches when the tolerance required for solution is not too tight, which is the case when solving LDs of IPs.

During the planning of communication networks, the routing decision process (distributed and online) often remains decoupled from the network design process, that is, resource installation and allocation-planning process (centralized and offline). To reconcile both processes and take into account demand variability, we generalize the capacitated multicommodity fixed charge network design class of problems by including different types of fixed costs (installation and maintenance costs) and variable costs (routing costs) but also variable traffic demands over multiple periods. However, conventional integer programming methods can typically solve only small to medium size instances of this problem. Two major difficulties are encountered when using commercial solvers to solve the associated mixed integer programs: (i) problems are large scale and even solving the linear relaxation of the problem can be challenging; and (ii) the solver hardly find good feasible solutions for medium to large scale instances. As an alternative, we propose a Lagrangian approach for computing a lower bound by relaxing the flow conservation constraints such that the Lagrangian subproblem itself decomposes by node. Though this approach yields one subproblem per network node, solving the Lagrangian dual by means of the bundle method remains a complex computational tasks. However, it always provides a lower bound on the optimal solution. Moreover, based on this relaxation, we propose a Lagrangian heuristic that makes the approach more robust than a black-box usage of a Mixed Integer Programming (MIP) solver. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 69(1), 110–123 2017

The Continuous Convex Separable Quadratic Knapsack problem (CQKnP) is an easy but useful model that has very many different applications. Although the problem can be solved quickly, it must typically be solved very many times within approaches to (much) more difficult models; hence an efficient solution approach is required. We present and discuss a small open-source library for its solution that we have recently developed and distributed.