After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming codes, running on the fastest computing hardware. Moreover, this is a trend that may well continue and intensify, as problem sizes escalate and the need for fast algorithms becomes more stringent. Traditionally, the focus in optimization algorithms, and in particular, in algorithms for linear programming, has been to solve problems "to optimality." In concrete implementations, this has always meant the solution ofproblems to some finite accuracy (for example, eight digits). An alternative approach would be to explicitly, and rigorously, trade o# accuracy for speed. One motivating factor is that in many practical applications, quickly obtaining a partially accurate solution is much preferable to obtaining a very accurate solution very slowly. A secondary (and independent) consideration is that the input data in many practical applications has limited accuracy to begin with. During the last ten years, a new body ofresearch has emerged, which seeks to develop provably good approximation algorithms for classes of linear programming problems. This work both has roots in fundamental areas of mathematical programming and is also framed in the context ofthe modern theory ofalgorithms. The result ofthis work has been a family ofalgorithms with solid theoretical foundations and with growing experimental success. In this manuscript we will study these algorithms, starting with some ofthe very earliest examples, and through the latest theoretical and computational developments.

To efficiently derive bounds for large-scale instances of the capacitated fixed-charge network design problem, Lagrangian relaxations appear promising. This paper presents the results of comprehensive experiments aimed at calibrating and comparing bundle and subgradient methods applied to the optimization of Lagrangian duals arising from two Lagrangian relaxations. This study substantiates the fact that bundle methods appear superior to subgradient approaches because they converge faster and are more robust relative to different relaxations, problem characteristics, and selection of the initial parameter values. It also demonstrates that effective lower bounds may be computed efficiently for large-scale instances of the capacitated fixed-charge network design problem. Indeed, in a fraction of the time required by a standard simplex approach to solve the linear programming relaxation, the methods we present attain very high quality solutions.

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In the last two decades, a number of algorithms for the linear single-commodity Min Cost Flow problem (MCF) have been proposed, and several efficient codes are available that implement different variants of the algorithms. The practical significance of the algorithms has been tested by comparing the time required by their implementations for solving "from scratch" instances of (MCF), of different classes, as the size of the problem (number of nodes and arcs) increases. However, in many applications several closely related instances of (MCF) have to be sequentially solved, so that reoptimization techniques can be used to speed up computations, and the most attractive algorithm is the one which minimizes the total time required to solve all the instances in the sequence. In this paper we compare the performances of four different efficient implementations of algorithms for (MCF) under cost reoptimization in the context of decomposition algorithms for the Multicommodity Min Cost Flow problem (MMCF), showing that for some classes of instances the relative performances of the codes doing "from scratch" optimization do not accurately predict the relative performances when reoptimization is used. Since the best solver depends both on the class and on the size of the instance, this work also shows the usefulness of a standard interface for (MCF) problem solvers that we have proposed and implemented.

Abstract—The paper presents a simple and effective Lagrangian relaxation approach for the solution of the optimal short-term unit commitment problem in hydrothermal power-generation systems. The proposed approach, based on a disaggregated Bundle method for the solution of the dual u spinning reserves Rt in each period t. I-rows T-columns matrix, whose rows are the T-dimensional arrays ui of the 0-1 variables ui,t indicating the commitment state of thermal unit i during period t. problem, with a new warm-starting procedure, achieves accurate solutions in few iterations. The adoption of a disaggregated Bundle method not only improves the convergence of the proposed approach but also provides information that are suitably exploited for generating a feasible solution of the primal pI pH I-rows T-columns matrix, whose rows are the T-dimensional arrays pi of production levels pi,t of thermal unit i during each period t. H-rows T-columns matrix, whose rows are problem and for obtaining an optimal hydro scheduling. A the T-dimensional arrays ph of production comparison between the proposed Lagrangian approach and other ones, based on sub-gradient and Bundle methods, is presented for a simple yet reasonable formulation of the Hydrothermal Unit Commitment problem. Index Terms—Power generation operation, Hydrothermal unit

The authors also thank two anonymous referees and an Associate Editor for their remarks, which improved the presentation of this paper. We consider a network interdiction problem on a multicommodity flow network, in which an attacker disables a set of network arcs in order to minimize the maxi-mum profit that can be obtained from shipping commodities across the network. The attacker is assumed to have some budget for destroying (or “interdicting”) arcs, and each arc is associated with a positive interdiction expense. In this paper, we exam-ine problems in which interdiction must be discrete (i.e., each arc must either be left alone or completely destroyed), and in which interdiction can be continuous (the ca-pacities of arcs may be partially reduced). For the discrete problem, we describe a linearized model for optimizing network interdiction that is similar to previous studies in the field, and compare it to a penalty model that does not require linearization con-straints. For the continuous case, we prescribe an optimal partitioning algorithm along with a heuristic procedure for estimating the optimal objective function value. We demonstrate on a set of randomly generated test data that our penalty model for the discrete interdiction problem significantly reduces computational time when compared to that consumed by the linearization model. 1

We study the coarse-grained parallelization of an efficient bundle-based costdecomposition algorithm for the solution of multicommodity min-cost flow (MMCF) problems. We show that a code exploiting only the natural parallelism inherent in the costdecomposition approach, i.e., solving the min-cost flow subproblems in parallel, obtains satisfactory efficiencies even with many processors on large, difficult MMCF problems with many commodities. This is exactly the class of instances where the decomposition approach attains its best results in sequential. The parallel code we developed is highly portable and flexible, and it can be used on different machines. We also show how to exploit a common characteristic of current supercomputer facilities, i.e., the side-to-side availability of massively parallel and vector supercomputers, to implement an asymmetric decomposition algorithm where each architecture is used for the tasks for which it is best suited.

The semimetric polytope is an important polyhedral structure lying at the heart of hard combinatorial problems. Therefore, linear optimization over the semimetric polytope is crucial for a number of relevant applications. Building on some recent polyhedral and algorithmic results about a related polyhedron, the rooted semimetric polytope, we develop and test several approaches, mainly based over Lagrangian relaxation and application of Non Differentiable Optimization algorithms, for linear optimization over the semimetric polytope. We show that some of these approaches can obtain very accurate primal and dual solutions in a small fraction of the time required for the same task by state-of-the-art general purpose linear programming technology. In some cases, good estimates of the dual optimal solution (but not of the primal solution) can be obtained even quicker.

A parallel implementation of the specialized interior-point algorithm for multicommodity network ows introduced in [5] is presented. In this algorithm, the positive denite systems of each iteration are solved through a scheme that combines direct factorization and a preconditioned conjugate gradient (PCG) method. Since the solution of at least k independent linear systems is required at each iteration of the PCG, k being the number of commodities, a coarse-grained parallellization of the algorithm naturally arises, where these systems are solved on dierent processors. Also, several other minor steps of the algorithm are easily parallelized by commodity. An extensive set of computational results on a shared memory machine is presented, using problems of up to 2.5 million variables and 260,000 constraints. The results show that the approach is especially competitive on large, dicult multicommodity ow problems. This work has been supported by the European Center for P...

Abstract. Recently, new cost-based filtering algorithms for shorter-path constraints have been developed. However, so far only the theoretical properties of shorter-path constraint filtering have been studied. We provide the first extensive experimental evaluation of the new algorithms in the context of the resource constrained shortest path problem. We show how reasoning about path-substructures in combination with CP-based Lagrangian relaxation can help to improve significantly over previously developed problem-tailored filtering algorithms and investigate the impact of required-edge detection, undirected versus directed filtering, and the choice of the algorithm optimizing the Lagrangian dual. 1