Dual Ascent Procedure Research Articles

A dual ascent heuristic for obtaining a lower bound of the generalized set partitioning problem with convexity constraints

In this paper we propose a dual ascent heuristic for solving the linear relaxation of the generalized set partitioning problem with convexity constraints, which often models the master problem of a column generation approach. The generalized set partitioning problem contains at the same time set covering, set packing and set partitioning constraints. The proposed dual ascent heuristic is based on a reformulation and it uses Lagrangian relaxation and subgradient method. It is inspired by the dual ascent procedure already proposed in literature, but it is able to deal with right hand side greater than one, together with under and over coverage. To prove its validity, it has been applied to the minimum sum coloring problem, the multi-activity tour scheduling problem, and some newly generated instances. The reported computational results show the effectiveness of the proposed method.

An exact method for shrinking pivot tables

Pivot tables are one of the most popular tools for data visualization in both business and research applications. Although they are in general easy to use, their comprehensibility becomes progressively lower when the quantity of cells to be visualized increases (i.e., information flooding problem). Pivot tables are largely adopted in OLAP, the main approach to multidimensional data analysis. To cope with the information flooding problem in OLAP, the shrink operation enables users to balance the size of query results with their approximation, exploiting the presence of multidimensional hierarchies. The only implementation of the shrink operator proposed in the literature is based on a greedy heuristic that, in many cases, is far from reaching a desired level of effectiveness.In this paper we propose a model for optimizing the implementation of the shrink operation which considers two possible problem types. The first type minimizes the loss of precision ensuring that the resulting data do not exceed the maximum allowed size. The second one minimizes the size of the resulting data ensuring that the loss of precision does not exceed a given maximum value. We model both problems as set partitioning problems with a side constraint. To solve the models we propose a dual ascent procedure based on a Lagrangian pricing approach, a Lagrangian heuristic, and an exact method. Experimental results show the effectiveness of the proposed approaches, that is compared with both the original greedy heuristic and a commercial general-purpose MIP solver.

Integrating combinatorial algorithms into a linear programming solver

While there are numerous linear (and nonlinear) solvers, as well as specialized algorithms for combinatorial problems, they are rarely used together. We wrote a new module for the XPRESS optimizer that lets us call the objects and functions of the LEMON C++ library. We tested this module by comparing two versions of the Dual Ascent Procedure (DAP) (Adams and Johnson in DIMACS Ser Discrete Math Theor Comput Sci 16:43–75, 1994). It is an iterative algorithm that produces lower bounds for the quadratic assignment problem (QAP). The DAP is based on the level-1 RLT-relaxation (reformulation-linearization technique) of the QAP. This formulation is significantly larger than the original QAP, but if we construct its Lagrangian dual, we can decompose the new relaxation to smaller linear assignment problems for a fixed Lagrange multiplier. The algorithm determines a better Lagrange multiplier in every iteration. We implemented two versions of the DAP with the XPRESS Optimization Software. In the first model we solved the smaller linear assignment problems with the linear programming methods of XPRESS, while in the second model we solved them by using the graph algorithms of the LEMON library via the new module. Using the instances of QAPLIB, the new module produced significantly better running times, which suggests that this direction of research might yield further results in the future.

A Lagrangean-based decomposition approach for the link constrained Steiner tree problem

The link constrained Steiner tree problem is a variant of the classic Steiner tree problem where the number of links to be activated must not exceed a pre-fixed value. We introduce a multi-start heuristic to obtain a quick feasible solution. The proposed heuristic is embedded into a decomposition framework based on Lagrangean relaxation. In particular, the relaxed problem is decomposed into two polynomially solvable subproblems and, to tackle the Lagrangean dual, we introduce a dual ascent procedure where just one multiplier at a time is updated. Our approach can be classified as a Lagrangean heuristic. In fact, at each iteration of the dual ascent procedure, the information derived from the solution of the relaxed problem is used to provide a feasible solution, by solving a restricted problem defined on an appropriate subgraph. Several versions of the proposed approach are defined and tested on instances drawn from the scientific literature.

A Graphics Processing Unit Algorithm to Solve the Quadratic Assignment Problem Using Level-2 Reformulation-Linearization Technique

The quadratic assignment problem (QAP) is a combinatorial optimization problem that arises in many real-world applications, such as equipment allocation in industry. The QAP is NP-hard and, in practice, one of the hardest combinatorial optimization problems to solve to optimality. Exact solutions of QAP are typically obtained by the branch-and-bound method. This method, however, potentially requires a high computational effort, and the use of good lower bounds is essential to prune the search tree. Branch-and-bound algorithms that use the dual-ascent procedure based on the level-2 reformulation linearization technique (RLT2) belong to the state of the art on exactly solving QAP. In this work, we propose a parallel implementation of that branch-and-bound algorithm. Our approach uses the Auction Algorithm of Bertsekas and Castañon to solve the linear assignment problems of RLT2, which allows us to take advantage of the massive parallel environment of graphics processing units to speed up the lower bound computation and implement some memory optimization techniques to address large-size problems. We report experimental results that show significant execution time reductions compared to previous works and allow us to provide, for the first time, exact solutions for two instances of QAP: tai35b and tai40b.

Lagrangian relaxation for the directional sensor coverage problem with continuous orientation

We present a mixed integer nonlinear programming formulation of the Directional Sensors Continuous Coverage Problem (DSCCP), where a given set of targets on a plane are to be covered by a set of sensors whose locations are known. Sensors are supposed to be directional, that is characterized by a discrete set of possible radii and aperture angles. The orientation (which is a continuous variable), the radius and the aperture angle of each sensor are to be decided, taking into account possibility of keeping one or more sensors switched off. The objective is to minimize cost coverage. We incorporate into the objective function penalty cost for possibly uncovered targets.We prove NP-hardness of DSCCP and introduce a Lagrangian relaxation model. We design a dual ascent procedure based on acting on a single multiplier at a time accompanied by a heuristics to find a feasible solution at each ascent iteration. We report also the results of the method on several test problems.

An algorithmic framework for the exact solution of tree-star problems

Many problems arising in the area of telecommunication ask for solutions with a tree-star topology. This paper proposes a general procedure for finding optimal solutions to a family of these problems. The family includes problems in the literature named as connected facility location, rent-or-buy and generalized Steiner tree-star. We propose a solution framework based on a branch-and-cut algorithm which also relies on sophisticated reduction and heuristic techniques. An important ingredient of this framework is a dual ascent procedure for asymmetric connected facility location. This paper shows how this procedure can be exploited in combination with various mixed integer programming formulations. Using the new framework, many benchmark instances in the literature for which only heuristic results were available so far, can be solved to provable optimality within seconds. To better assess the computational performance of the new approach, we additionally consider larger instances and provide optimal solutions for most of them too.

Lower bounds for the Quadratic Minimum Spanning Tree Problem based on reduced cost computation

The Minimum Spanning Tree Problem (MSTP) is one of the most known combinatorial optimization problems. It concerns the determination of a minimum edge-cost subgraph spanning all the vertices of a given connected graph. The Quadratic Minimum Spanning Tree Problem (QMSTP) is a variant of the MSTP whose cost considers also the interaction between every pair of edges of the tree. In this paper we review different strategies found in the literature to compute a lower bound for the QMSTP and develop new bounds based on a reformulation scheme and some new mixed 0–1 linear formulations that result from a reformulation–linearization technique (RLT). The new bounds take advantage of an efficient way to retrieve dual information from the MSTP reduced cost computation. We compare the new bounds with the other bounding procedures in terms of both overall strength and computational effort. Computational experiments indicate that the dual-ascent procedure applied to the new RLT formulation provides the best bounds at the price of increased computational effort, while the bound obtained using the reformulation scheme seems to reasonably tradeoff between the bound tightness and computational effort.

On a class of subadditive duals for the uncapacitated facility location problem

A family of subadditive functions which contains the optimal solution of subadditive duals for integer programs has been presented by Klabjan (2007). In this paper we present a subadditive dual ascent procedure to find an optimal subadditive dual function based on Klabjan’s generator subadditive function. Then we use the proposed method to solve the so called uncapacitated facility location problem (UFLP). Establishing an explicit formulation for generator subadditive functions in the columns of the coefficient matrix of the UFLP, we derive some variable fixing rules for the problem. Some computational results are also presented.

A set covering based matheuristic for a real‐world city logistics problem

AbstractThis paper describes an application of a matheuristic algorithm to a real‐world city logistics problem for a mid‐sized town, whose core could be modeled as a multitrip vehicle routing problem with time windows, pickup and deliveries, and heterogeneous fleet. The proposed matheuristic is based on a dual ascent procedure applied to an extended set covering model (SCC) and on a randomized constructive heuristic. It starts by constructing an initial set of feasible solutions. The corresponding routes are the starting subset of columns of SCC. At each iteration, the dual ascent based on a Lagrangian optimization of the SCC computes a near‐optimal dual solution that is returned along with a primal solution obtained by a Lagrangian heuristic. The dual costs are used for generating new feasible solutions by means of the constructive heuristic, and the corresponding routes are the new columns added to the SCC. At the end of the algorithm, the best heuristic solution is pruned to get a feasible solution for the city logistics problem. The matheuristic could actually solve different city logistics scenarios to be used as a basis for a successive stakeholders concertation process. A further interesting byproduct of this research is the dual ascent procedure that also proved effective on the classical set covering problem, in addition to its extension considered in this paper.

An application of a Lagrangian-type relaxation for the uncapacitated facility location problem

In this paper a surrogate version of semi-Lagrangian relaxation (SLR) of an integer program is introduced. The corresponding dual problem of the proposed relaxation, has properties similar to SLR. In particular, it closes the duality gap. As an application of the surrogate version, a heuristic dual-ascent procedure is proposed for the uncapacitated facility location problem. Numerical tests on some classical instances and real-life applications show that the algorithm provides near-optimal solutions in reasonable time.

An exact solution framework for a broad class of vehicle routing problems

This paper presents an exact solution framework for solving some variants of the vehicle routing problem (VRP) that can be modeled as set partitioning (SP) problems with additional constraints. The method consists in combining different dual ascent procedures to find a near optimal dual solution of the SP model. Then, a column-and-cut generation algorithm attempts to close the integrality gap left by the dual ascent procedures by adding valid inequalities to the SP formulation. The final dual solution is used to generate a reduced problem containing all optimal integer solutions that is solved by an integer programming solver. In this paper, we describe how this solution framework can be extended to solve different variants of the VRP by tailoring the different bounding procedures to deal with the constraints of the specific variant. We describe how this solution framework has been recently used to derive exact algorithms for a broad class of VRPs such as the capacitated VRP, the VRP with time windows, the pickup and delivery problem with time windows, all types of heterogeneous VRP including the multi depot VRP, and the period VRP. The computational results show that the exact algorithm derived for each of these VRP variants outperforms all other exact methods published so far and can solve several test instances that were previously unsolved.

A dual ascent procedure for the set partitioning problem

The set partitioning problem is a fundamental model for many important real-life transportation problems, including airline crew and bus driver scheduling and vehicle routing. In this paper we propose a new dual ascent heuristic and an exact method for the set partitioning problem. The dual ascent heuristic finds an effective dual solution of the linear relaxation of the set partitioning problem and it is faster than traditional simplex based methods. Moreover, we show that the lower bound achieved dominates the one achieved by the classic Lagrangean relaxation of the set partitioning constraints. We describe a simple exact method that uses the dual solution to define a sequence of reduced set partitioning problems that are solved by a general purpose integer programming solver. Our computational results indicate that the new bounding procedure is fast and produces very good dual solutions. Moreover, the exact method proposed is easy to implement and it is competitive with the best branch and cut algorithms published in the literature so far.

An acceleration of Erlenkotter-Körkel’s algorithms for the uncapacitated facility location problem

This contribution is focused on an acceleration of branch and bound algorithms for the uncapacitated facility location problem. Our approach is based on the well-known Erlenkotters’ procedures and Korkels’ multi-ascent and multi-adjustment algorithms, which have proved to be the efficient tools for solving the large-sized instances of the uncapacitated facility location problem. These two original approaches were examined and a thorough analysis of their performance revealed how each particular procedure contributes to the computational time of the whole algorithms. These analyses helped us to focus our effort on the most frequent procedures. The unique contribution of this paper is a new dual ascent procedure. This procedure leads to considerable acceleration of the lower bound computation process and reduces the resulting computational time. To demonstrate more efficient performance of amended algorithms we present the results of extensive numerical experiments.

An algorithm for the generalized quadratic assignment problem

This paper reports on a new algorithm for the Generalized Quadratic Assignment problem (GQAP). The GQAP describes a broad class of quadratic integer programming problems, wherein M pair-wise related entities are assigned to N destinations constrained by the destinations’ ability to accommodate them. This new algorithm is based on a Reformulation Linearization Technique (RLT) dual ascent procedure. Experimental results show that the runtime of this algorithm is as good or better than other known exact solution methods for problems as large as M=20 and N=15.

More on ‘Models and algorithms for the dynamic-demand joint replenishment problem’

This paper presents tight mathematical programming formulations for the dynamic demand joint replenishment problem (DJRP). Experimental studies using general-purpose software document the improved computational efficiency of the new formulations versus the earlier models of Boctor et al. (Boctor, F.F., Laporte, G. and Renaud, J., Models and algorithms for the dynamic-demand joint replenishment problem. Int. J. Prod. Res., 2004, 42, 2667–2678.) and Robinson and Gao (Robinson, E.P. and Gao, L., A dual-ascent procedure for multi-product dynamic demand coordinated replenishment with backlogging. Manag. Sci., 1996, 42, 1556–1564.). The findings encourage the development of specialized algorithms for their solution. The paper also evaluates the performance of the perturbation and dual-ascent heuristics for solving DJRP finding the perturbation heuristic is superior at relatively low set-up cost ratios and high joint set-up cost levels, while the dual-ascent heuristic strongly dominates at relatively high set-up cost ratios and low joint set-up costs. Considering that both heuristics are computationally efficient, the best application strategy is to solve the problem with both heuristics and implement the best found solution.

A note on “a dual-ascent approach to the fixed-charge capacitated network design problem”

We show by a counterexample that the dual-ascent procedure proposed by Herrmann, Ioannou, Minis and Proth in a 1996 issue of the European Journal of Operational Research is incorrect in the sense that it does not generate a valid lower bound to the optimal value of fixed-charge capacitated network design problems. We provide a correct dual-ascent procedure based on the same ideas and we give an interpretation of it in terms of a simple Lagrangean relaxation. Although correct, this procedure is not effective, as in general, it provides a less tighter bound than the linear programming relaxation.

A dual-ascent procedure for the file allocation and join site selection problem on a telecommunications network

In this paper, a model and a solution procedure is developed for the File Allocation and Join Site Selection Problem with 2-way Join [FAJSP-2], defined on a telecommunications network. This problem attempts to integrate the file allocation and query optimization aspects of a distributed computing system. By allowing for queries that require processing up to two file types, this problem is designed to determine simultaneously the number and location of file types and the location of join operations. This problem is modeled as a mixed-integer linear program, for which a fast dual-ascent approximation procedure is developed. Extensive computational results are presented which demonstrate that our dual-ascent procedure is able to solve even large-scale problems to near optimality quickly.

Capacitated selection problem

Optimizing the selection of resources to accomplish a set of tasks involves evaluating the tradeoffs between the cost of maintaining the resources necessary to accomplish the tasks and the penalty cost associated with unfinished tasks. We consider the case where resources are categorized into types, and limits (capacity) are imposed on the number of each type that can be selected. The objective is to minimize the sum of penalty costs and resource costs. This problem has several practical applications including production planning, new product design, menu selection and inventory management. We develop a branch-and-bound algorithm to find exact solutions to the problem. To generate bounds, we utilize a dual ascent procedure which exploits the special structure of the problem. Information from the dual and recovered primal solutions are used to select branching variables. We generate strong valid inequalities and use them to fix other variables at each branching step. Results of tests performed on reasonably sized problems are presented. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 19–37, 1999

A Dual Ascent Procedure with Valid Inequalities for Designing Hierarchical Network Topologies

Topological design of communication networks has been well examined in the literature. However, most of these studies focus on nonhierarchical networks where all nodes are considered equivalent from a routing perspective. It is well recognized, however, that hierarchical topologies offer several advantages for remote office internetworking. In this article, we provide a mixed integer programming formulation for designing hierarchical topologies. We also provide an interesting solution strategy that incorporates valid inequalities within a dual ascent framework to obtain tight lower bounds and feasible solutions to the problem. Computational experiments with a Fortran implementation of the procedure are also reported.