Tabu Search for Frequency Assignment in Mobile Radio Networks

The main goal of the Frequency Assignment Problem in mobile radio networks consists of assigning a limited number of frequencies to each radio cell in a cellular network while minimizing electromagnetic interference due to the reuse of frequencies. This problem, known to be NP-hard, is of great importance in practice since better solutions will allow a telecommunications operator to manage larger cellular networks. This paper presents a new Tabu Search algorithm for this application. The algorithm is tested on realistic and large problem instances and compared with other methods based on simulated annealing, constraint programming and graph coloring algorithms. Empirical evidence shows that the Tabu algorithm is very competitive by giving the best solutions to the tested instances.     

Integer programming formulations and efficient local search for relaxed correlation clustering

Relaxed correlation clustering (RCC) is a vertex partitioning problem that aims at minimizing the so-called relaxed imbalance in signed graphs. RCC is considered to be an NP-hard unsupervised learning problem with applications in biology, economy, image recognition and social network analysis. In order to solve it, we propose two linear integer programming formulations and a local search-based metaheuristic. The latter relies on auxiliary data structures to efficiently perform move evaluations during the search process. Extensive computational experiments on existing and newly proposed benchmark instances demonstrate the superior performance of the proposed approaches when compared to those available in the literature. While the exact approaches obtained optimal solutions for open problems, the proposed heuristic algorithm was capable of finding high quality solutions within a reasonable CPU time. In addition, we also report improving results for the symmetrical version of the problem. Moreover, we show the benefits of implementing the efficient move evaluation procedure that enables the proposed metaheuristic to be scalable, even for large-size instances.

     

A partitioning algorithm for the network loading problem

This paper proposes a Benders-like partitioning algorithm to solve the network loading problem. The approach is an iterative method in which the integer programming solver is not used to produce the best integer point in the polyhedral relaxation of the set of feasible capacities. Rather, it selects an integer solution that is closest to the best known integer solution. Contrary to previous approaches, the method does not exploit the original mixed integer programming formulation of the problem. The effort of computing integer solutions is entirely left to a pure integer programming solver while valid inequalities are generated by solving standard nonlinear multicommodity flow problems. The method is compared to alternative approaches proposed in the literature and appears to be efficient for computing good upper bounds.     

A discrete dynamic convexized method for the max-cut problem

The max-cut problem is a classical NP-hard problem in graph theory. In this paper, we adopt a local search method, called MCFM, which is a simple modification of the Fiduccia-Mattheyses heuristic method in Fiduccia and Mattheyses (Proc. ACM/IEEE DAC, pp. 175?C181, 1982) for the circuit partitioning problem in very large scale integration of circuits and systems. The method uses much less computational cost than general local search methods. Then, an auxiliary function is presented which has the same global maximizers as the max-cut problem. We show that maximization of the function using MCFM can escape successfully from previously converged discrete local maximizers by taking increasing values of a parameter. An algorithm is proposed for the max-cut problem, by maximizing the auxiliary function using MCFM from random initial solutions. Computational experiments were conducted on three sets of standard test instances from the literature. Experimental results show that the proposed algorithm is effective for the three sets of standard test instances.     

A New Method,the Fusion Fission,for the Relaxed <Emphasis Type="Italic">k</Emphasis>-way Graph Partitioning Problem,and Comparisons with Some Multilevel Algorithms

In this paper a new graph partitioning problem is introduced, the relaxed k-way graph partitioning problem. It is close to the k-way, also called multi-way, graph partitioning problem, but with relaxed imbalance constraints. This problem arises in the air traffic control area. A new graph partitioning method is presented, the Fusion Fission, which can be used to resolve the relaxed k-way graph partitioning problem. The Fusion Fission method is compared to classical Multilevel packages and with a Simulated Annealing algorithm. The Fusion Fission algorithm and the Simulated Annealing algorithm, both require a longer computation time than the Multilevel algorithms, but they also find better partitions. However, the Fusion Fission algorithm partitions the graph with a smaller imbalance and a smaller cut than Simulated Annealing does.     

Local search heuristics for single-machine scheduling with batching to minimize the number of late jobs

Local search heuristics are developed for a problem of scheduling jobs on a single machine. Jobs are partitioned into families, and a set-up time is necessary when there is a switch in processing jobs from one family to jobs of another family. The objective is to minimize the number of late jobs. Four alternative local search methods are proposed: multi-start descent, simulated annealing, tabu search and a genetic algorithm. The performance of these heuristics is evaluated on a large set of test problems. The best results are obtained with the genetic algorithm; multi-start descent also performs quite well.     

Applying local search to the feedback vertex set problem

The feedback vertex set problem (FVSP) consists in making a given directed graph acyclic by removing as few vertices as possible. In spite of the importance of this NP-hard problem, no local search approach had been proposed so far for tackling it. Building on a property of acyclic graphs, we suggest in this paper a new representation of the solutions of the FVSP (feedback sets). Thanks to this solution representation, we are able to design a local transformation (equivalent to a neighborhood) that changes a feedback set into a new one. Based on this neighborhood, we have developed a simulated annealing algorithm for the FVSP. Our experiments show that our algorithm outperforms the best existing heuristic, namely the greedy adaptive search procedure by Pardalos et al.     

The continuous grey pattern problem

A new location problem is formulated and solved. It is the continuous version of the grey pattern problem which is a special case of the Quadratic Assignment Problem. The problem is a minimization of a convex function subject to non-convex constraints and has infinitely many optimal solutions. We propose several mathematical programming formulations that are suitable for a multi-start heuristic algorithm. In addition to solving these formulations by the Solver in Excel and Mathematica, a special Nelder–Mead algorithm is proposed. This special algorithm provided the best results. One suggested modification may improve the performance of the Nelder–Mead algorithm for other optimization problems as well.     

Lock-Gain Based Graph Partitioning

We propose a new heuristic for the graph partitioning problem. Based on the traditional iterative improvement framework, the heuristic uses a new type of gain in selecting vertices to move between partitions. The new type of gain provides a good explanation for the performance difference of tie-breaking strategies in KL-based iterative improvement graph partitioning algorithms. The new heuristic performed excellently. Theoretical arguments supporting its efficacy are also provided. As the proposed heuristic is considered a good candidate for local optimization engines in metaheuristics, we combined it with a genetic algorithm as a sample case and obtained a surprising result that even the average results over 1,000 runs equalled the best known for most graphs.     

Global optimization of multilevel electricity market models including network design and graph partitioning

We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in Grimm et al. (2019) that does not include a network design problem. The two classical discrete optimization problems of network design and graph partitioning together with nonlinearities due to economic modeling yield extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof including novel primal-dual bound tightening techniques, whereas the second is a tailored generalized Benders decomposition. For the latter, we strengthen the Benders cuts of Grimm et al. (2019) by using the structure of the newly introduced network design subproblem. We prove for both methods that they yield global optimal solutions. Afterward, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation. Finally, we present a case study that illustrates the economic effects that are captured in our model.     

An equivalent subproblem relaxation for improving the solution of a class of transportation scheduling problems

In this paper an algorithm is presented for determining the K best paths that may contain cycles in a directed network.The basic idea behind the algorithm is quite simple. Once the best path has been determined it is excluded from the network in such a way that no new path is formed and no more paths are excluded. This step leads to an enlarged network where all the paths, but the best one, can be determined. The method is repeated until the desired paths have been computed.The proposed algorithm can be used not only for the classical K shortest paths problem but also for ranking paths under a nonlinear objective function, provided that an algorithm to determine the best path exists.Computational results are presented and comparisons with other approaches for the classical problem are made.     

Metaheuristic algorithms hybridised with variable neighbourhood search for solving the response time variability problem

The response time variability problem (RTVP) is a scheduling problem with a wide range of real-world applications: mixed-model assembly lines, multi-threaded computer systems, network environments, broadcast of commercial videotapes and machine maintenance, among others. The RTVP arises whenever products, clients or jobs need to be sequenced in such a way that the variability in the time between the points at which they receive the necessary resources is minimised. Since the RTVP is NP-hard, several heuristic and metaheuristic techniques are needed to solve non-small instances. The published metaheuristic procedure that obtained the best solutions, on average, for non-small RTVP instances is an algorithm based on a variant of the variable neighbourhood search (VNS), called Reduced VNS (RVNS). We propose hybridising RVNS with three existing algorithms based on tabu search, multi-start and particle swarm optimisation. The aim is to combine the strengths of the metaheuristics. A computational experiment is carried out and it is shown that, on average, all proposed hybrid methods are able to improve the best published solutions.     

A tabu-based large neighbourhood search methodology for the capacitated examination timetabling problem

Neighbourhood search algorithms are often the most effective known approaches for solving partitioning problems. In this paper, we consider the capacitated examination timetabling problem as a partitioning problem and present an examination timetabling methodology that is based upon the large neighbourhood search algorithm that was originally developed by Ahuja and Orlin. It is based on searching a very large neighbourhood of solutions using graph theoretical algorithms implemented on a so-called improvement graph. In this paper, we present a tabu-based large neighbourhood search, in which the improvement moves are kept in a tabu list for a certain number of iterations. We have drawn upon Ahuja–Orlin's methodology incorporated with tabu lists and have developed an effective examination timetabling solution scheme which we evaluated on capacitated problem benchmark data sets from the literature. The capacitated problem includes the consideration of room capacities and, as such, represents an issue that is of particular importance in real-world situations. We compare our approach against other methodologies that have appeared in the literature over recent years. Our computational experiments indicate that the approach we describe produces the best known results on a number of these benchmark problems.     

Speeding up continuous GRASP

Continuous GRASP (C-GRASP) is a stochastic local search metaheuristic for finding cost-efficient solutions to continuous global optimization problems subject to box constraints (Hirsch et al., 2007). Like a greedy randomized adaptive search procedure (GRASP), a C-GRASP is a multi-start procedure where a starting solution for local improvement is constructed in a greedy randomized fashion. In this paper, we describe several improvements that speed up the original C-GRASP and make it more robust. We compare the new C-GRASP with the original version as well as with other algorithms from the recent literature on a set of benchmark multimodal test functions whose global minima are known. Hart’s sequential stopping rule (1998) is implemented and C-GRASP is shown to converge on all test problems.     

An exact algorithm for graph partitioning

An exact algorithm is presented for solving edge weighted graph partitioning problems. The algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem. Lower bounds are obtained by decomposing the objective function into convex and concave parts and replacing the concave part by an affine underestimate. It is shown that the best affine underestimate can be expressed in terms of the center and the radius of the smallest sphere containing the feasible set. The concave term is obtained either by a constant diagonal shift associated with the smallest eigenvalue of the objective function Hessian, or by a diagonal shift obtained by solving a semidefinite programming problem. Numerical results show that the proposed algorithm is competitive with state-of-the-art graph partitioning codes.     

A two-stage intelligent search algorithm for the two-dimensional strip packing problem

This paper presents a two-stage intelligent search algorithm for a two-dimensional strip packing problem without guillotine constraint. In the first stage, a heuristic algorithm is proposed, which is based on a simple scoring rule that selects one rectangle from all rectangles to be packed, for a given space. In the second stage, a local search and a simulated annealing algorithm are combined to improve solutions of the problem. In particular, a multi-start strategy is designed to enhance the search capability of the simulated annealing algorithm. Extensive computational experiments on a wide range of benchmark problems from zero-waste to non-zero-waste instances are implemented. Computational results obtained in less than 60 seconds of computation time show that the proposed algorithm outperforms the supposedly excellent algorithms reported recently, on average. It performs particularly better for large instances.     

Breakout local search for the Steiner tree problem with revenue,budget and hop constraints

The Steiner tree problem (STP) is one of the most popular combinatorial optimization problems with various practical applications. In this paper, we propose a Breakout Local Search (BLS) algorithm for an important generalization of the STP: the Steiner tree problem with revenue, budget and hop constraints (STPRBH), which consists of determining a subtree of a given undirected graph which maximizes the collected revenues, subject to both budget and hop constraints. Starting from a probabilistically constructed initial solution, BLS uses a Neighborhood Search (NS) procedure based on several specifically designed move operators for local optimization, and employs an adaptive diversification strategy to escape from local optima. The diversification mechanism is implemented by adaptive perturbations, guided by dedicated information of discovered high-quality solutions. Computational results based on 240 benchmarks show that BLS produces competitive results with respect to several previous approaches. For the 56 most challenging instances with unknown optimal results, BLS succeeds in improving 49 and matching one best known results within reasonable time. For the 184 instances which have been solved to optimality, BLS can also match 167 optimal results.     

On the Efficiency of a Global Non-differentiable Optimization Algorithm Based on the Method of Optimal Set Partitioning

The examined algorithm for global optimization of the multiextremal non-differentiable function is based on the following idea: the problem of determination of the global minimum point of the function f(x) on the set (f(x) has a finite number of local minima in this domain) is reduced to the problem of finding all local minima and their attraction spheres with a consequent choice of the global minimum point among them. This reduction is made by application of the optimal set partitioning method. The proposed algorithm is evaluated on a set of well-known one-dimensional, two-dimensional and three-dimensional test functions. Recommendations for choosing the algorithm parameters are given.     

Traditional heuristic versus Hopfield neural network approaches to a car sequencing problem

This paper considers the problem of optimally sequencing different car models along an assembly line according to some contiguity constraints, while ensuring that the demands for each of the models are satisfied. This car sequencing problem (CSP) is a practical NP-hard combinatorial optimisation problem. The CSP is formulated as a nonlinear integer programming problem and it is shown that exact solutions to the problem are difficult to obtain due to the indefinite quadratic form of the CSP objective function. Two traditional heuristics (steepest descent and simulated annealing) are employed to solve the CSP approximately. Several Hopfield neural network (HNN) approaches are also presented. The process of mapping an optimisation problem onto a HNN is demonstrated explicitly, and modifications to the existing neural approaches are presented which guarantee feasibility of solutions. Further modifications are proposed to improve the solution quality by permitting escape from local minima in an attempt to locate the global optimum. Results from all solutions techniques are compared on a set of instances of the CSP, and conclusions drawn.     

Concurrent stochastic methods for global optimization

The global optimization problem, finding the lowest minimizer of a nonlinear function of several variables that has multiple local minimizers, appears well suited to concurrent computation. This paper presents a new parallel algorithm for the global optimization problem. The algorithm is a stochastic method related to the multi-level single-linkage methods of Rinnooy Kan and Timmer for sequential computers. Concurrency is achieved by partitioning the work of each of the three main parts of the algorithm, sampling, local minimization start point selection, and multiple local minimizations, among the processors. This parallelism is of a coarse grain type and is especially well suited to a local memory multiprocessing environment. The paper presents test results of a distributed implementation of this algorithm on a local area network of computer workstations. It also summarizes the theoretical properties of the algorithm.Research supported by AFOSR grant AFOSR-85-0251, ARO contract DAAG 29-84-K-0140, NSF grant DCR-8403483, and NSF cooperative agreement DCR-8420944.