Large Combinatorial Optimization Problems Research Articles

PHH: Policy-Based Hyper-Heuristic With Reinforcement Learning

Hyper-heuristics have a high level of generality and adaptability, allowing them to effectively solve a wide range of complex optimization problems. With reinforcement learning, hyper-heuristics can use experience and knowledge gained to tackle unforeseen problems, allowing hyper-heuristics to adapt and improve over time. Our paper proposes a framework for using policy-based reinforcement learning to improve the performance of hyper-heuristics. The framework trains hyper-heuristic agents to select the best generalized constructive low-level heuristics to solve combinatorial optimization problems. The framework evaluation was performed using three benchmarking problems: traveling salesman, capacitated vehicle routing, and bin packing problems. The results showed that the proposed framework can outperform existing meta-heuristic and hyper-heuristic-based algorithms for all large problem instances in all problem domains. The proposed framework was also evaluated by applying it to a cost optimization problem for workflow scheduling on a hybrid cloud with a deadline constraint. Eight agents were trained on medium-sized workflows with two deadlines and tested against traditional meta-heuristic and hyper-heuristic methods to solve smaller and larger workflows with unforeseen deadlines. Four workflow applications, three workflow sizes, and three deadlines were used in the evaluation. The results showed that our proposed framework provided significantly better solutions of up to 98% for benchmarking problems, and up to 22% for cost optimization in workflow scheduling. Moreover, trained with small problem instances, the framework performed well for unforeseen larger problem instances implying its generalization. The proposed framework, thus, has the potential to improve both the generality and performance of solving large combinatorial optimization problems.

A Hybrid Optimization Algorithm for Solving of the Unit Commitment Problem Considering Uncertainty of the Load Demand

Unit commitment problem (UCP) is classified as a mixed-integer, large combinatorial, high-dimensional and nonlinear optimization problem. This paper suggests solving the UCP under deterministic and stochastic load demand using a hybrid technique that includes the modified particle swarm optimization (MPSO) along with equilibrium optimizer (EO), termed as MPSO-EO. The proposed approach is tested firstly on 15 benchmark test functions, and then it is implemented to solve the UCP under two test systems. The results are basically compared to that of standard EO and previously applied optimization techniques in solving the UCP. In test system 1, the load demand is deterministic. The proposed technique is in the best three solutions for the 10-unit system with cost savings of 309.95 USD over standard EO and for the 20-unit system it shows the best results over all algorithms in comparison with cost savings of 1951.5 USD over standard EO. In test system 2, the load demand is considered stochastic, and only the 10-unit system is studied. The proposed technique outperforms the standard EO with cost savings of 40.93 USD. The simulation results demonstrate that MPSO-EO has fairly good performance for solving the UCP with significant total operating cost savings compared to standard EO compared with other reported techniques.

A pointer network based deep learning algorithm for unconstrained binary quadratic programming problem

Combinatorial optimization problems have been widely used in various fields. And many types of combinatorial optimization problems can be generalized into the model of unconstrained binary quadratic programming (UBQP). Therefore, designing an effective and efficient algorithm for UBQP problems will also contribute to solving other combinatorial optimization problems. Pointer network is an end-to-end sequential decision structure and combines with deep learning technology. With the utilization of the structural characteristics of combinatorial optimization problems and the ability to extract the rule behind the data by deep learning, pointer network has been successfully applied to solve several classical combinatorial optimization problems. In this paper, a pointer network based algorithm is designed to solve UBQP problems. The network model is trained by supervised learning (SL) and deep reinforcement learning (DRL) respectively. Trained pointer network models are evaluated by self-generated benchmark dataset and ORLIB dataset respectively. Experimental results show that pointer network model trained by SL has strong learning ability to specific distributed dataset. Pointer network model trained by DRL can learn more general distribution data characteristics. In other words, it can quickly solve problems with great generalization ability. As a result, the framework proposed in this paper for UBQP has great potential to solve large scale combinatorial optimization problems.

Towards ultra-scale Branch-and-Bound using a high-productivity language

Due to the highly irregular nature and prohibitive execution times of Branch-and-Bound (B&B) algorithms applied to combinatorial optimization problems (COPs), their parallelization has received the se two last decades great attention. Indeed, significant efforts have been made to revisit the parallelization of B&B following the rapid evolution of high-performance computing technologies dealing with their associated scientific and technical challenges. However, these parallelization efforts have always been guided by the performance objective setting aside programming productivity. Nevertheless, this latter is crucial for designing ultra-scale algorithms able to harness modern supercomputers which are increasingly complex, including millions of processing cores and heterogeneous building-block devices. In this paper, we investigate the partitioned global address space (PGAS)-based approach using Chapel for the productivity-aware design and implementation of distributed B&B for solving large COPs. The proposed algorithms are intensively experimented using the Flow-shop scheduling problem as a test-case. The Chapel-based implementation is compared to its MPI+X-based traditionally used counterpart in terms of performance, scalability, and productivity. The results show that Chapel is much more expressive and up to 7.8× more productive than MPI+Pthreads. In addition, the Chapel-based search presents performance equivalent to MPI+Pthreads for its best results on 1024 cores and reaches up to 84% of the linear speedup. However, there are cases where the built-in load balancing provided by Chapel cannot produce regular load among computer nodes. In such cases, the MPI-based search can be up to 4.2× faster and reaches speedups up to 3× higher than its Chapel-based counterpart. Thorough feedback on the experience is given, pointing out the strengths and limitations of the two opposite approaches (Chapel vs. MPI+X). To the best of our knowledge, the present study is pioneering within the context of exact parallel optimization.

Testing evolutionary algorithms for optimization of water distribution networks

Water distribution networks (WDNs) are one of the most important elements of urban infrastructure and require large investment for construction. Design of WDNs is classified as a large combinatorial discrete nonlinear optimization problem. The main concerns associated with the optimization of such networks are the nonlinearity of the discharge-head loss relationships for pipes and the discrete nature of pipe sizes. Due to these issues, this problem is widely considered to be a benchmark problem for testing and evaluating the performance of nonlinear and heuristic optimization algorithms. This paper compares different techniques, all based on evolutionary algorithms (EAs), which yield optimal solutions for least-cost design of WDNs. All of these algorithms search for the global optimum starting from populations of solutions, rather than from a single solution, as in Newton-based search methods. They use different operators to improve the performance of many solutions over repeated iterations. Ten EAs, four of them for the first time, are applied to the design of three networks and their performance in terms of the least cost, under different stopping criteria, are evaluated. Statistical information for 20 executions of the ten algorithms is summarized, and Friedman tests are conducted. Results show that, for the two-loop benchmark network, the particle swarm optimization gravitational search and biology and bioinformatics global optimization algorithms efficiently converge to the global optimum, but perform poorly for large networks. In contrast, given a sufficient number of function evaluations, the covariance matrix adaptation evolution strategy and soccer league competition algorithm consistently converge to the global optimum, for large networks.

Efficiently Targeted Billboard Advertising Using Crowdsensing Vehicle Trajectory Data

Different from online promotion, the outdoor billboard advertising industry suffers from a lack of audience-targeted delivery and quantitative dissemination evaluation, which undermine its impact in practice and hinder it from fast development. To bridge this gap, in this paper, we leverage crowdsensing vehicle trajectory data to empower audience-targeted billboard advertising. More specifically, by integrating the information of mobility transition, traffic conditions (traffic volume and average speed), and advertisement semantic topics, we propose a quantitative model to quantify advertisement influence spread, with a special consideration on influence overlapping among mobile users. Based on it, an influence maximization-targeted billboard advertising problem is formulated to find k advertising units over spatiotemporal dimensions, with the goal of maximizing the total expected advertisement influence spread. To tackle the efficiency issue for solving large combinatorial optimization problem, we employ a divide-and-conquer mechanism, and propose a utility evaluation-based optimal searching approach. Extensive experiments on real-world taxicab trajectories clearly validate the effectiveness and efficiency of our proposed approach.

Biased randomization of heuristics using skewed probability distributions: A survey and some applications

Randomized heuristics are widely used to solve large scale combinatorial optimization problems. Among the plethora of randomized heuristics, this paper reviews those that contain biased-randomized procedures (BRPs). A BRP is a procedure to select the next constructive ‘movement’ from a list of candidates in which their elements have different probabilities based on some criteria (e.g., ranking, priority rule, heuristic value, etc.). The main idea behind biased randomization is the introduction of a slight modification in the greedy constructive behavior that provides a certain degree of randomness while maintaining the logic behind the heuristic. BRPs can be categorized into two main groups according to how choice probabilities are computed: (i) BRPs using an empirical bias function; and (ii) BRPs using a skewed theoretical probability distribution. This paper analyzes the second group and illustrates, throughout a series of numerical experiments, how these BRPs can benefit from parallel computing in order to significantly outperform heuristics and even simple metaheuristic approaches, thus providing reasonably good solutions in ‘real time’ to different problems in the areas of transportation, logistics, and scheduling.

A Heuristic Initialized Stochastic Memetic Algorithm for MDPVRP With Interdependent Depot Operations.

The vehicle routing problem (VRP) is a widely studied combinatorial optimization problem. We introduce a variant of the multidepot and periodic VRP (MDPVRP) and propose a heuristic initialized stochastic memetic algorithm to solve it. The main challenge in designing such an algorithm for a large combinatorial optimization problem is to avoid premature convergence by maintaining a balance between exploration and exploitation of the search space. We employ intelligent initialization and stochastic learning to address this challenge. The intelligent initialization technique constructs a population by a mix of random and heuristic generated solutions. The stochastic learning enhances the solutions' quality selectively using simulated annealing with a set of random and heuristic operators. The hybridization of randomness and greediness in the initialization and learning process helps to maintain the balance between exploration and exploitation. Our proposed algorithm has been tested extensively on the existing benchmark problems and outperformed the baseline algorithms by a large margin. We further compared our results with that of the state-of-the-art algorithms working under MDPVRP formulation and found a significant improvement over their results.

Fine-grained parallelization of fitness functions in bioinformatics optimization problems: gene selection for cancer classification and biclustering of gene expression data.

BackgroundMetaheuristics are widely used to solve large combinatorial optimization problems in bioinformatics because of the huge set of possible solutions. Two representative problems are gene selection for cancer classification and biclustering of gene expression data. In most cases, these metaheuristics, as well as other non-linear techniques, apply a fitness function to each possible solution with a size-limited population, and that step involves higher latencies than other parts of the algorithms, which is the reason why the execution time of the applications will mainly depend on the execution time of the fitness function. In addition, it is usual to find floating-point arithmetic formulations for the fitness functions. This way, a careful parallelization of these functions using the reconfigurable hardware technology will accelerate the computation, specially if they are applied in parallel to several solutions of the population.ResultsA fine-grained parallelization of two floating-point fitness functions of different complexities and features involved in biclustering of gene expression data and gene selection for cancer classification allowed for obtaining higher speedups and power-reduced computation with regard to usual microprocessors.ConclusionsThe results show better performances using reconfigurable hardware technology instead of usual microprocessors, in computing time and power consumption terms, not only because of the parallelization of the arithmetic operations, but also thanks to the concurrent fitness evaluation for several individuals of the population in the metaheuristic. This is a good basis for building accelerated and low-energy solutions for intensive computing scenarios.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-016-1200-9) contains supplementary material, which is available to authorized users.

Comparative study of diversity based parallel dual population genetic algorithm for unconstrained function optimisations

The genetic algorithms GAs metaheuristic deals with large scale combinatorial optimisation problems. It is biologically inspired by the method, based on the principle of survival of the fittest. In GAs, the concept of multiple populations offers an advantage of diversity. However, as the population evolves, the GA loses its diversity and sometimes it cannot avoid the local optima problem also known as premature convergence. The dual population genetic algorithm DPGA uses an extra population called the reserve population to provide additional diversity to the main population through crossbreeding. Crossbreeding solves the problem of premature convergence and helps to converge early. This paper is the empirical study of the Binary encoded parallel DPGA PDPGA. It is compared with metaheuristics given in literature based on reliability, efficacy and efficiency. The performance of PDPGA is competitive over other nature-inspired optimisation methods like genetic algorithm GA, particle swarm optimisation PSO, differential evolution DE, ANTS, bee colony, grenade explosion method GEM and bee colony optimisation BCO, but not better than artificial bee colony ABC and teaching-learning-based optimisation TLBO.

Impact of the Objective Function on the Construction of Internal Grids of Wind Farms Using Genetic Algorithm

The macro complex of the construction industry is energy intensive. Solutions that enable the supply of this demand while meeting the principles of sustainability are needed. The construction of wind farms has been a strategy employed by many countries to produce clean energy. An increase in the construction of wind farms has also been witnessed in Brazil. This calls for different activities, such as the design and construction of infrastructure. This article focuses on the design of internal medium voltage distribution grids for wind farms. The purpose is to find a radial configuration that connects a set of wind generators to the substation, in an optimum way, minimizing operational and construction costs, reducing loss and therefore contributing to sustainability. In large farms, the project design consists of a large combinatorial optimization problem, given the large number of possible configurations to be deployed. Finding the best solution for the internal grid depends on the criterion adopted for the objectives pursued. This article analyzes the different criteria that can be adopted in the design of the wind farm’s internal grid using a methodology based on genetic algorithm (GA). Its aim is to identify their influence on the solution of the problem and help decision-making by finding the most adequate criterion for the objectives pursued. The results show that the design of the internal grid is sensitive to the criteria adopted for the objective function. In addition, the degree of sensitivity is analyzed, showing that, in some cases, the solutions are not economically attractive and do not contribute to the reduction of losses.

Reliability Based Optimum Reservoir Design by Hybrid ACO-LP Algorithm

Optimal design of irrigation and water supply reservoirs under reliability constraints may be categorized as large combinatorial optimization problems. In this paper, the reliability based optimum design of a single water supply reservoir is formulated as a mixed integer programming and a hybrid algorithm is introduced for its solution. To eliminate iterative procedures in reliability-based reservoir design and operation, the reliability requirements are directly embedded into the modeling framework and treated as different sets of constraints. Adaptive penalty method is used for constraint handling in the solution methodology. The proposed algorithm couples an ant colony optimization (ACO) optimizer with a virtual linear programing (LP) model for the solution of the resulted NP-hard mixed integer nonlinear programming problem. Dez reservoir for irrigation water supply with 480 months of inflow is used to demonstrate the method and its performance. The structure and solution methodology is verified by the solution to the inverse problem. It is shown that the proposed hybrid model can efficiently solve the problem for various combinations of reliability measures in a multiple period modeling scheme. It is illustrated that under some circumstances and specific reliability values, the mixed integer nonlinear programming (MINLP) solver may even fail to address a feasible and local optimal solution. Although operating rule is not included in the operational scheme, the procedure is capable of identifying coefficients for decision rules with any proposed structure.

Parallel ant colony optimization for resource constrained job scheduling

In mining supply chains, large combinatorial optimization problems arise. These are NP-hard and typically require a large number of computing resources to solve them. In particular, the run-time overheads can become increasingly prohibitive with increasing problem sizes. Parallel methods provide a way to manage such run-time issues by utilising several processors in independent or shared memory architectures. However it is not obvious how to adapt serial optimisation algorithms to perform best in a parallel environment. Here, we consider a resource constrained scheduling problem which is motivated in mining supply chains and present two popular meta-heuristics, ant colony optimization (ACO) and simulated annealing and investigate how best to parallelize these methods on a shared memory architecture consisting of several cores. ACO’s solution construction framework is inherently parallel allowing a relatively straightforward parallel implementation. However, for best performance, ACO needs an element of local search. This significantly complicates the paralellization. Several alternative schemes for parallel ACO with elements of local search are considered and evaluated empirically. We find that ACO with local search is the most effective single-threaded algorithm. The best parallel implementation can obtain similar quality results to the serial method in significantly less elapsed time.

Soccer league competition algorithm: A novel meta-heuristic algorithm for optimal design of water distribution networks

Water distribution networks are one of the most important elements in the urban infrastructure system and require huge investment for construction. Optimal design of water systems is classified as a large combinatorial discrete non-linear optimization problem. The main concern associated with optimization of water distribution networks is related to the nonlinearity of discharge-head loss equation, availability of the discrete nature of pipe sizes, and constraints, such as conservation of mass and energy equations. This paper introduces an efficient technique, entitled Soccer League Competition (SLC) algorithm, which yields optimal solutions for design of water distribution networks. Fundamental theories of the method are inspired from soccer leagues and based on the competitions among teams and players. Like other meta-heuristic methods, the proposed technique starts with an initial population. Population individuals (players) are in two types: fixed players and substitutes that all together form some teams. The competition among teams to take the possession of the top ranked positions in the league table and the internal competitions between players in each team for personal improvements are used for simulation purpose and convergence of the population individuals to the global optimum. Results of applying the proposed algorithm in three benchmark pipe networks show that SLC converges to the global optimum more reliably and rapidly in comparison with other meta-heuristic methods.

Production Scheduling of Open Pit Mines Using Particle Swarm Optimization Algorithm

Determining an optimum long term production schedule is an important part of the planning process of any open pit mine; however, the associated optimization problem is demanding and hard to deal with, as it involves large datasets and multiple hard and soft constraints which makes it a large combinatorial optimization problem. In this paper a procedure has been proposed to apply a relatively new and computationally less expensive metaheuristic technique known as particle swarm optimization (PSO) algorithm to this computationally challenging problem of the open pit mines. The performance of different variants of the PSO algorithm has been studied and the results are presented.

Solving the three dimensional quadratic assignment problem on a computational grid

The exact resolution of large instances of combinatorial optimization problems, such as three dimensional quadratic assignment problem (Q3AP), is a real challenge for grid computing. Indeed, it is necessary to reconsider the resolution algorithms and take into account the characteristics of such environments, especially large scale and dynamic availability of resources, and their multi-domain administration. In this paper, we revisit the design and implementation of the branch and bound algorithm for solving large combinatorial optimization problems such as Q3AP on the computational grids. Such gridification is based on new ways to efficiently deal with some crucial issues, mainly dynamic adaptive load balancing and fault tolerance. Our new approach allowed the exact resolution on a nation-wide grid of a difficult Q3AP instance. To solve this instance, an average of 1,123 computing cores were used for less than 12 days with a peak of around 3,427 computing cores.

Combining multi-core and GPU computing for solving combinatorial optimization problems

In this paper, we revisit the design and implementation of Branch-and-Bound (B&B) algorithms for solving large combinatorial optimization problems on GPU-enhanced multi-core machines. B&B is a tree-based optimization method that uses four operators (selection, branching, bounding and pruning) to build and explore a highly irregular tree representing the solution space. In our previous works, we have proposed a GPU-accelerated approach in which only a single CPU core is used and only the bounding operator is performed on the GPU device. Here, we extend the approach (LL-GB&B) in order to minimize the CPU–GPU communication latency and thread divergence. Such an objective is achieved through a GPU-based fine-grained parallelization of the branching and pruning operators in addition to the bounding one. The second contribution consists in investigating the combination of a GPU with multi-core processing. Two scenarios have been explored leading to two approaches: a concurrent (RLL-GB&B) and a cooperative one (PLL-GB&B). In the first one, the exploration process is performed concurrently by the GPU and the CPU cores. In the cooperative approach, the CPU cores prepare and off-load to GPU pools of tree nodes using data streaming while the GPU performs the exploration. The different approaches have been extensively experimented on the Flowshop scheduling problem. Compared to a single CPU-based execution, LL-GB&B allows accelerations up to (×160) for large problem instances. Moreover, when combining multi-core and GPU, we figure out that using RLL-GB&B is not beneficial while PLL-GB&B enables an improvement up to 36% compared to LL-GB&B.

Combinatorial complexity problem reduction by the use of artificial vaccines

A general method to reduce computing time for large combinatorial optimization problems by the use of a novel proposal is presented. It is based on reducing the problem complexity by the systematic application of vaccines, it is inspired in the concept of immunization derived from Artificial Immune Systems. The method can be applied practically to any combinatorial problem program solver such as genetic algorithms, memetic algorithms, artificial immune systems, ant colony optimization, the Dantzig–Fulkerson–Johnson algorithm, etc., providing optimal and suboptimal routes outperforming the selected algorithm itself. As a direct consequence of reducing problem complexity, the method provides a means to bring combinatorial optimization open problems that are too big to be treated by known techniques to a tractable point where acceptable solutions can be obtained. To demonstrate the proposed methodology the Traveling Salesman Problem for huge quantity of cities was used, we tested the method with modern evolutionary algorithms and the Concorde program. Comparative experiments that shows the effectiveness of the method are presented.

Research on MPI-Based Parallel Max-Min Ant System

When solving large scale combinatorial optimization problems, Max-Min Ant System requires long computation time. MPI-based Parallel Max-Min Ant System described in this paper can ensure the quality of the solution, as well as reduce the computation time. Numerical experiments on the multi-node cluster system show that when solving the traveling salesman problem, MPI-based Parallel Max-Min Ant System can get better computational efficiency.

Hybrid particle swarm optimization and differential evolution for optimal design of water distribution systems

Water distribution system design belongs to a class of large combinatorial non-linear optimization problems, involving complex implicit constraints, such as conservation of mass and energy equations, which are commonly satisfied through the use of hydraulic simulation solvers. Recently, many researchers have shifted the focus from traditional optimization methods to the use of meta-heuristic approaches for handling this complexity. This paper proposes a hybrid particle swarm optimization (PSO) and differential evolution (DE) method, linked to the hydraulic simulator, EPANET, for minimizing the cost design of water distribution systems. The performance of the proposed PSO-DE algorithm is demonstrated using three well-known benchmark water distribution system problems, the two-loop network, the Hanoi network and the New York Tunnels network. The results are compared to that of standard PSO and previously applied optimization methods. It is found that PSO-DE is a promising method for solving water distribution system design problems as it outperforms standard PSO and other algorithms previously presented in the literature for the three case studies considered.