Set Of Benchmark Problems Research Articles

An integrated coupled oscillator network to solve optimization problems

Solving combinatorial optimization problems is essential in scientific, technological, and engineering applications, but can be very time and energy-consuming using classical algorithms executed on digital processors. Oscillator-based Ising machines offer a promising alternative by exploiting the analog coupling between electrical oscillators to solve such optimization problems more efficiently. Here we present the design and the capabilities of our scalable approach to solve Ising and quadratic unconstrained binary optimization problems. This approach includes routable oscillator connections to simplify the time-consuming embedding of the problem into the oscillator network. Our manufactured silicon chip, featuring 1440 oscillators implemented in a 28 nm technology, demonstrates the ability to solve optimization problems in 950 ns while consuming typically 319 μW per node. A frequency, phase, and delay calibration ensures robustness against manufacturing variations. The system is evaluated with multiple sets of benchmark problems to analyze the sensitivity for parameters such as the coupling strength or frequency.

Minimising makespan in distributed assembly hybrid flowshop scheduling problems

To enhance the manufacturing flexibility, resilience, and production efficiency, the integration of scheduling for distributed manufacturing with assembly systems has become a pivotal driver of production planning evolution. In this research endeavour, we present a Mixed-Integer Linear Programming model and an innovative Iterated Epsilon-Greedy Reinforcement Learning algorithm to address the distributed assembly hybrid flowshop scheduling problem. Empirical validation, conducted through computational experiments on a benchmark problem set, is used to gain important managerial insights. The computational results demonstrate that the proposed algorithms significantly reduce the makespan for the addressed problem. This study has the potential to make valuable contributions to ongoing research endeavours within the realm of multi-stage shop scheduling, an area that continues to warrant progressive advancement.

The traveling purchaser problem for perishable foods

This paper addresses the well-known traveling purchaser problem (TPP) considering restrictions related to perishable food, which is called the traveling purchaser problem for perishable foods (TPP-PF). In addition to the main assumptions of the capacitated TPP, the TPP-PF takes into account the release times of perishable foods at markets. In this context, a product can only be purchased at a market after it becomes available for sale. The TPP-PF also considers that the quality of perishable products decreases due to waiting on the market shelves and during transportation, and it additionally takes the deterioration cost of the foods into account. The aim of the problem is to find the best procurement and route plan for the purchaser to minimize the total transportation cost of the temperature-controlled vehicle, the damage cost of perishable foods, and the purchasing cost. Since the deterioration cost of products based on the waiting time is determined through exponential functions, the problem is formulated as a mixed-integer non-linear programming model. To find efficient results for the problem, an adaptive large neighborhood search (ALNS) algorithm is introduced with new problem-specific destroy/repair operators. Furthermore, an additional mechanism is included in the ALNS to change the direction of the search when the algorithm cannot improve the best solution after a number of iterations. In order to analyze the performance of the proposed ALNS, a new benchmark problem set is generated using a well-known TPP benchmark set. The proposed ALNS is first performed for the TPP-PF instances and compared to GUROBI and a simulated annealing algorithm. Following the TPP-PF experiments, the ALNS is carried out for the TPP instances and compared to six state-of-the-art solution approaches. The computational results show that the proposed ALNS outperforms those approaches by finding better results for almost all instances.

Informed Down-Sampled Lexicase Selection: Identifying productive training cases for efficient problem solving.

Genetic Programming (GP) often uses large training sets and requires all individuals to be evaluated on all training cases during selection. Random down-sampled lexicase selection evaluates individuals on only a random subset of the training cases allowing for more individuals to be explored with the same amount of program executions. However, sampling randomly can exclude important cases from the down-sample for a number of generations, while cases that measure the same behavior (synonymous cases) may be overused. In this work, we introduce Informed Down-Sampled Lexicase Selection. This method leverages population statistics to build down-samples that contain more distinct and therefore informative training cases. Through an empirical investigation across two different GP systems (PushGP and Grammar-Guided GP), we find that informed down-sampling significantly outperforms random down-sampling on a set of contemporary program synthesis benchmark problems. Through an analysis of the created down-samples, we find that important training cases are included in the down-sample consistently across independent evolutionary runs and systems. We hypothesize that this improvement can be attributed to the ability of Informed Down-Sampled Lexicase Selection to maintain more specialist individuals over the course of evolution, while still benefiting from reduced per-evaluation costs.

The comparison of different PDP-type self-adaptive schemes for the cooperation of GA, DE, and PSO algorithms

Many global optimization problems are presented as a black-box model, in which there is no information on the objective function properties. Traditional optimization algorithms usually can't effectively solve that kind of problems. Different heuristics and metaheuristics are usually applied in that case. Evolutionary algorithms are one of the most popular and effective approaches to black-box optimization problems. However, it's hard to choose one specific method that will solve the given problem better than other algorithms. For dealing with this issue, self-adaptive schemes are usually implemented. In this paper we have investigated the performance of different PDP-type adaptive schemes using such popular evolutionary-based algorithms as Genetic Algorithm, Differential Evolution, and Particle Swarm Optimization. The experimental results on a set of benchmark problems have shown that investigated schemes can improve the performance compared with the performance of a stand-alone evolutionary algorithm. At the same time the choice of a scheme and its parameters affect the results.

Towards explainable metaheuristics: Feature extraction from trajectory mining

AbstractExplaining the decisions made by population‐based metaheuristics can often be considered difficult due to the stochastic nature of the mechanisms employed by these optimisation methods. As industries continue to adopt these methods in areas that increasingly require end‐user input and confirmation, the need to explain the internal decisions being made has grown. In this article, we present our approach to the extraction of explanation supporting features using trajectory mining. This is achieved through the application of principal components analysis techniques to identify new methods of tracking population diversity changes post‐runtime. The algorithm search trajectories were generated by solving a set of benchmark problems with a genetic algorithm and a univariate estimation of distribution algorithm and retaining all visited candidate solutions which were then projected to a lower dimensional sub‐space. We also varied the selection pressure placed on high fitness solutions by altering the selection operators. Our results show that metrics derived from the projected sub‐space algorithm search trajectories are capable of capturing key learning steps and how solution variable patterns that explain the fitness function may be captured in the principal component coefficients. A comparative study of variable importance rankings derived from a surrogate model built on the same dataset was also performed. The results show that both approaches are capable of identifying key features regarding variable interactions and their influence on fitness in a complimentary fashion.

Differential Evolution with a Level-Based Learning Strategy for Multimodal Optimization

Multimodal optimization aims at efficiently finding multiple optimal solutions of a problem. Owing to the population-based search mechanism, evolutionary algorithms (EAs) are becoming increasingly popular in solving multimodal optimization problems (MOPs). Most existing work focuses on designing and incorporating niching techniques into EAs so that multiple subpopulations can be formed and assigned to locate different optima. To further enhance the exploration and exploitation abilities of existing EAs, this paper developed a multimodal level-based learning strategy. The basic idea is that individuals should be treated differently according to their positions in the subpopulation. In the evolutionary process, a subpopulation is formed for each candidate solution by grouping its neighboring solutions. Then, individuals in the subpopulation are sorted according to their fitness. Subsequently, the multimodal level-based learning strategy applies different mutation operators to different individuals according to their rankings. Experiments are conducted on a set of benchmark problems to verify the efficacy of the multimodal level-based learning strategy. The results show that the proposed learning strategy can significantly enhance the performance of the existing algorithm. In addition, the algorithm integrated with the proposed strategy is applied to the task of finding multiple roots of nonlinear equation systems (NESs). The results indicate that with the support of the proposed learning strategy, the integrated algorithm compares favorably with state-of-the-art root finding algorithms.

On the hybridization of geometric semantic GP with gradient-based optimizers

Geometric semantic genetic programming (GSGP) is a popular form of GP where the effect of crossover and mutation can be expressed as geometric operations on a semantic space. A recent study showed that GSGP can be hybridized with a standard gradient-based optimized, Adam, commonly used in training artificial neural networks.We expand upon that work by considering more gradient-based optimizers, a deeper investigation of their parameters, how the hybridization is performed, and a more comprehensive set of benchmark problems. With the correct choice of hyperparameters, this hybridization improves the performances of GSGP and allows it to reach the same fitness values with fewer fitness evaluations.

REIN: Reliability Estimation via Importance sampling with Normalizing flows

We introduce a novel framework called REIN: Reliability Estimation by learning an Importance sampling (IS) distribution with Normalizing flows (NFs). The NFs learn probability space maps that transform the probability distribution of the input random variables into a quasi-optimal IS distribution. NFs stack together invertible neural networks to construct differentiable bijections with efficiently computed Jacobian determinants. The NF ‘pushes forward’ a realization from the input probability distribution into a realization from the IS distribution, with importance weights calculated using the change of variables formula. We also propose a loss function to learn a NF map that minimizes the reverse Kullback–Leibler divergence between the ‘pushforward’ distribution and a sequentially updated target distribution obtained by modifying the optimal IS distribution. We demonstrate REIN’s efficacy on a set of benchmark problems that feature very low failure rates, multiple failure modes and high dimensionality, while comparing against other variance reduction methods. We also consider two simple applications, the reliability analyses of a thirty-four story building and a cantilever tube, to demonstrate the applicability of REIN to practical problems of interest. As compared to other methods, REIN is shown to be useful for high-dimensional reliability estimation problems with very small failure probabilities.

Solving travelling thief problems using coordination based methods

A travelling thief problem (TTP) is a proxy to real-life problems such as postal collection. TTP comprises an entanglement of a travelling salesman problem (TSP) and a knapsack problem (KP) since items of KP are scattered over cities of TSP, and a thief has to visit cities to collect items. In TTP, city selection and item selection decisions need close coordination since the thief’s travelling speed depends on the knapsack’s weight and the order of visiting cities affects the order of item collection. Existing TTP solvers deal with city selection and item selection separately, keeping decisions for one type unchanged while dealing with the other type. This separation essentially means very poor coordination between two types of decision. In this paper, we first show that a simple local search based coordination approach does not work in TTP. Then, to address the aforementioned problems, we propose a human designed coordination heuristic that makes changes to collection plans during exploration of cyclic tours. We further propose another human designed coordination heuristic that explicitly exploits the cyclic tours in item selections during collection plan exploration. Lastly, we propose a machine learning based coordination heuristic that captures characteristics of the two human designed coordination heuristics. Our proposed coordination based approaches help our TTP solver significantly outperform existing state-of-the-art TTP solvers on a set of benchmark problems. Our solver is named Cooperation Coordination (CoCo) and its source code is available from https://github.com/majid75/CoCo.

A dual surrogate assisted evolutionary algorithm based on parallel search for expensive multi/many-objective optimization

Numerous optimization problems in the real world involve multi-objective and computationally expensive simulations (i.e., expensive multi-objective optimization problems). This paper purposes a dual surrogates-assisted evolutionary algorithm (SAEA) based on parallel search, termed DSAEA-PS, for this issue. Approximation and classification are two main implementation forms of surrogate models, but the existing methods of expensive multi-objective optimization only apply one kind of them, and scarce works have paid attention to combining approximation and classification to improve the optimization performance. In the proposed algorithm, to enhance the prediction accuracy and reliability, both the approximation model and classification model are applied to cooperate to provide the quality and uncertainty information of candidate solutions. Meanwhile, the parallel search based on heterogeneous multi-objective evolutionary algorithms is introduced for better exploration of the decision space. In addition, combined with the strengthened dominance relation (SDR), a sampling strategy that comprehensively considers the quality of candidate solutions and their uncertainty information is proposed. Experimental results with five peer competitors on a set of widely-used benchmark problems demonstrate the ability of DSAEA-PS. Furthermore, DSAEA-PS is adopted for a five-objective blended-wing-body underwater glider design problem that involves time-consuming simulations of fluid dynamics and structural strength. A series of high-performance solutions obtained from DSAEA-PS verifies its effectiveness on engineering applications.

A framework of global exploration and local exploitation using surrogates for expensive optimization

In solving expensive optimization problems, most research focuses on using either a single global surrogate or multiple local surrogates in surrogate-assisted evolutionary algorithms (SAEAs) or evolutionary Bayesian optimization (EBO). It is challenging for single global or multiple local surrogates to balance the exploration and exploitation of algorithms. To address this issue, the paper proposes a framework of global exploration and local exploitation with surrogates for solving expensive optimization problems. In detail, a scalable Gaussian process (GP) is used to fit the global fitness landscape and guide algorithms towards global exploration. A radial basis function network (RBFN) guides algorithms for local exploitation by capturing the local patterns identified by the global surrogate. In the local exploitation, a local search region is proposed to ensure that the search region contains the bilateral region of the current optimum solution. This way, the exploration and exploitation of algorithms can be balanced cooperatively by global and local surrogates. The effectiveness of the proposed method has been investigated and analyzed on a set of expensive benchmark problems.

A Discrete Cooperative Control Method for Production Scheduling Problem of Assembly Manufacturing System

With a sharp decrease in resource utilization of the complex production process, integrated scheduling methods are urgently needed in assembly manufacturing industries. To this end, this paper presents an integrated scheduling with jobs processing and assembly sequence (ISJPAS) problem where jobs and assemblies can be processed simultaneously. As the first contribution to this work, we develop a mixed integer nonlinear programming model (MINLP) that aims at minimizing maximum completion time and determining the earliness and tardiness of jobs and resulting inventory time. The structured property of the optimal scheduling solution is analyzed with regard to job sequencing and assembly sequencing. The NP-hard nature of the problem is proved, which provides upper and lower bounds on the optimal solution. Second, an efficient discrete assembly time and arrival time control (DAATC) method is presented based on continuous time variable control models. From a modified set of benchmark problems, the proposed method is tested by comparing four assembly association levels in real applications. The comparisons indicate the potentiality of our method to satisfy the due date. Lastly, relevance of practical applications are discussed, and several future research avenues are emphasized.

Towards a benchmark framework for model order reduction in the Mathematical Research Data Initiative (MaRDI)

AbstractThe race for the most efficient, accurate, and universal algorithm in scientific computing drives innovation. At the same time, this healthy competition is only beneficial if the research output is actually comparable to prior results. Fairly comparing algorithms can be a complex endeavor, as the implementation, configuration, compute environment, and test problems need to be well‐defined. Due to the increase in computer‐based experiments, new infrastructure for facilitating the exchange and comparison of new algorithms is also needed. To this end, we propose a benchmark framework as a set of generic specifications for comparing implementations of algorithms using test cases native to a community. Its value lies in its ability to fairly compare and validate existing methods for new applications, as well as compare newly developed methods with existing ones. As a prototype for a more general framework, we have begun building a benchmark tool for the model order reduction (MOR) community. The data basis of the tool is the collection of the Model Order Reduction Wiki (MORWiki). The wiki features three main categories: benchmarks, methods, and software. An editorial board curates submissions and patrols edited entries. Data sets for linear and parametric‐linear models are already well represented in the existing collection. Data sets for non‐linear or procedural models, for which only evaluation data, or codes/algorithmic descriptions, rather than equations, are available, are being added and extended. Properties and interesting characteristics used for benchmark selection and later assessments are recorded in the model metadata. Our tool, the Model Order Reduction Benchmarker (MORB), is under active development for linear time‐invariant systems and solvers. An ontology (MORBO) and knowledge graph are being developed in parallel. They catalog benchmark problem sets and their metadata and will also be integrated into the Mathematical Research Data Initiative (MaRDI) Portal to help improve the findability of such data sets. MORB faces a number of technical and field‐specific challenges, and we hope to recruit community input and feedback while presenting some initial results.

A flexible forcing immersed boundary scheme-based one-step simplified lattice Boltzmann method for two-dimensional fluid-solid interaction problems

In this study, we coupled the one-step simplified lattice Boltzmann method (OSLBM) [1] with flexible forcing immersed boundary (FFIB) scheme [2] and proposed a computationally efficient FFIB-OSLBM solver for complex two-dimensional fluid-solid interaction problems. The adopted OSLBM algorithm has the benefits of lower virtual memory requirement and easy implementation of physical boundary conditions over the traditional lattice Boltzmann (LBM) solver and is also computationally more efficient than the simplified lattice Boltzmann method (SLBM) [3]. Unlike in SLBM, the macroscopic variable is updated in one-step in OSLBM, thus avoiding the predictor and corrector steps and reducing the computational time. Furthermore, the adopted FFIB algorithm uses an implicit approach to calculate the body force density, which accurately satisfies the no-slip velocity condition at the boundary of the immersed solid. Moreover, this method requires less computational effort since it discards the time-consuming, complex matrix inversion process associated with the conventional immersed boundary (IB) method. We have conducted the numerical accuracy test and noticed that the proposed FFIB-OSLBM solver is nearly second-order accurate in space, which is higher than the typical coupled IB-LBM solvers. In addition, the applicability of the present solver is tested through a set of benchmark problems, and the results obtained from the present solver show an excellent agreement with the data available in the references. Compared to the FFIB-LBM [2], the present solver consumes at least 20% less computational memory and CPU time for all the validation problems investigated in this study.

Error Bounds for Discrete-Continuous Free Flight Trajectory Optimization

Two-stage methods addressing continuous shortest path problems start local minimization from discrete shortest paths in a spatial graph. The convergence of such hybrid methods to global minimizers hinges on the discretization error induced by restricting the discrete global optimization to the graph, with corresponding implications on choosing an appropriate graph density. A prime example is flight planning, i.e., the computation of optimal routes in view of flight time and fuel consumption under given weather conditions. Highly efficient discrete shortest path algorithms exist and can be used directly for computing starting points for locally convergent optimal control methods. We derive a priori and localized error bounds for the flight time of discrete paths relative to the optimal continuous trajectory, in terms of the graph density and the given wind field. These bounds allow designing graphs with an optimal local connectivity structure. The properties of the bounds are illustrated on a set of benchmark problems. It turns out that localization improves the error bound by four orders of magnitude, but still leaves ample opportunities for tighter error bounds by a posteriori estimators.

Surrogate-Assisted Automatic Parameter Adaptation Design for Differential Evolution

In this study, parameter adaptation methods for differential evolution are automatically designed using a surrogate approach. In particular, Taylor series are applied to model the searched dependence between the algorithm’s parameters and values, describing the current algorithm state. To find the best-performing adaptation technique, efficient global optimization, a surrogate-assisted optimization technique, is applied. Three parameters are considered: scaling factor, crossover rate and population decrease rate. The learning phase is performed on a set of benchmark problems from the CEC 2017 competition, and the resulting parameter adaptation heuristics are additionally tested on CEC 2022 and SOCO benchmark suites. The results show that the proposed approach is capable of finding efficient adaptation techniques given relatively small computational resources.

Novel Algorithm for Linearly Constrained Derivative Free Global Optimization of Lipschitz Functions

This paper introduces an innovative extension of the DIRECT algorithm specifically designed to solve global optimization problems that involve Lipschitz continuous functions subject to linear constraints. Our approach builds upon recent advancements in DIRECT-type algorithms, incorporating novel techniques for partitioning and selecting potential optimal hyper-rectangles. A key contribution lies in applying a new mapping technique to eliminate the infeasible region efficiently. This allows calculations to be performed only within the feasible region defined by linear constraints. We perform extensive tests using a diverse set of benchmark problems to evaluate the effectiveness and performance of the proposed algorithm compared to existing DIRECT solvers. Statistical analyses using Friedman and Wilcoxon tests demonstrate the superiority of a new algorithm in solving such problems.

An assessment of the total Lagrangian material point method: Comparison to conventional MPM, higher order basis, and treatment of near-incompressibility

The total Lagrangian material point method (TLMPM) is a recent modification to the conventional material point method (MPM). It poses the weak form in the undeformed configuration, and has shown great potential in solid mechanics applications because it seamlessly eliminates the so-called cell-crossing instability and overcomes numerical fracture. Despite its promise, it has not been widely adopted by the MPM community yet, and only a handful of research groups have employed it. At the same time, due to TLMPM’s recent appearance, there are open research questions, such as the subject of near-incompressibility that has not been thoroughly studied within its context. In this paper we attempt to strengthen TLMPM’s position in the field by providing an independent assessment of the method in a set of benchmark problems. Furthermore, we evaluate the need of using higher order spline shape functions, which have been shown to be crucial for the success of the conventional MPM. Finally, we present a simple projection technique for addressing solid mechanics problems that fall within the near-incompressible regime. The numerical results show that TLMPM is indeed a strong alternative to conventional MPM, whereas our near-incompressibility approach produces solutions that are free of volumetric locking and pressure oscillations, while introducing very minor modifications to existing codes.

A numerical two-scale approach for nonlinear hyperelastic beams and beam networks

We introduce a numerical framework for modeling hyperelastic slender trusses, oftentimes used as elementary building blocks in architected materials, which accounts for both the geometric nonlinearity inherent in thin structures and the nonlinear constitutive behavior of the base material. Akin to the FE2 method in homogenization, our approach is based on a formal, two-scale expansion. We decompose the three-dimensional (3D) description of a slender structure into a macroscale problem solving for the deformation of the beam center-line (based on an effective strain energy density, which depends on the stretching, bending, and torsional strains of the beam’s center-line) and a series of two-dimensional (2D) microscale boundary value problems (defined over the beam cross-section). We solve a series of 2D problems (covering a range of macroscopic strain combinations) for a given cross-sectional geometry and material distribution in an offline, pre-processing step. Using this pre-computed energy landscape, we solve the macroscopic boundary value problem within a geometrically exact, nonlinear discrete beam framework. We demonstrate the accuracy of this approach through a set of benchmark problems highlighting the nonlinear effects of the cross-sectional geometry and constitutive material, including material heterogeneity and pre-strains. We further illustrate how this technique is applied to truss-based architected materials consisting of networks of slender struts, which are typically made of polymeric base materials and thus allow for large nonlinear elastic deformation.