Benchmark Optimization Problems Research Articles

Tight Lieb–Robinson Bound for approximation ratio in quantum annealing

Quantum annealing (QA) holds promise for optimization problems in quantum computing, especially for combinatorial optimization. This analog framework attracts attention for its potential to address complex problems. Its gate-based homologous, QAOA with proven performance, has attracted a lot of attention to the NISQ era. Several numerical benchmarks try to compare these two metaheuristics, however, classical computational power highly limits the performance insights. In this work, we introduce a parametrized version of QA enabling a precise 1-local analysis of the algorithm. We develop a tight Lieb–Robinson bound for regular graphs, achieving the best-known numerical value to analyze QA locally. Studying MaxCut over cubic graph as a benchmark optimization problem, we show that a linear-schedule QA with a 1-local analysis achieves an approximation ratio over 0.7020, outperforming any known 1-local algorithms.

Benchmark problems for large-scale constrained multi-objective optimization with baseline results

The interests in evolutionary constrained multiobjective optimization are rapidly increasing during the past two decades. However, most related studies are limited to small-scale problems, despite the fact that many practical problems contain large-scale decision variables. Although several large-scale constrained multi-objective evolutionary algorithms (CMOEAs) have been developed, they are still tested on benchmarks that are designed for small-scale problems without the features of large-scale problems. To promote the research on large-scale constrained multi-objective optimization (LSCMO), this paper proposes a new LSCMO benchmark based on the design principles of large-scale multi-objective optimization and constrained multi-objective optimization. In this benchmark, more realistic features are considered, such as mixed linkages between constraint variables and unconstrained variables, imbalanced contributions of variables to the objectives, varying number constraint functions. Besides, to better solve the proposed benchmark, a bidirectional sampling strategy is proposed, where a convergence direction sampling and a diversity direction sampling are used to accelerate the convergence and maintain diversity respectively. Furthermore, the proposed bidirectional sampling strategy is embedded into an existing CMOEA to improve the search ability of algorithm in the large-scale search space with constraints. In experiments, the proposed algorithm is compared with several latest peer algorithms, and the results verify that the designed benchmark functions can effectively test the performance of algorithms and the proposed algorithm can better tackle the new benchmark. Finally, the proposed algorithm is used to solve the network structure control-based personalized drug target recognition problems with more than 2000 decision variables, and results show its superiority.

Elk herd optimizer: a novel nature-inspired metaheuristic algorithm

This paper proposes a novel nature-inspired swarm-based optimization algorithm called elk herd optimizer (EHO). It is inspired by the breeding process of the elk herd. Elks have two main breeding seasons: rutting and calving. In the rutting season, the elk herd splits into different families of various sizes. This division is based on fighting for dominance between bulls, where the stronger bull can form a family with large numbers of harems. In the calving season, each family breeds new calves from its bull and harems. This inspiration is set in an optimization context where the optimization loop consists of three operators: rutting season, calving season, and selection season. During the selection season, all families are merged, including bulls, harems, and calves. The fittest elk herd will be selected for use in the upcoming rutting and calving seasons. In simple words, EHO divides the population into a set of groups, each with one leader and several followers in the rutting season. The number of followers is determined based on the fitness value of its leader group. Each group will generate new solutions based on its leader and followers in the calving season. The members of all groups including leaders, followers, and new solutions are combined and the fittest population is selected in the selection season. The performance of EHO is assessed using 29 benchmark optimization problems utilized in the CEC-2017 special sessions on real-parameter optimization and four traditional real-world engineering design problems. The comparative results were conducted against ten well-established metaheuristic algorithms and showed that the proposed EHO yielded the best results for almost all the benchmark functions used. Statistical testing using Friedman’s test post-hocked by Holm’s test function confirms the superiority of the proposed EHO when compared to other methods. In a nutshell, EHO is an efficient nature-inspired swarm-based optimization algorithm that can be used to tackle several optimization problems.

Coupling of bond-based peridynamics and continuous density-based topology optimization methods for effective design of three-dimensional structures with discontinuities

AbstractThis study proposes continuous density-based three-dimensional topology optimization (TO) approaches developed by coupling the peridynamic theory (PD) with optimality criteria (OC) and proportional approach (PROP). These frameworks, abbreviated as PD-OC-TO and PD-PROP-TO, can be practically utilized to enhance the fracture toughness of the structures during the optimization process by taking critical regions into account as pre-defined cracks. Breaking the non-local interactions (bonds) between relevant PD particles enables us to readily model cracks. Utilizing this advantage, we solve several benchmark optimization problems including different numbers, positions, and alignments of the cracks. The major differences between the proposed methods are examined by comparing optimum topologies for various cracked scenarios. Moreover, the mechanical behaviour of the optimized structures is investigated under dynamic loads to prove the significant improvements achieved by the present approach in the final designs. The results of dynamic analyses reveal the viability of both PD-TO methods for increasing the fracture toughness of the structure in the optimization stage. Overall, the proposed approach is confirmed as a superior design and optimization tool for future engineering structures. Graphical abstract

Minimization of the weight of the blade of the aerial installation by an adaptive hybrid optimization method

In various fields of engineering practice, such as turbine building, power engineering, elements of thin-walled structures are widely used, which operate under increased loads when interacting with air or water. These include blades of impellers of radial-axial and rotary-blade hydraulic turbines, blades of wind power plants. Designing highly efficient machines and structures with the required level of reliability requires determining their optimal characteristics. The article proposes an adaptive method for finding the minimum of an arbitrary smooth multivariable function. The method was used to solve the benchmark optimization problem of a function in the form of a valley. The essence of the proposed algorithm lies in the sequential approach to the bottom of the valley and the subsequent movement in the direction of decreasing the objective function. Comparison of the results of calculating the minimum point of the function is performed using both non-gradient and gradient methods, namely: Powell, Hook-Jeeves, the steepest descent method and the developed method. It was found that the effectiveness of the proposed method is greater than the usual search algorithms, but it is not without its drawbacks. The method is proposed that represents a number of hybrid methods, which form a hybrid coalition Proposed hybrid algorithm does not provide a satisfactory result in the "single" search. The search algorithm reaches a point where all the values of the function at the surrounding points are greater than the values at the obtained point, and the algorithm cannot overcome the barrier. To solve the problem, it is necessary to take the obtained point as a new starting point and repeat the algorithm for finding the minimum of the function, that is, use the multistart method. The proposed method was used to solve the problem of optimizing the blade of an air installation, which was reduced to the problem of unconditional optimization using the method of penalty functions, but the goal function had a significantly valley structure. The optimal values of section thicknesses were obtained, which made it possible to build a blade with improved characteristics.

A Data Stream Ensemble Assisted Multifactorial Evolutionary Algorithm for Offline Data-Driven Dynamic Optimization.

Existing work on offline data-driven optimization mainly focuses on problems in static environments, and little attention has been paid to problems in dynamic environments. Offline data-driven optimization in dynamic environments is a challenging problem because the distribution of collected data varies over time, requiring surrogate models and optimal solutions tracking with time. This paper proposes a knowledge-transfer-based data-driven optimization algorithm to address these issues. First, an ensemble learning method is adopted to train surrogate models to leverage the knowledge of data in historical environments as well as adapt to new environments. Specifically, given data in a new environment, a model is constructed with the new data, and the preserved models of historical environments are further trained with the new data. Then, these models are considered to be base learners and combined as an ensemble surrogate model. After that, all base learners and the ensemble surrogate model are simultaneously optimized in a multitask environment for finding optimal solutions for real fitness functions. In this way, the optimization tasks in the previous environments can be used to accelerate the tracking of the optimum in the current environment. Since the ensemble model is the most accurate surrogate, we assign more individuals to the ensemble surrogate than its base learners. Empirical results on six dynamic optimization benchmark problems demonstrate the effectiveness of the proposed algorithm compared with four state-of-the-art offline data-driven optimization algorithms. Code is available at https://github.com/Peacefulyang/DSE_MFS.git.

Grey Wolf Cuckoo Search Algorithm for Training Feedforward Neural Network and Logic Gates Design

This paper presents a new hybrid Swarm Intelligence (SI) algorithm based on the Cuckoo Search Algorithm (CSA) and Grey Wolf Optimizer (GWO) called the Grey Wolf Cuckoo Search (GWCS) algorithm. The GWCS algorithm extracts and combines CSA and GWO features for efficient optimization. To carry out the comprehensive validation, the developed algorithm is applied to three different scenarios with their counterparts. The first validation is carried out on standard optimization benchmark problems. Further, they are used to train Feedforward Neural Networks and finally applied to design logic gates. The comprehensive results are presented and it is found that the proposed GWCS algorithms perform better compared to the state-of-the-art.

A novel variational inequality approach for modeling the optimal equilibrium in multi-tiered supply chain networks

We present a novel variational inequality model (VIM) to capture the complex real decision-making process in multi-tiered supply chain networks (MSCN) without strictly limiting the features of related functions. The VIM is formulated with the equilibrium conditions on links as the optimization goal and the flow conservation condition as the main constraints. We transform the VIM into a series of equivalent Non-Linear Programming Models (NLPMs) to solve. To address this challenge, we propose a novel population-based heuristic algorithm called the Multiscale Model Learning Algorithm (MMLA). The MMLA is inspired by the learning behavior of individuals in a group and can converge to an optimal equilibrium state of the MSCN. The MMLA has two key operations: zooming in on the search field and learning search in a learning stage. The excellent performers, called medalists, are imitated by other learners. With the increase in learning stages, the learning efficiency is improved, and the searching energy is concentrated in a more promising area. We employ sixteen benchmark optimization problems and two supply chain networks to demonstrate the effectiveness of the MMLA and the rationality of the equilibrium models. The results obtained by MMLA for the NLPM show that the MMLA can solve the equilibrium model effectively, and multiple optimal equilibrium states may exist for an MSCN. The flexibility of the NLPM makes it possible to consider more complicated decision-making mechanisms in the model.

Choice of benchmark optimization problems does matter

Various benchmark sets have already been proposed to facilitate comparison between metaheuristics, or Evolutionary Algorithms. During the competition, typically algorithms are allowed either to run until the allowed number of function calls is exhausted (and one compares the quality of solutions found), or until a required objective function value is obtained (one compares the speed in reaching the required solution). During the last 20 years several problem sets were defined using the first approach. In this study, we test 73 optimization algorithms proposed between the 1960′s and 2022 on nine competitions based on four sets of problems (CEC 2011, CEC 2014, CEC 2017, and CEC 2020) with different dimensionalities. We intend to test the original versions of 73 algorithms “as they are”, with control parameters proposed by the authors of the particular method. The recent benchmark set, CEC 2020, includes fewer problems and allows much more function calls than the former sets. As a result, one group of algorithms perform best on older, a different one on the more recent (CEC 2020) benchmark sets. Almost all algorithms that perform best on CEC 2020 set achieve moderate-to-poor performance on older sets, including real-world problems from CEC 2011. Algorithms that perform best on older sets are more flexible than those that perform best on CEC 2020 benchmark. The choice of the benchmark may have a crucial impact on the final ranking of algorithms. The lack of tuning may affect the results that were obtained in this study, hence it is highly recommended to repeat a similar large-scale comparison with control parameters of each algorithm tuned, best by different methods, separately for each benchmark set.

Hybridizing interval method with a heuristic for solving real-world constrained engineering optimization problems

Practical issues in real-world engineering problems limit the use of traditional optimization methods in finding solutions to design optimization problems. Although several advanced optimization approaches have been developed recently, a unified way to solve almost every problem effectively is yet to arise. To address this, we present a new method for optimizing several structure engineering design problems using our unique hybrid optimization approach inspired by the integration of branch and bound-like concepts of interval analysis with heuristics, and it differs from other methods in the literature. We hybridize the Split-Detect-Discard-Shrink technique and the Sophisticated ABC (SDDS-SABC) algorithm. The advantage of the SDDS process is that it shrinks the entire search region through recursive breakdown and improves computational effort to focus on subregion covering potential solutions for further decomposition. SABC plays a vital role in extracting the best solutions from subregions whose values help detect the promising subregion. SDDS and SABC are sequentially repeated until global/ close to the global solution(s) is reached. The ranking and selection rules are applied to assist in an optimistic decision-making process. In the SABC algorithm, we introduce Latin Hypercube, a new initialization scheme for food sources. This enhances the computational approach towards the optimal solution. Develop a dual strategy employed bees phase to explore their neighbourhoods better. Introduce a new Dynamic penalty method free from extra parameters/ factors. Here 57 real-world complex constrained optimization benchmark problems (CEC2020) featuring problems with Industrial Chemical Processes, Synthesis and Design, Mechanical Engineering, Power System, Power Electronics, and Livestock Feed Ration Optimization are presented, where our method is applied to optimize the decision parameters. Finally, our optimal results are compared in terms of their statistical significance using Friedman and Wilcoxon rank tests with three well-known optimizer algorithms in the literature. The results show that SDDS-SABC outperforms most tested competitors and demonstrates the practicability of SDDS-SABC in solving challenging real-world problems. Moreover, the SDDS-SABC method is numerically stable and suitable for the illustrated optimization examples. The key novelty of the presented approach is the ability to perform a static and better optimal solution in most runs despite the problem being too complex.

An Improved Sparrow Search Algorithm for Global Optimization with Customization-Based Mechanism

To solve the problems of the original sparrow search algorithm’s poor ability to jump out of local extremes and its insufficient ability to achieve global optimization, this paper simulates the different learning forms of students in each ranking segment in the class and proposes a customized learning method (CLSSA) based on multi-role thinking. Firstly, cube chaos mapping is introduced in the initialization stage to increase the inherent randomness and rationality of the distribution. Then, an improved spiral predation mechanism is proposed for acquiring better exploitation. Moreover, a customized learning strategy is designed after the follower phase to balance exploration and exploitation. A boundary processing mechanism based on the full utilization of important location information is used to improve the rationality of boundary processing. The CLSSA is tested on 21 benchmark optimization problems, and its robustness is verified on 12 high-dimensional functions. In addition, comprehensive search capability is further proven on the CEC2017 test functions, and an intuitive ranking is given by Friedman's statistical results. Finally, three benchmark engineering optimization problems are utilized to verify the effectiveness of the CLSSA in solving practical problems. The comparative analysis shows that the CLSSA can significantly improve the quality of the solution and can be considered an excellent SSA variant.

Surrogate-assisted distributed swarm optimisation for computationally expensive geoscientific models

Evolutionary algorithms provide gradient-free optimisation which is beneficial for models that have difficulty in obtaining gradients; for instance, geoscientific landscape evolution models. However, such models are at times computationally expensive and even distributed swarm-based optimisation with parallel computing struggle. We can incorporate efficient strategies such as surrogate-assisted optimisation to address the challenges; however, implementing inter-process communication for surrogate-based model training is difficult. In this paper, we implement surrogate-based estimation of fitness evaluation in distributed swarm optimisation over a parallel computing architecture. We first test the framework on a set of benchmark optimisation problems and then apply to a geoscientifc model that features landscape evolution model. Our results demonstrate very promising results for benchmark functions and the Badlands landscape evolution model. We obtain a reduction in computationally time while retaining optimisation solution accuracy through the use of surrogates in a parallel computing environment. The major contribution of the paper is in the application of surrogate-based optimisation for geoscientific models which can in the future help in better understanding of paleoclimate and geomorphology.

Improved Cascade Chaotic Invasive Weed Optimization Algorithm (ICCIWO), application to controller tuning and optimization

ABSTRACT An Improved Cascade Chaotic Invasive Weed Optimization Algorithm (ICCIWO) is proposed, employing cascade chaotic maps in the structure of a metaheuristic optimisation algorithm. The main aim of this modification is to provide a new algorithm for tuning the parameters of controllers. Conventionally, invasive weed optimisation algorithms (IWO) have been utilised in the literature to solve optimisation problems. The IWO has merits, namely fast convergence and high extraction capability. However, the IWO suffers from a lack of population diversity and poor exploration. Two novel improvements are proposed here to eliminate these disadvantages, based on utilising chaotic behaviours in the population generation and in the standard deviation update rule. The final proposed mechanism is applied to tune the parameters of a PID controller and solve several benchmark optimisation problems. Simulation analysis compares the performance of the proposed method with a recent similar approach. The simulation results validate the high performance of the proposed novel cascade chaotic scheme.

Developments on metaheuristic-based optimization for numerical and engineering optimization problems: Analysis, design, validation, and applications

Optimization problems are prevalent in a variety of real-world applications, including medical, engineering, chemical, and others, and must be precisely solved to enhance the performance of these applications. Unfortunately, finding near-optimal solutions to these problems is regarded as a hard challenge due to their various characteristics. In a new attempt to solve these problems, this paper presents a new variant of the artificial gorilla troops optimizer (GTO) called ranking-based GTO (RGTO). This variant uses two strategies known as the ranking-based update strategy and the convergence acceleration strategy to improve both the classical GTO's exploitation and exploration capabilities. The first strategy is proposed to enhance each gorilla's local and global search abilities, whereas the latter is intended to enhance GTO's global search abilities to reach better solutions as quickly as possible. First, a recent and challenging benchmark, namely CEC-2017, is utilized to assess the RGTO's explorative and exploitative capabilities. After that, RGTO is used to solve three engineering optimization problems, including parameter estimation problems for both photovoltaic (PV) models and proton exchange membrane fuel cells (PEMFCs), as well as some engineering design problems, to demonstrate how well it performs for real-world optimization problems. Compared to several rival optimizers, the proposed algorithm provides outstanding outcomes for the three engineering optimization benchmark problems considered.

Self-Adaptive Spherical Search With a Low-Precision Projection Matrix for Real-World Optimization.

Since the last three decades, numerous search strategies have been introduced within the framework of different evolutionary algorithms (EAs). Most of the popular search strategies operate on the hypercube (HC) search model, and search models based on other hypershapes, such as hyper-spherical (HS), are not investigated well yet. The recently developed spherical search (SS) algorithm utilizing the HS search model has been shown to perform very well for the bound-constrained and constrained optimization problems compared to several state-of-the-art algorithms. Nevertheless, the computational burdens for generating an HS locus are higher than that for an HC locus. We propose an efficient technique to construct an HS locus by approximating the orthogonal projection matrix to resolve this issue. As per our empirical experiments, this technique significantly improves the performance of the original SS with less computational effort. Moreover, to enhance SS's search capability, we put forth a self-adaptation technique for choosing the effective values of the control parameters dynamically during the optimization process. We validate the proposed algorithm's performance on a plethora of real-world and benchmark optimization problems with and without constraints. Experimental results suggest that the proposed algorithm remains better than or at least comparable to the best-known state-of-the-art algorithms on a wide spectrum of problems.

Improved Teaching Learning Algorithm with Laplacian operator for solving nonlinear engineering optimization problems

Teaching Learning Algorithm (TLA) is a recently developed nature-inspired optimization technique applicable to complex optimization problems. This paper proposes an improved TLA version using the Laplacian operator of the Genetic Algorithm (GA), named LX-TLA. The proposed algorithm is tested on benchmark optimization problems, including unimodal and multimodal problems. The numerical results are obtained in the form of objective function values, and a t-test is applied to compare the performance of LX-TLA and basic TLA. Convergence plots are given to provide insight into the convergence behavior of LX-TLA. The results reveal that proposed algorithm provides effective and efficient performance in solving benchmark test functions. The proposed algorithm is also applied to engineering design problems, such as Tuned Mass Damper (TMD), truss structure, welded beam, tension string, and pressure vessel. The results obtained using LX-TLA are compared with other nature-inspired optimization algorithms. The results demonstrate that the proposed algorithm is a robust and effective tool for solving complex optimization problems.

A pairwise ranking estimation model for surrogate-assisted evolutionary algorithms

Surrogate-assisted evolutionary algorithms (SAEAs) have attracted considerable attention for reducing the computation time required by an EA on computationally expensive optimization problems. In such algorithms, a surrogate model estimates the solution evaluation with a low computing cost and is used to obtain promising solutions to which the accurate evaluation with an expensive computation cost is then applied. This study proposes a novel pairwise ranking surrogate model called the Extreme Learning-machine-based DirectRanker (ELDR). ELDR integrates two machine learning models: extreme learning machine (ELM) and DirectRanker (DR). ELM is a single-layer neural network capable of fast learning, whereas DR uses pairwise learning to rank using a neural network developed mainly for information retrieval. To investigate the effectiveness of the proposed surrogate model, this study first examined the estimation accuracy of ELDR. Subsequently, ELDR was incorporated into a state-of-the-art SAEA and compared with existing SAEAs on well-known real-valued optimization benchmark problems. The experimental results revealed that ELDR has a high estimation accuracy even on high-dimensional problems with a small amount of training data. In addition, the SAEA using ELDR exhibited a high search performance compared with other existing SAEAs, especially on high-dimensional problems.

GLISp-r: a preference-based optimization algorithm with convergence guarantees

AbstractPreference-based optimization algorithms are iterative procedures that seek the optimal calibration of a decision vector based only on comparisons between couples of different tunings. At each iteration, a human decision-maker expresses a preference between two calibrations (samples), highlighting which one, if any, is better than the other. The optimization procedure must use the observed preferences to find the tuning of the decision vector that is most preferred by the decision-maker, while also minimizing the number of comparisons. In this work, we formulate the preference-based optimization problem from a utility theory perspective. Then, we propose , an extension of a recent preference-based optimization procedure called . The latter uses a Radial Basis Function surrogate to describe the tastes of the decision-maker. Iteratively, proposes new samples to compare with the best calibration available by trading off exploitation of the surrogate model and exploration of the decision space. In , we propose a different criterion to use when looking for new candidate samples that is inspired by , a popular procedure in the black-box optimization framework. Compared to , is less likely to get stuck on local optima of the preference-based optimization problem. We motivate this claim theoretically, with a proof of global convergence, and empirically, by comparing the performances of and on several benchmark optimization problems.

A Hybrid Algorithm Based on NSGA-II and MOPSO for Multi-Objective Designs of Electromagnetic Devices

In this paper, a hybrid algorithm is proposed by combining the NSGA-II with MOPSO algorithm. The original NSGA-II is improved by using Logistic mapping initialization and a dynamic selection mechanism of crossover and mutation operators is proposed. The performance of the proposed hybrid algorithm is verified using standard test functions and it is applied to the multi-objective optimization benchmark problem TEAM 22. Numerical results demonstrate the effectiveness and superiority of the proposed hybrid algorithm.

A Computational Comparison of Three Nature-Inspired, Population-Based Metaheuristic Algorithms for Modelling-to-Generate Alternatives

In “real life” decision-making situations, inevitably, there are numerous unmodelled components, not incorporated into the underlying mathematical programming models, that hold substantial influence on the overall acceptability of the solutions calculated. Under such circumstances, it is frequently beneficial to produce a set of dissimilar–yet “good”–alternatives that contribute very different perspectives to the original problems. The approach for creating maximally different solutions is known as modelling-to-generate alternatives (MGA). Recently, a data structure that permits MGA using any population-based solution procedure has been formulated that can efficiently construct sets of maximally different solution alternatives. This new approach permits the production of an overall best solution together with n locally optimal, maximally different alternatives in a single computational run. The efficacy of this novel computational approach is tested on four benchmark optimization problems.