Benchmark Optimization Problems Research Articles

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.

Reduced order modeling for compressible cake filtration processes using proper orthogonal decomposition

This work aims to increase the computational efficiency of a complex mathematical cake-filtration model with strong nonlinearities representing cake compression. To this end, we employ a hybrid data driven approach using the technique of proper orthogonal decomposition. Hereby, a few sample simulations from the initial system of partial differential equations are used as the foundation to find optimal, globally defined basis functions, which in return offer the possibility to build a reduced-order model. In summary, the dimension of the reduced order model is diminished by ≈98% compared to the full order model, which translates to a net decrease of ≈90 % computational time needed to solve a benchmark optimization problem. This significant numerical speed-up offers the possibility to use the reduced order model in further advanced process control and optimization methods.

The Mountain Gazelle Optimizer for truss structures optimization

<abstract> <p>Computational tools have been used in structural engineering design for numerous objectives, typically focusing on optimizing a design process. We first provide a detailed literature review for optimizing truss structures with metaheuristic algorithms. Then, we evaluate an effective solution for designing truss structures used in structural engineering through a method called the mountain gazelle optimizer, which is a nature-inspired meta-heuristic algorithm derived from the social behavior of wild mountain gazelles. We use benchmark problems for truss optimization and a penalty method for handling constraints. The performance of the proposed optimization algorithm will be evaluated by solving complex and challenging problems, which are common in structural engineering design. The problems include a high number of locally optimal solutions and a non-convex search space function, as these are considered suitable to evaluate the capabilities of optimization algorithms. This work is the first of its kind, as it examines the performance of the mountain gazelle optimizer applied to the structural engineering design field while assessing its ability to handle such design problems effectively. The results are compared to other optimization algorithms, showing that the mountain gazelle optimizer can provide optimal and efficient design solutions with the lowest possible weight.</p> </abstract>

A penalty-based algorithm proposal for engineering optimization problems.

This paper presents a population-based evolutionary computation model for solving continuous constrained nonlinear optimization problems. The primary goal is achieving better solutions in a specific problem type, regardless of metaphors and similarities. The proposed algorithm assumes that candidate solutions interact with each other to have better fitness values. The interaction between candidate solutions is limited with the closest neighbors by considering the Euclidean distance. Furthermore, Tabu Search Algorithm and Elitism selection approach inspire the memory usage of the proposed algorithm. Besides, this algorithm is structured on the principle of the multiplicative penalty approach that considers satisfaction rates, the total deviations of constraints, and the objective function value to handle continuous constrained problems very well. The performance of the algorithm is evaluated with real-world engineering design optimization benchmark problems that belong to the most used cases by evolutionary optimization researchers. Experimental results show that the proposed algorithm produces satisfactory results compared to the other algorithms published in the literature. The primary purpose of this study is to provide an algorithm that reaches the best-known solution values rather than duplicating existing algorithms through a new metaphor. We constructed the proposed algorithm with the best combination of features to achieve better solutions. Different from similar algorithms, constrained engineering problems are handled in this study. Thus, it aims to prove that the proposed algorithm gives better results than similar algorithms and other algorithms developed in the literature.

Improved multi-objective structural optimization with adaptive repair-based constraint handling

Engineering optimization typically involves a large number of nonlinear constraints; therefore, effective constraint handling techniques (CHTs) are sought for metaheuristic optimization algorithms. Modified repair-based CHT is proposed here for a multi-objective evolutionary algorithm based on decomposition (MOEA/D). This CHT is: (1) adaptive to the share of infeasible solutions in a population; (2) free of problem-specific heuristics that users typically need to provide for repair; and (3) without control parameters. Infeasible solutions with superior decomposition function value are repaired using information contained in the neighbourhoods of the current population. The approach is tested on four multi-objective problems: a common mathematical optimization benchmark problem, two truss optimization problems and a real-world structural design of a tanker ship. A few prominent CHTs and metaheuristic algorithms are used for comparison. With the proposed CHT, MOEA/D shows improved convergence speed and spread of the Pareto front, providing competitive results in comparison to the other algorithms.

Boosted sooty tern optimization algorithm for global optimization and feature selection

Feature selection (FS) represents an optimization problem that aims to simplify and improve the quality of highly dimensional datasets through selecting prominent features and eliminating redundant and irrelevant data to classify results better. The goals of FS comprise dimensionality reduction and enhancing the classification accuracy in general, accompanied by great significance in different fields like data mining applications, pattern classification, and data analysis. Using powerful optimization algorithms is crucial to obtaining the best subsets of information in FS. Different metaheuristics, such as the Sooty Tern Optimization Algorithm (STOA), help to optimize the FS problem. However, such kind of techniques tends to converge in sub-optimal solutions. To overcome this problem in the STOA, an improved version called mSTOA is introduced. It employs the balancing exploration/exploitation strategy, self-adaptive of the control parameters strategy, and population reduction strategy. The proposed approach is proposed for solving the FS problem, but also it has been validated over benchmark optimization problems from the CEC 2020. To assess the performance of the mSTOA, it has also been tested with different algorithms. The experiments in terms of FS provide qualitative and quantitative evidence of the capabilities of the mSTOA for extracting the optimal subset of features. Besides, statistical analyses and no-parametric tests were also conducted to validate the result obtained by the mSTOA in optimization.

A multi-objective multi-tasking evolutionary algorithm based inverse mapping and adaptive transformation strategy: IM-MFEA

Multi-tasking optimization algorithm attracts much attention because the knowledge transfer between tasks enables the algorithm to process multiple related tasks simultaneously. However, negative knowledge transfer occasionally occurs, which may weaken the performance of the algorithm. To reduce the impact of negative knowledge transfer, a multi-objective multi-tasking optimization algorithm (IM-MFEA) based on inverse model mapping and an objective transformation strategy is proposed. First, correlation analysis is applied in an inverse mapping strategy to improve the accuracy of the inverse mapping model. Then, following the pattern of using the source domain solutions to assist the optimization of the target domain, the adaptive transformation strategy is used to improve the quality of the source domain solution in the objective space. These transformed solutions are reconstructed through the inverse mapping strategy. Finally, these reconstructed source domain solutions are mated with the target domain solutions to generate competitive offspring individuals for the target domain. To verify the effectiveness of the IM-MFEA, comprehensive experiments were conducted on nine multi-objective multi-factorial optimization (MFO) benchmark problems. Empirical results demonstrate that IM-MFEA is superior to other algorithms in 90% of test instances by inverted generational distance (IGD) and hypervolume (HV) value indicators.

Interactive knowledge discovery and knowledge visualization for decision support in multi-objective optimization

In many practical applications, the end-goal of multi-objective optimization is to select an implementable solution that is close to the Pareto-optimal front while satisfying the decision maker’s preferences. The decision making process is challenging since it involves the manual consideration of all solutions. The field of multi-criteria decision making offers many methods that help the decision maker in this process. However, most methods only focus on analyzing the solutions’ objective values. A more informed decision generally requires the additional knowledge of how different preferences affect the variable values. One difficulty in realizing this is that while the preferences are often expressed in the objective space, the knowledge required to implement a preferred solution exists in the decision space. In this paper, we propose a decision support system that allows interactive knowledge discovery and knowledge visualization to support practitioners by simultaneously considering preferences in the objective space and their impact in the decision space. The knowledge discovery step can use either of two recently proposed data mining techniques for extracting decision rules that conform to given preferences, while the extracted knowledge is visualized via a novel graph-based approach that allows the discovery of important variables, their values and their interactions with other variables. The result is an intuitive and interactive decision support system that aids the entire decision making process — from solution visualization to knowledge visualization. We demonstrate the usefulness of this system on benchmark optimization problems up to 10 objectives and real-world problems with up to six objectives.

The novel combination lock algorithm for improving the performance of metaheuristic optimizers

Nature-inspired population-based metaheuristics are promising search methods for solving optimization problems. In this paper, a novel systematic pre-processing approach, called the combination lock algorithm, for obtaining a good starting point for population-based algorithms is proposed. The proposed algorithm is tested on 32 benchmark unconstrained multidimensional optimization problems of different characteristics that are either unimodal or multimodal, continuous or non-continuous, separable or non-separable, differentiable or non-differentiable. The tests also include four engineering constrained optimization benchmark problems. The experimental results of applying the proposed algorithm for the Particle Swarm Optimization, the Ant Colony Optimization for Continuous Domain, and the Grey Wolf Optimization were compared with the results obtained from the conventional approach of initializing the starting population of population-based metaheuristic methods. The simulation results show the potential of the proposed algorithm as an efficient and reliable approach to enhance the performance of population-based optimization algorithms such that, overall, 50% and up to 100% of the tested problems “across various population size” settings, had either improved or equalled the optimal values when the proposed algorithm was applied.

Large-scale multiobjective optimization with adaptive competitive swarm optimizer and inverse modeling

Competitive swarm optimizer (CSO) is an efficient algorithm for solving larger-scale multiobjective optimization problems (LSMOPs). However, the winners are not updated in original algorithm, and the random parameters model affects the convergence speed. To improve the performance of CSO for solving LSMOPs, an adaptive competitive swarm optimizer with inverse modeling is developed. In the method, an adaptive parameter model is designed to accelerate the convergence speed when the population not traps in local convergence, and the random parameter model is used to help the losers jump out of local convergence when the population traps in stagnation. Moreover, inverse modeling is used to update the winners, and the number of them is maintained by a new designed environment selection method. The exploitation ability of algorithm is improved and the drawback that convergence speed of the algorithm might be slowed by few winners in the early stage of evolution is avoided. In addition, an adjacent individual competition method is proposed to improve the distribution of solutions. Finally, the proposed algorithm is tested on nine benchmark optimization problems, and the results indicate that the average ranks in terms of IGD and HV of the proposed algorithm outperform those of seven compared algorithms.

Bayesian Optimization for Expensive Smooth-Varying Functions

Bayesian optimization (BO) is a powerful class of data-driven techniques for the maximization of expensive-to-evaluate objective functions. These techniques construct a Gaussian process (GP) regression for representing the objective function according to the latest available function evaluations and sequentially select samples and evaluate the function by maximizing an acquisition function. The primary assumption in most BO policies is that the objective function has a uniform level of smoothness over the input space, modeled by a kernel function. However, the uniform smoothness assumption is likely to be violated in a wide range of practical problems, primary domains in which the objective function is evaluated differently at various regions of input space (e.g., through different experiments, software, or approximators). This article develops a BO framework capable of optimizing expensive smooth-varying functions. Unlike the existing techniques that rely on a single GP model, the proposed framework constructs a set of local and global GP models to represent the objective function. The predictive mean and variance at any given sample in the input space are computed according to the posterior probabilities of the local and global GP models. Local and global models are adaptively controlled through a single parameter, which can be optimized along with other GP models’ parameters during the optimization process. Using the predicted local and global values, the expected improvement acquisition function is employed as one of the possible acquisition functions for the selection process. The performance of the proposed framework is assessed extensively through two optimization benchmark problems.

Adaptive hybrid optimization method for valley functions in weight minimization problems for wind turbine blades

The article proposes an adaptive method for finding the minimum of an arbitrary smooth multivariable function. The method has been used to solve the benchmark optimization problem of a valley function. 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. The comparison of the results of calculating the minimum point of the function is performed by using both non-gradient and gradient methods, namely: Powell, Hook-Jeeves, the steepest descent method and the method developed. It has been found that the effectiveness of the proposed method is greater than the usual search algorithms, but it is not without its drawbacks. The method that represents a number of hybrid methods, which form a hybrid coalition is proposed. The 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 has been used to solve the problem of optimizing the blade of a wind turbine, which was reduced to the problem of unconditional optimization by using the method of penalty functions, but the goal function had a significantly valley structure. The optimal values of section thicknesses have been obtained, which makes it possible to build a blade with improved characteristics.

Cooperative co-evolutionary algorithm for multi-objective optimization problems with changing decision variables

Multi-objective optimization problems (MOPs) with changing decision variables exist in the actual industrial production and daily life, which have changing Pareto sets and complex relations among decision variables and are difficult to solve. In this study, we present a cooperative co-evolutionary algorithm by dynamically grouping decision variables to effectively tackle MOPs with changing decision variables. In the presented algorithm, decision variables are grouped into a series of groups using maximum entropic epistasis (MEE) at first, with decision variables in different groups owning a weak dependency. Subsequently, a sub-population is generated to solve decision variables in each group with an existing multi-objective evolutionary algorithm (MOEA). Further, a complete solution including all the decision variables is achieved through the cooperation among sub-populations. Finally, when a decision variable is added or deleted from the existing problem, the grouping of decision variables is dynamically adjusted based on the correlation between the changed decision variable and existing groups. To verify the performance of the developed method, the presented method is compared with five popular methods by tackling eight benchmark optimization problems. The experimental results reveal that the presented method is superior in terms of diversity, convergence, and spread of solutions on most benchmark optimization problems.

Stress Optimization of Vent Holes with Different Shapes Using Efficient Switching Delayed PSO Algorithm

An effective integrated design optimization method is developed to reduce the maximum von Mises stress around vent holes of a high-pressure turbine sealing disk. It mainly includes four different shape designs (circular, elliptical, race-track, and four-arc) for holes, an updated self-developed modelling and meshing tool, an APDL-based strength analysis, and a self-proposed efficient switching delayed particle swarm optimization (SDPSO) algorithm. The main idea of SDPSO is: (1) by evaluating an evolutionary factor and utilizing a probability transition matrix, a non-homogeneous Markov chain is determined and auto-updated in each generation; (2) the evolutionary factor and the Markov chain are used to adaptively select the inertia weight, acceleration coefficients, and delayed information to adjust the particle’s velocity. The performance of SDPSO is evaluated through two benchmark optimization problems with constraints. The results show that SDPSO is superior to two well-known PSO algorithms in optimization capability, numerical robustness, and convergence speed. Furthermore, SDPSO is used for the stress optimization of vent holes with four different shapes. The results show that: (1) SDPSO is suitable and valuable for practical engineering optimization problems with constraints; (2) the developed integrated design optimization method is effective and advanced for reducing the maximum von Mises stress around the vent holes; and (3) the four-arc hole has more tremendous advantages in reducing the maximum von Mises stress, followed by the elliptical hole, the race-track hole, and the circular hole.

A diversity preservation method for expensive multi-objective combinatorial optimization problems using Novel-First Tabu Search and MOEA/D

Expensive multi-objective combinatorial optimization problems have constraints in the number of objective function evaluations due to time, financial, or resource restrictions. As most combinatorial problems, they are subject to a high number of duplicated solutions. Given the fact that expensive environments limit the number of objective function evaluations, the existence of duplicated solutions heavily impacts the optimization process due to poor diversity and low convergence speed. This paper proposes the Novel-First Tabu Search, a greedy-strategy mechanism that uses Knowledge-Assisted Local Search methods to preserve the population diversity and increase the exploration and exploitation ability of MOEA/D. Experiments are conducted on constrained, unconstrained, multimodal, deceptive, linear, convex, and non-convex Pareto Front multi-objective combinatorial optimization benchmark problems. This paper also conducts an experiment on the real-world, expensive problem of Well Placement Optimization using a benchmark case based on the Namorado oil field, located in the Campos Basin, Brazil. The experimental results and performance comparison with state-of-the-art algorithms demonstrate that the proposed design significantly preserves diversity and increases convergence without violating the constraint in the number of objective function evaluations.