Algorithm For Multi-objective Optimization Problems Research Articles

The Weighted p-Norm Weight Set Decomposition for Multiobjective Discrete Optimization Problems

Many solution algorithms for multiobjective optimization problems are based on scalarization methods that transform the multiobjective problem into a scalar-valued optimization problem. In this article, we study the theory of weighted p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-norm scalarizations. These methods minimize the distance induced by a weighted p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-norm between the image of a feasible solution and a given reference point. We provide a comprehensive theory of the set of eligible weights and, in particular, analyze the topological structure of the normalized weight set. This set is composed of connected subsets, called weight set components which are in a one-to-one relation with the set of optimal images of the corresponding weighted p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-norm scalarization. Our work generalizes and complements existing results for the weighted sum and the weighted Tchebycheff scalarization and provides new insights into the structure of the set of all Pareto optimal solutions.

A dual-population-based evolutionary algorithm for multi-objective optimization problems with irregular Pareto fronts

When solving multi-objective optimization problems (MOPs) with irregular Pareto fronts (e.g., disconnected, degenerated, inverted) via evolutionary algorithms, a critical issue is how to obtain a set of well-distributed Pareto optimal solutions. To remedy this issue, we propose a dual-population-based evolutionary algorithm with individual exploitation and weight vector adaptation, named DPEA-IEAW. Specifically, the two populations, termed globPop and locPop, individually evolve by decomposition-based and Pareto dominance-based methods, responsible for global evolution and local evolution, respectively. These two populations collaborate through substantial information exchange, thereby facilitating each other’s evolution. Firstly, the distribution of the two populations is analyzed and an individual exploitation operation is designed for locPop to exploit some promising areas that are undeveloped in globPop. Then, the guide-position is devised for globPop to indicate the optimal point for a subproblem on the Pareto front (PF). By using the guide-position, a strategy for generating uniform weight vectors is proposed to improve the population diversity. Finally, comprehensive experiments on 37 widely used test functions and 2 real-world problems demonstrate that the proposed DPEA-IEAW outperforms comparison algorithms in solving MOPs with various PFs.

Hyperbolic Augmented Lagrangian algorithm for multiobjective optimization problems

Hyperbolic Augmented Lagrangian algorithm for multiobjective optimization problems

Exploring the solution space for adaptive curriculum sequencing: Study of a multi-objective approach

Adaptive Curriculum Sequencing (ACS) is an important issue in personalized learning. In ACS problems, one desires the best sequence of learning materials that meet the profile of a given student. To do so, multiple features of the students and the materials used are necessary to generate good solutions. In fact, understanding the students’ goals, motivation, and preferences is not an easy task and, consequently, different Internet of Things (IoT) approaches to gather this information during the learning process have been proposed. Actually, some works from the literature consider five objectives and, in this case, one has a many-objective optimization problem. Instead of solving the optimization problem considering the multiple objectives individually, the usual approach is to obtain solutions for a weighted sum of the objective values using search approaches for mono-objective optimization problems. However, this kind of approach may bias the search and limits the capacity of finding good results. Here, we solve the multi-objective ACS problem considering five objective functions. NSGA-II, a well-known Genetic Algorithm for multi-objective optimization problems, was used. In addition, the aggregation trees were employed to reduce the number of objectives to two and three due to the large number of objectives in the original problem. ACS problems from the literature were used to comparatively evaluate the proposed methods and the results obtained were compared to those found by the traditional approach of summing the objective values. According to these results, the best curriculum sequences were reached when using the proposal.

Application of cell mapping to control optimization for an antenna servo system on a disturbed carrier

Abstract. The cell-mapping method, due to its global optimality, has been applied to solve multi-objective optimization problems (MOPs) and optimal control problems. However, the curse of dimensionality limits its application in high-dimensional systems. In this paper, the multi-parameter sensitivity analysis is investigated to reduce the parameter space dimension, which broadens the application of cell mapping to MOPs in high-dimensional parameter space. A post-processing algorithm for MOPs is introduced to help choose proper control parameters from the Pareto set. The proposed scheme is applied successfully in the control parameter optimization of an adaptive nonsingular terminal sliding-mode control for an antenna servo system on a disturbed carrier. Moreover, as the existing global optimal tracking control with an adjoining cell-mapping method may generate tracking-phase differences, an optimal-sliding-mode combined-control strategy is proposed. By using the combined-control strategy, the azimuth and pitch angles of the antenna system are controlled to catch up to a target trajectory with the minimum cost function and to keep high-precision tracking after that.

Multi-objective boxing match algorithm for multi-objective optimization problems

In the last two decades, due to having fast computation after inventing computers and also considering real-world optimization problems, research on developing new algorithms for problem having more than one objective have been one of the appealing and attractive topics both for academia and industrial practitioners. By this motivation, we introduce a Multi-Objective Boxing Match Algorithm (MOBMA) in this paper. The proposed algorithm studies the multi-objective version of the Boxing Match Algorithm (BMA) by incorporating a unique search strategy and new solutions-producing mechanism, enhancing the algorithm’s capability for exploration and exploitation phases. Besides, its performance is analyzed with famous and capable multi-objective metaheuristics. We consider ten multi-objective benchmarks and three classical engineering problems. Statistical analyses are also conducted on the benchmark test functions from three engineering design problems. This study shows the superior performance of the proposed algorithm, considering both quantitative and qualitative analyses.

An improved estimation of distribution algorithm for multi-objective optimization problems with mixed-variable

An improved estimation of distribution algorithm for multi-objective optimization problems with mixed-variable

A two-stage diversity enhancement differential evolution algorithm for multi-objective optimization problem

In order to solve the premature convergence of multi-objective evolutionary algorithm, a two-stage diversity enhancement differential evolution algorithm for multi-objective optimization problem(TSDE) is proposed. The offspring with better performance needs the generation of high-quality parent generation. In this paper, an improved cell density method is used to screen for the high quality parents by estimating the global distribution of the objective space. Moreover, Principal Component Analysis operator is introduced to the external archive to perturb the non-dominated solution, which not only ensures the convergence but also improves the diversity. In order to verify the effectiveness of the algorithm, TSDE and other advanced methods are run on 19 test functions. The results show that TSDE performs better than other algorithms.

A multi-resolution grid-based bacterial foraging optimization algorithm for multi-objective optimization problems

In recent years, bacterial foraging optimization (BFO) has been used to solve multiobjective optimization problems (MOPs). However, BFO has not fully developed its potentials on MOPs for the reason of lacking of in-depth research on the optimization mechanisms and the diversity maintenance strategies. To solve it, this paper develops a multi-resolution grid-based BFO algorithm (called as MRBFO). MRBFO redesigns four tailored optimization mechanisms for MOPs including chemotaxis, conjugation, reproduction, and elimination and dispersal to search optimal nondominated solutions. Moreover, MRBFO defines a multi-resolution grid strategy to produce well-distributed diverse nondominated solutions. The performance of MRBFO is comprehensively evaluated by comparing it with several state-of-the-art algorithms on many benchmark test problems. The empirical results have sufficiently verified the advantages of MRBFO.

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.

Modified teaching-learning-based optimization algorithm for multi-objective optimization problems

When solving multi-objective optimization problems, an important issue is how to promote convergence and distribution of solution set simultaneously. To address the above issue, a novel optimization algorithm, named as multi-objective modified teaching-learning-based optimization (MOMTLBO), is proposed. Firstly, a grouping teaching strategy based on pareto dominance relationship is proposed to strengthen the convergence efficiency. Afterward, a diversified learning strategy is presented to enhance the distribution. Meanwhile, differential operations are incorporated to the proposed algorithm. By the above process, the search ability of the algorithm can be encouraged. Additionally, a set of well-known benchmark test functions including ten complex problems proposed for CEC2009 is used to verify the performance of the proposed algorithm. The results show that MOMTLBO exhibits competitive performance against other comparison algorithms. Finally, the proposed algorithm is applied to the aerodynamic optimization of airfoils.

Enhanced Jaya Algorithm for Multi-objective Optimisation Problems

Evolutionary algorithms are suitable techniques for solving complex problems. Many improvements have been made on the original structure algorithm in order to obtain more desirable solutions. The current study intends to enhance multi-objective performance with benchmark optimisation problems by incorporating the chaotic inertia weight into the current multi-objective Jaya (MOJaya) algorithm. Essentially, Jaya is a recently established population-oriented algorithm. Exploitation proves to be more dominant in MOJaya following its propensity to capture local optima. This research addressed the aforementioned shortcoming by refining the MOJaya algorithm solution to update the equation for exploration-exploitation balance, enhancing divergence, and deterring premature convergence to retain the algorithm fundamentals while simultaneously sustaining its parameter-free component. The recommended chaotic inertia weight-multi-objective Jaya (MOiJaya) algorithm was assessed using well-known ZDT benchmark functions with 30 variables, whereas the convergence matrix (CM) and divergence matrix (DM) analysed the suggested MOiJaya algorithm performances are inspected. As such, this algorithm enhanced the exploration-exploitation balance and substantially prevented premature convergence. Then, the proposed algorithm is compared with a few other algorithms. Based on the comparison, the convergence metric and diversity metric results show that the recommended MOiJaya algorithm potentially resolved multi-objective optimisation problems better than the other algorithms.

An approximation algorithm for multi-objective optimization problems using a box-coverage

For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the nondominated set. However, often these approximations are used for further evaluations. For those applications a polyhedron is a structure that is not easy to handle. In this paper, we introduce an approximation with a simpler structure respecting the natural ordering. In particular, we compute a box-coverage of the nondominated set. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. The algorithm is guaranteed to stop with a finite number of boxes, each being sufficiently thin.

A multi-objective elitist feedback teaching–learning-based optimization algorithm and its application

Students generally attempt to improve themselves with different manners in their spare time. Inspired by this fact, this paper presents a multi-objective elitist feedback teaching–learning-based optimization (MEFTO) algorithm for multi-objective optimization problems. To promote the exploration capacity and convergence of the basic teaching–learning-based optimization (TLBO), a feedback phase that simulates the spare-time learning phenomenon is introduced at the end of the learner phase. Students should compare their score with the average class score to select an appropriate means for further improvement. Poorly performing students can learn from the teacher directly for rapid improvement, whereas high-performing students prefer to motivate themselves for reinforcement learning. Non-dominated sorting is incorporated to permit this heuristic to solve problems with several objectives. Crowding distance calculation is adopted to maintain the diversity of the obtained solution set in a single run. Elitism strategy is also employed to provide a great improvement for the algorithm. The performance of the MEFTO algorithm is compared with seven other well-known algorithms by using a set of unconstrained benchmark test problems and three constrained engineering optimization problems. The qualitative and quantitative results indicate that the proposed algorithm can provide considerably competitive results and outperforms the other algorithms.

Approximation Methods for Multiobjective Optimization Problems: A Survey

Algorithms for approximating the nondominated set of multiobjective optimization problems are reviewed. The approaches are categorized into general methods that are applicable under mild assumptions and, thus, to a wide range of problems, and into algorithms that are specifically tailored to structured problems. All in all, this survey covers 52 articles published within the last 41 years, that is, between 1979 and 2020. Summary of Contribution: In many problems in operations research, several conflicting objective functions have to be optimized simultaneously, and one is interested in finding Pareto optimal solutions. Because of the high complexity of finding Pareto optimal solutions and their usually very large number, however, the exact solution of such multiobjective problems is often very difficult, which motivates the study of approximation algorithms for multiobjective optimization problems. This research area uses techniques and methods from algorithmics and computing in order to efficiently determine approximate solutions to many well-known multiobjective problems from operations research. Even though approximation algorithms for multiobjective optimization problems have been investigated for more than 40 years and more than 50 research articles have been published on this topic, this paper provides the first survey of this important area at the intersection of computing and operations research.

A Survey of Evolutionary Algorithms for Multi-Objective Optimization Problems With Irregular Pareto Fronts

Evolutionary algorithms have been shown to be very successful in solving multi-objective optimization problems (MOPs). However, their performance often deteriorates when solving MOPs with irregular Pareto fronts. To remedy this issue, a large body of research has been performed in recent years and many new algorithms have been proposed. This paper provides a comprehensive survey of the research on MOPs with irregular Pareto fronts. We start with a brief introduction to the basic concepts, followed by a summary of the benchmark test problems with irregular problems, an analysis of the causes of the irregularity, and real-world optimization problems with irregular Pareto fronts. Then, a taxonomy of the existing methodologies for handling irregular problems is given and representative algorithms are reviewed with a discussion of their strengths and weaknesses. Finally, open challenges are pointed out and a few promising future directions are suggested.

Ions motion optimization algorithm for multiobjective optimization problems

This paper offers a novel multiobjective approach – Multiobjective Ions Motion Optimization (MOIMO) algorithm stimulated by the movements of ions in nature. The main inspiration behind this approach is the force of attraction and repulsion between anions and cations. A storage and leader selection strategy is combined with the single objective Ions Motion Optimization (IMO) approach to estimate the Pareto optimum front for multiobjective optimization. The proposed method is applied to 18 different benchmark test functions to confirm its efficiency in finding optimal solutions. The outcomes are compared with three novel and well-accepted techniques in the literature using five performance parameters quantitatively and obtained Pareto fronts qualitatively. The comparison proves that MOIMO can approximate Pareto optimal solutions with good convergence and coverage with minimum computational time.

SETNDS: A SET-Based Non-Dominated Sorting Algorithm for Multi-Objective Optimization Problems

Non-dominated sorting, used to find pareto solutions or assign solutions to different fronts, is a key but time-consuming process in multi-objective evolutionary algorithms (MOEAs). The best-case and worst-case time complexity of non-dominated sorting algorithms currently known are O(MNlogN) and O(MN2); M and N represent the number of objectives and the population size, respectively. In this paper, a more efficient SET-based non-dominated sorting algorithm, shorted to SETNDS, is proposed. The proposed algorithm can greatly reduce the number of comparisons on the promise of ensuring a shorter running time. In SETNDS, the rank of a solution to be sorted is determined by only comparing with the one with the highest rank degree in its dominant set. This algorithm is compared with six generally existing non-dominated sorting algorithms—fast non-dominated sorting, the arena’s principle sort, the deductive sort, the corner sort, the efficient non-dominated sort and the best order sort on several kinds of datasets. The compared results show that the proposed algorithm is feasible and effective and its computational efficiency outperforms other existing algorithms.

A New Hybrid Metaheuristic Algorithm for Multiobjective Optimization Problems

The elitist nondominated sorting genetic algorithm (NSGA-II) is hybridized with the sine-cosine algorithm (SCA) in this paper to solve multiobjective optimization problems. The proposed hybrid algorithm is named nondominated sorting sine-cosine genetic algorithm (NS-SCGA). The main idea of this algorithm is the following: NS-SCGA integrates the merits of exploitation capability of NSGA-II and exploration capability of SCA for a better search ability and speeds up the searching process. The performance of NS-SCGA is tested on the set of benchmark functions provided for CEC09. The NS-SCGA results are compared with other recently developed multiobjective algorithms in terms of convergence, spacing, and spread of the obtained nondominated solutions to the true Pareto front. The statistical analysis of the results obtained is performed by nonparametric Friedman and Wilcoxon signed-rank tests. The results prove that NS-SCGA is superior to or competitive with other multiobjective optimization algorithms considered in the comparison. Furthermore, the economic emission dispatch problem (EEDP) is solved by NS-SCGA. The operating cost (fuel cost) and pollutant emission of the standard IEEE 30-bus network with six generating units are minimized simultaneously by the NS-SCGA considering the losses. The results show the superiority of NS-SCGA and confirm its ability in solving EEDP. Finally, TOPSIS technique is applied to choose the best compromise solution from the obtained Pareto-optimal solutions of EEDP according to the decision-maker’s preference.

An Improved Bare Bone Multi-Objective Particle Swarm Optimization Algorithm for Solar Thermal Power Plants

Solar energy has many advantages, such as being abundant, clean and environmentally friendly. Solar power generation has been widely deployed worldwide as an important form of renewable energy. The solar thermal power generation is one of a few popular forms to utilize solar energy, yet its modelling is a complicated problem. In this paper, an improved bare bone multi-objective particle swarm optimization algorithm (IBBMOPSO) is proposed based on the bare bone multi-objective particle swarm optimization algorithm (BBMOPSO). The algorithm is first tested on a set of benchmark problems, confirming its efficacy and the convergency speed. Then, it is applied to optimize two typical solar power generation systems including the solar Stirling power generation and the solar Brayton power generation; the results show that the proposed algorithm outperforms other algorithms for multi-objective optimization problems.