Showing papers on "Multi-objective optimization published in 2019"

IGD Indicator-Based Evolutionary Algorithm for Many-Objective Optimization Problems

Abstract: Inverted generational distance (IGD) has been widely considered as a reliable performance indicator to concurrently quantify the convergence and diversity of multiobjective and many-objective evolutionary algorithms. In this paper, an IGD indicator-based evolutionary algorithm for solving many-objective optimization problems (MaOPs) has been proposed. Specifically, the IGD indicator is employed in each generation to select the solutions with favorable convergence and diversity. In addition, a computationally efficient dominance comparison method is designed to assign the rank values of solutions along with three newly proposed proximity distance assignments. Based on these two designs, the solutions are selected from a global view by linear assignment mechanism to concern the convergence and diversity simultaneously. In order to facilitate the accuracy of the sampled reference points for the calculation of IGD indicator, we also propose an efficient decomposition-based nadir point estimation method for constructing the Utopian Pareto front (PF) which is regarded as the best approximate PF for real-world MaOPs at the early stage of the evolution. To evaluate the performance, a series of experiments is performed on the proposed algorithm against a group of selected state-of-the-art many-objective optimization algorithms over optimization problems with 8-, 15-, and 20-objective. Experimental results measured by the chosen performance metrics indicate that the proposed algorithm is very competitive in addressing MaOPs.

Two-Archive Evolutionary Algorithm for Constrained Multiobjective Optimization

Abstract: When solving constrained multiobjective optimization problems, an important issue is how to balance convergence, diversity, and feasibility simultaneously. To address this issue, this paper proposes a parameter-free constraint handling technique, a two-archive evolutionary algorithm, for constrained multiobjective optimization. It maintains two collaborative archives simultaneously: one, denoted as the convergence-oriented archive (CA), is the driving force to push the population toward the Pareto front; the other one, denoted as the diversity-oriented archive (DA), mainly tends to maintain the population diversity. In particular, to complement the behavior of the CA and provide as much diversified information as possible, the DA aims at exploring areas under-exploited by the CA including the infeasible regions. To leverage the complementary effects of both archives, we develop a restricted mating selection mechanism that adaptively chooses appropriate mating parents from them according to their evolution status. Comprehensive experiments on a series of benchmark problems and a real-world case study fully demonstrate the competitiveness of our proposed algorithm, in comparison to five state-of-the-art constrained evolutionary multiobjective optimizers.

A survey on handling computationally expensive multiobjective optimization problems with evolutionary algorithms

Abstract: Evolutionary algorithms are widely used for solving multiobjective optimization problems but are often criticized because of a large number of function evaluations needed. Approximations, especially function approximations, also referred to as surrogates or metamodels are commonly used in the literature to reduce the computation time. This paper presents a survey of 45 different recent algorithms proposed in the literature between 2008 and 2016 to handle computationally expensive multiobjective optimization problems. Several algorithms are discussed based on what kind of an approximation such as problem, function or fitness approximation they use. Most emphasis is given to function approximation-based algorithms. We also compare these algorithms based on different criteria such as metamodeling technique and evolutionary algorithm used, type and dimensions of the problem solved, handling constraints, training time and the type of evolution control. Furthermore, we identify and discuss some promising elements and major issues among algorithms in the literature related to using an approximation and numerical settings used. In addition, we discuss selecting an algorithm to solve a given computationally expensive multiobjective optimization problem based on the dimensions in both objective and decision spaces and the computation budget available.

Coevolutionary Particle Swarm Optimization With Bottleneck Objective Learning Strategy for Many-Objective Optimization

Abstract: The application of multiobjective evolutionary algorithms to many-objective optimization problems often faces challenges in terms of diversity and convergence. On the one hand, with a limited population size, it is difficult for an algorithm to cover different parts of the whole Pareto front (PF) in a large objective space. The algorithm tends to concentrate only on limited areas. On the other hand, as the number of objectives increases, solutions easily have poor values on some objectives, which can be regarded as poor bottleneck objectives that restrict solutions’ convergence to the PF. Thus, we propose a coevolutionary particle swarm optimization with a bottleneck objective learning (BOL) strategy for many-objective optimization. In the proposed algorithm, multiple swarms coevolve in distributed fashion to maintain diversity for approximating different parts of the whole PF, and a novel BOL strategy is developed to improve convergence on all objectives. In addition, we develop a solution reproduction procedure with both an elitist learning strategy (ELS) and a juncture learning strategy (JLS) to improve the quality of archived solutions. The ELS helps the algorithm to jump out of local PFs, and the JLS helps to reach out to the missing areas of the PF that are easily missed by the swarms. The performance of the proposed algorithm is evaluated using two widely used test suites with different numbers of objectives. Experimental results show that the proposed algorithm compares favorably with six other state-of-the-art algorithms on many-objective optimization.

Multiobjective Cloud Workflow Scheduling: A Multiple Populations Ant Colony System Approach

Abstract: Cloud workflow scheduling is significantly challenging due to not only the large scale of workflow but also the elasticity and heterogeneity of cloud resources. Moreover, the pricing model of clouds makes the execution time and execution cost two critical issues in the scheduling. This paper models the cloud workflow scheduling as a multiobjective optimization problem that optimizes both execution time and execution cost. A novel multiobjective ant colony system based on a co-evolutionary multiple populations for multiple objectives framework is proposed, which adopts two colonies to deal with these two objectives, respectively. Moreover, the proposed approach incorporates with the following three novel designs to efficiently deal with the multiobjective challenges: 1) a new pheromone update rule based on a set of nondominated solutions from a global archive to guide each colony to search its optimization objective sufficiently; 2) a complementary heuristic strategy to avoid a colony only focusing on its corresponding single optimization objective, cooperating with the pheromone update rule to balance the search of both objectives; and 3) an elite study strategy to improve the solution quality of the global archive to help further approach the global Pareto front. Experimental simulations are conducted on five types of real-world scientific workflows and consider the properties of Amazon EC2 cloud platform. The experimental results show that the proposed algorithm performs better than both some state-of-the-art multiobjective optimization approaches and the constrained optimization approaches.

Push and pull search for solving constrained multi-objective optimization problems

Abstract: This paper proposes a push and pull search (PPS) framework for solving constrained multi-objective optimization problems (CMOPs). To be more specific, the proposed PPS divides the search process into two different stages: push and pull search stages. In the push stage, a multi-objective evolutionary algorithm (MOEA) is used to explore the search space without considering any constraints, which can help to get across infeasible regions very quickly and to approach the unconstrained Pareto front. Furthermore, the landscape of CMOPs with constraints can be probed and estimated in the push stage, which can be utilized to conduct the parameter setting for the constraint-handling approaches to be applied in the pull stage. Then, a modified form of a constrained multi-objective evolutionary algorithm (CMOEA), with improved epsilon constraint-handling, is applied to pull the infeasible individuals achieved in the push stage to the feasible and non-dominated regions. To evaluate the performance regarding convergence and diversity, a set of benchmark CMOPs and a real-world optimization problem are used to test the proposed PPS (PPS-MOEA/D) and state-of-the-art CMOEAs, including MOEA/D-IEpsilon, MOEA/D-Epsilon, MOEA/D-CDP, MOEA/D-SR, C-MOEA/D and NSGA-II-CDP. The comprehensive experimental results show that the proposed PPS-MOEA/D achieves significantly better performance than the other six CMOEAs on most of the tested problems, which indicates the superiority of the proposed PPS method for solving CMOPs.

Modeling and Planning for Dual-Objective Selective Disassembly Using and / or Graph and Discrete Artificial Bee Colony

Abstract: Disassembly sequencing is important for remanufacturing and recycling used or discarded products. AND / OR graphs (AOGs) have been applied to describe practical disassembly problems by using “ AND ” and “ OR ” nodes. An AOG-based disassembly sequence planning problem is an NP-hard combinatorial optimization problem. Heuristic evolution methods can be adopted to handle it. While precedence and “ AND ” relationship issues can be addressed, OR (exclusive OR ) relations are not well addressed by the existing heuristic methods. Thus, an ineffective result may be obtained in practice. A conflict matrix is introduced to cope with the exclusive OR relation in an AOG graph. By using it together with precedence and succession matrices in the existing work, this work proposes an effective triple-phase adjustment method to produce feasible disassembly sequences based on an AOG graph. Energy consumption is adopted to evaluate the disassembly efficiency. Its use with the traditional economical criterion leads to a novel dual-objective optimization model such that disassembly profit is maximized and disassembly energy consumption is minimized. An improved artificial bee colony algorithm is developed to effectively generate a set of Pareto solutions for this dual-objective disassembly optimization problem. This methodology is employed to practical disassembly processes of two products to verify its feasibility and effectiveness. The results show that it is capable of rapidly generating satisfactory Pareto results and outperforms a well-known genetic algorithm.

Handling Constrained Multiobjective Optimization Problems With Constraints in Both the Decision and Objective Spaces

Abstract: Constrained multiobjective optimization problems (CMOPs) are frequently encountered in real-world applications, which usually involve constraints in both the decision and objective spaces. However, current artificial CMOPs never consider constraints in the decision space (i.e., decision constraints) and constraints in the objective space (i.e., objective constraints) at the same time. As a result, they have a limited capability to simulate practical scenes. To remedy this issue, a set of CMOPs, named DOC, is constructed in this paper. It is the first attempt to consider both the decision and objective constraints simultaneously in the design of artificial CMOPs. Specifically, in DOC, various decision constraints (e.g., inequality constraints, equality constraints, linear constraints, and nonlinear constraints) are collected from real-world applications, thus making the feasible region in the decision space have different properties (e.g., nonlinear, extremely small, and multimodal). On the other hand, some simple and controllable objective constraints are devised to reduce the feasible region in the objective space and to make the Pareto front have diverse characteristics (e.g., continuous, discrete, mixed, and degenerate). As a whole, DOC poses a great challenge for a constrained multiobjective evolutionary algorithm (CMOEA) to obtain a set of well-distributed and well-converged feasible solutions. In order to enhance current CMOEAs’ performance on DOC, a simple and efficient two-phase framework, named ToP, is proposed in this paper. In ToP, the first phase is implemented to find the promising feasible area by transforming a CMOP into a constrained single-objective optimization problem. Then in the second phase, a specific CMOEA is executed to obtain the final solutions. ToP is applied to four state-of-the-art CMOEAs, and the experimental results suggest that it is quite effective.

Accelerating Large-Scale Multiobjective Optimization via Problem Reformulation

Abstract: In this paper, we propose a framework to accelerate the computational efficiency of evolutionary algorithms on large-scale multiobjective optimization. The main idea is to track the Pareto optimal set (PS) directly via problem reformulation. To begin with, the algorithm obtains a set of reference directions in the decision space and associates them with a set of weight variables for locating the PS. Afterwards, the original large-scale multiobjective optimization problem is reformulated into a low-dimensional single-objective optimization problem. In the reformulated problem, the decision space is reconstructed by the weight variables and the objective space is reduced by an indicator function. Thanks to the low dimensionality of the weight variables and reduced objective space, a set of quasi-optimal solutions can be obtained efficiently. Finally, a multiobjective evolutionary algorithm is used to spread the quasi-optimal solutions over the approximate Pareto optimal front evenly. Experiments have been conducted on a variety of large-scale multiobjective problems with up to 5000 decision variables. Four different types of representative algorithms are embedded into the proposed framework and compared with their original versions, respectively. Furthermore, the proposed framework has been compared with two state-of-the-art algorithms for large-scale multiobjective optimization. The experimental results have demonstrated the significant improvement benefited from the framework in terms of its performance and computational efficiency in large-scale multiobjective optimization.

Multidirectional Prediction Approach for Dynamic Multiobjective Optimization Problems

Abstract: Various real-world multiobjective optimization problems are dynamic, requiring evolutionary algorithms (EAs) to be able to rapidly track the moving Pareto front of an optimization problem once an environmental change occurs. To this end, several methods have been developed to predict the new location of the moving Pareto set (PS) so that the population can be reinitialized around the predicted location. In this paper, we present a multidirectional prediction strategy to enhance the performance of EAs in solving a dynamic multiobjective optimization problem (DMOP). To more accurately predict the moving location of the PS, the population is clustered into a number of representative groups by a proposed classification strategy, where the number of clusters is adapted according to the intensity of the environmental change. To examine the performance of the developed algorithm, the proposed prediction strategy is compared with four state-of-the-art prediction methods under the framework of particle swarm optimization as well as five popular EAs for dynamic multiobjective optimization. Our experimental results demonstrate that the proposed algorithm can effectively tackle DMOPs.

Multi-objective optimization for energy-efficient flexible job shop scheduling problem with transportation constraints

Abstract: Manufacturing enterprises nowadays face huge environmental challenges because of energy consumption and associated environmental impacts. One of the effective strategies to reduce energy consumption is by employing intelligent scheduling techniques. Production scheduling can have significant impact on energy saving in manufacturing system from the operation management point of view. Resource flexibility and complex constraints in flexible manufacturing system make production scheduling a complicated nonlinear programming problem. To this end, a multi-objective optimization model with the objective of minimizing energy consumption and makespan is formulated for a flexible job shop scheduling problem with transportation constraints. Then, an enhanced genetic algorithm is developed to solve the problem. Finally, comprehensive experiments are carried out to evaluate the performance of the proposed model and algorithm. The experimental results revealed that the proposed model and algorithm can solve the problem effectively and efficiently. This may provide a basis for the decision makers to consider energy-efficient scheduling in flexible manufacturing system.

Evolutionary Constrained Multiobjective Optimization: Test Suite Construction and Performance Comparisons

Abstract: For solving constrained multiobjective optimization problems (CMOPs), many algorithms have been proposed in the evolutionary computation research community for the past two decades. Generally, the effectiveness of an algorithm for CMOPs is evaluated by artificial test problems. However, after a brief review of current artificial test problems, we have found that they are not well-designed and fail to reflect the characteristics of real-world applications (e.g., small feasibility ratio). Thus, in this paper, we first propose a new constraint construction method to facilitate the systematic design of test problems. Then, on the basis of this method, we design a new test suite consisting of 14 instances, which covers diverse characteristics extracted from real-world CMOPs and can be divided into four types. Considering that the comprehensive performance comparisons among the constraint-handling techniques (CHTs) remain scarce, we choose several representative CHTs and compare their performance on our test suite. The performance comparisons identify the strengths and weaknesses of different CHTs on different types of CMOPs and provide guidelines on how to select/design a CHT in a specific scenario.

Pareto Multi-Task Learning

Abstract: Multi-task learning is a powerful method for solving multiple correlated tasks simultaneously. However, it is often impossible to find one single solution to optimize all the tasks, since different tasks might conflict with each other. Recently, a novel method is proposed to find one single Pareto optimal solution with good trade-off among different tasks by casting multi-task learning as multiobjective optimization. In this paper, we generalize this idea and propose a novel Pareto multi-task learning algorithm (Pareto MTL) to find a set of well-distributed Pareto solutions which can represent different trade-offs among different tasks. The proposed algorithm first formulates a multi-task learning problem as a multiobjective optimization problem, and then decomposes the multiobjective optimization problem into a set of constrained subproblems with different trade-off preferences. By solving these subproblems in parallel, Pareto MTL can find a set of well-representative Pareto optimal solutions with different trade-off among all tasks. Practitioners can easily select their preferred solution from these Pareto solutions, or use different trade-off solutions for different situations. Experimental results confirm that the proposed algorithm can generate well-representative solutions and outperform some state-of-the-art algorithms on many multi-task learning applications.

Comparison between MOEA/D and NSGA-III on a set of novel many and multi-objective benchmark problems with challenging difficulties

Abstract: Currently, evolutionary multiobjective optimization (EMO) algorithms have been successfully used to find a good approximation of many-objective optimization problems (MaOPs). To measure the performance of EMO algorithms, many benchmark multiobjective test problems have been constructed. Among them, DTLZ and WFG are two representative test suites with the scalability to the number of variables and objectives. It should be pointed out that MaOPs can be more challenging if they are involved with difficult problem features, such as objective scalability, complicated Pareto set, bias, disconnection, or degeneracy. In this paper, a set of ten new test problems with above-mentioned difficulties are constructed. Some experimental results on these test problems found by two popular EMO algorithms, i.e., MOEA/D and NSGA-III, are reported and analyzed. Moreover, the performance of these two EMO algorithms with different population sizes on these test problems are also studied.

Multimodal multiobjective optimization with differential evolution

Abstract: This paper proposes a multimodal multiobjective Differential Evolution optimization algorithm (MMODE). The technique is conceived for deployment on problems with a Pareto multimodality, where the Pareto set comprises multiple disjoint subsets, all of which map to the same Pareto front. A new contribution is the formulation of a decision-variable preselection scheme that promotes diversity of solutions in both the decision and objective space. A new mutation-bound process is also introduced as a supplement to a classical mutation scheme in Differential Evolution methods, where offspring that lie outside the search bounds are given a second opportunity to mutate, hence reducing the density of individuals on the boundaries of the search space. New multimodal multiobjective test functions are designed, along with analytical expressions for their Pareto sets and fronts. Some test functions introduce more complicated Pareto-front shapes and allow for decision-space dimensions greater than two. The performance of the MMODE algorithm is compared with five other state-of-the-art methods. The results show that MMODE realizes superior performance by finding more and better distributed Pareto solutions.

jMetalPy: A Python framework for multi-objective optimization with metaheuristics

Abstract: This paper describes jMetalPy, an object-oriented Python-based framework for multi-objective optimization with metaheuristic techniques. Building upon our experiences with the well-known jMetal framework, we have developed a new multi-objective optimization software platform aiming not only at replicating the former one in a different programming language, but also at taking advantage of the full feature set of Python, including its facilities for fast prototyping and the large amount of available libraries for data processing, data analysis, data visualization, and high-performance computing. As a result, jMetalPy provides an environment for solving multi-objective optimization problems focused not only on traditional metaheuristics, but also on techniques supporting preference articulation, constrained and dynamic problems, along with a rich set of features related to the automatic generation of statistical data from the results generated, as well as the real-time and interactive visualization of the Pareto front approximations produced by the algorithms. jMetalPy offers additionally support for parallel computing in multicore and cluster systems. We include some use cases to explore the main features of jMetalPy and to illustrate how to work with it.

Heterogeneous Ensemble-Based Infill Criterion for Evolutionary Multiobjective Optimization of Expensive Problems

Abstract: Gaussian processes (GPs) are the most popular model used in surrogate-assisted evolutionary optimization of computationally expensive problems, mainly because GPs are able to measure the uncertainty of the estimated fitness values, based on which certain infill sampling criteria can be used to guide the search and update the surrogate model. However, the computation time for constructing GPs may become excessively long when the number of training samples increases, which makes it inappropriate to use them as surrogates in evolutionary optimization. To address this issue, this paper proposes to use ensembles as surrogates and infill criteria for model management in evolutionary optimization. A heterogeneous ensemble consisting of a least square support vector machine and two radial basis function networks is constructed to enhance the reliability of ensembles for uncertainty estimation. In addition to the original decision variables, a selected subset of the decision variables and a set of transformed variables are used as inputs of the heterogeneous ensemble to further promote the diversity of the ensemble. The proposed heterogeneous ensemble is compared with a GP and a homogeneous ensemble for infill sampling criteria in evolutionary multiobjective optimization. Experimental results demonstrate that the heterogeneous ensemble is competitive in performance compared with GPs and much more scalable in computational complexity to the increase in search dimension.

A Clustering-Based Adaptive Evolutionary Algorithm for Multiobjective Optimization With Irregular Pareto Fronts

Abstract: Existing multiobjective evolutionary algorithms (MOEAs) perform well on multiobjective optimization problems (MOPs) with regular Pareto fronts in which the Pareto optimal solutions distribute continuously over the objective space. When the Pareto front is discontinuous or degenerated, most existing algorithms cannot achieve good results. To remedy this issue, a clustering-based adaptive MOEA (CA-MOEA) is proposed in this paper for solving MOPs with irregular Pareto fronts. The main idea is to adaptively generate a set of cluster centers for guiding selection at each generation to maintain diversity and accelerate convergence. We investigate the performance of CA-MOEA on 18 widely used benchmark problems. Our results demonstrate the competitiveness of CA-MOEA for multiobjective optimization, especially for problems with irregular Pareto fronts. In addition, CA-MOEA is shown to perform well on the optimization of the stretching parameters in the carbon fiber formation process.

A Collaborative Resource Allocation Strategy for Decomposition-Based Multiobjective Evolutionary Algorithms

Abstract: Decomposition of a multiobjective optimization problem (MOP) into several simple multiobjective subproblems, named multiobjective evolutionary algorithm based on decomposition (MOEA/D)-M2M, is a new version of multiobjective optimization-based decomposition. However, it fails to consider different contributions from each subproblem but treats them equally instead. This paper proposes a collaborative resource allocation (CRA) strategy for MOEA/D-M2M, named MOEA/D-CRA. It allocates computational resources dynamically to subproblems based on their contributions. In addition, an external archive is utilized to obtain the collaborative information about contributions during a search process. Experimental results indicate that MOEA/D-CRA outperforms its peers on 61% of the test cases in terms of three metrics, thereby validating the effectiveness of the proposed CRA strategy in solving MOPs.

Comprehensive Sensitivity Analysis and Multiobjective Optimization Research of Permanent Magnet Flux-Intensifying Motors

Abstract: In this paper, two permanent magnet flux-intensifying motors are designed and optimized to realize the characteristic of Ld > Lq and the flux-intensifying effect. Compared with the conventional interior permanent magnet motor with the characteristic of Ld , the advantages of a low irreversible demagnetization risk and a wide speed range can be obtained in the motors. To meet multiple design requirements effectively, a comprehensive sensitivity analysis method is first implemented to evaluate the influence of each design variable on the selected optimized objectives of output torque, reverse saliency ratio, and torque ripple. Second, a sequential nonlinear programming algorithm is used for realizing the multiobjective optimizations of strong-sensitive design variables. Then, the performances of the two optimal motors are analyzed and compared with the initial permanent magnet motor in detail. Finally, two prototype motors are manufactured and tested. Both the simulation and experimental results verify the validity of the new flux-intensifying motors and the optimization method.

Multi-objective sine-cosine algorithm (MO-SCA) for multi-objective engineering design problems

Abstract: This paper proposes a novel and an effective multi-objective optimization algorithm named multi-objective sine-cosine algorithm (MO-SCA) which is based on the search technique of sine-cosine algorithm (SCA). MO-SCA employs the elitist non-dominated sorting and crowding distance approach for obtaining different non-domination levels and to preserve the diversity among the optimal set of solutions, respectively. The effectiveness of the method is measured by implementing it on multi-objective benchmark problems that have various characteristics of Pareto front such as convex, non-convex and discrete. This proposed algorithm is also checked for the multi-objective engineering design problems with distinctive features. Furthermore, we show the proposed algorithm effectively generates the Pareto front and is easy to implement and algorithmically simple.