Distributed Pareto Optimal Solutions Research Articles

Manifold Interpolation for Large-Scale Multiobjective Optimization via Generative Adversarial Networks.

Large-scale multiobjective optimization problems (LSMOPs) are characterized as optimization problems involving hundreds or even thousands of decision variables and multiple conflicting objectives. To solve LSMOPs, some algorithms designed a variety of strategies to track Pareto-optimal solutions (POSs) by assuming that the distribution of POSs follows a low-dimensional manifold. However, traditional genetic operators for solving LSMOPs have some deficiencies in dealing with the manifold, which often results in poor diversity, local optima, and inefficient searches. In this work, a generative adversarial network (GAN)-based manifold interpolation framework is proposed to learn the manifold and generate high-quality solutions on the manifold, thereby improving the optimization performance of evolutionary algorithms. We compare the proposed approach with several state-of-the-art algorithms on various large-scale multiobjective benchmark functions. The experimental results demonstrate that significant improvements have been achieved by the proposed framework in solving LSMOPs.

HYBRID METHOD BASED ON EXPONENTIAL PENALTY FUNCTION AND MOMA-PLUS METHOD FOR MULTIOBJECTIVE OPTIMIZATION

In this paper, we propose a modified version of the MOMA-plus method to solve multiobjective optimization problems. We use an exponential penalty function instead of the Lagrangian penalty function in the initial version of MOMA-Plus in order to improve the convergence and distribution to the Pareto optimal solutions. The theoretical and numerical results show that this new version improves the quality of the obtained solutions compared to the last version. Six test problems have been successfully resolved, allowing us to highlight the good convergence and good distribution of Pareto optimal solutions.

A niching backtracking search algorithm with adaptive local search for multimodal multiobjective optimization

• Propose a niching backtracking search algorithm with adaptive local search. • Backtracking search algorithm is applied in this field for the first time. • Develop a novel mutation based on the affinity propagation clustering. • Develop an adaptive local search strategy for the proposed algorithm. Multimodal multiobjective optimization problems,which widely exist in real-world applications, have multiple Pareto optimal sets in decision space corresponding to the same Pareto front in objective space. The key to handling such problems is locating and maintaining all Pareto optimal solutions in decision space simultaneously. The distribution of Pareto optimal solutions obtained by some existing multimodal multiobjective evolutionary algorithms is still not satisfactory. This paper proposes a niching backtracking search algorithm with adaptive local search to solve such problems. In the proposed algorithm, the affinity propagation clustering method as a parameter-free automatic niching technique is adopted to form multiple niches. A novel mutation based on affinity propagation clustering is then developed to search for more Pareto solutions within each niche. In addition, an adaptive local search strategy is designed in each niche to improve the search efficiency and the accuracy of Pareto optimal solutions. The proposed algorithm is compared with seven state-of-the-art multimodal multiobjective evolutionary algorithms on a multimodal multiobjective optimization test suite with 22 benchmark functions from CEC 2019 competition and a map-based practical application problem. The experimental results show that the proposed algorithm outperforms its competitors on 14 out of 22 benchmark functions in terms of the reciprocal of Pareto sets proximity and inverted generational distance in decision space metrics. Also, the proposed method is more effective and competitive than its competitors when solving the map-based practical application problem.

Multicriteria optimization of variable thickness plates using adaptive weighted sum method

In this paper, a multicriteria design framework for variable thickness isotropic plates using the adaptive weighted sum method is developed. The design objectives are the minimization of weight and static displacement and the design variables are the elemental thicknesses of plates modelled using finite elements. Here, the multicriteria optimization framework is constructed by integrating the finite element method, analytical sensitivity technique along with optimization algorithms. The first-order shear deformation theory is used in the static and dynamic analyses of plates. Both single and multiobjective optimization studies are conducted to study the optimal thickness distributions of variable thickness plates under static and dynamic constraints. To study multicriteria optimization of plates, the weighted sum method is first applied which gives sparsely distributed Pareto optimal solutions. Then, the adaptive weighted sum method is employed where a coarser representation of Pareto optimal solutions is generated using the weighted sum method and less populated regions are identified for further refinement. The suboptimization problems are solved in these regions to determine a new set of Pareto optimal solutions. The Pareto optimal curves obtained using the adaptive weighted sum method are also compared with the conventional weighted sum method under different constraints. The effect of boundary conditions on the Pareto optimal solutions and thickness distributions of plates is also investigated.

Multiobjective optimization of the flaxseed mucilage extraction process using normal-boundary intersection approach

PurposeThe mucilage of the Linum usitatissimum L. seed (Linseed) is one of the natural mucilages that presents a great potential to provide a food hydrocolloid with potential applications in both food and pharmaceutical industries. To increase the yield and quality of linseed oil during its production process, it is necessary to previously extract its polysaccharides. Because of this, flax mucilage production can be made viable as a byproduct of oil extraction process, which is already a product of high commercial value consolidated in the market. Thus, the purpose of this work is to optimize the mucilage extraction process of L. usitatissimum L. using the normal-boundary intersection (NBI) multiobjective optimization method.Design/methodology/approachCurrently, the variables of the process of polysaccharide extraction from different sources are optimized using the response surface methodology. However, when the optimal points of the responses are conflicting it is necessary to study the best conditions to achieve a balance between these conflicting objectives (trade-offs) and to explore the available options it is necessary to formulate an optimization problem with multiple objectives. The multiobjective optimization method used in this work was the NBI developed to find uniformly distributed and continuous Pareto optimal solutions for a nonlinear multiobjective problem.FindingsThe optimum extraction point to obtain the maximum fiber concentration in the extracted material was pH 3.81, temperature of 46°C, time of 13.46 h. The maximum extraction yield of flaxseed was pH 6.45, temperature of 65°C, time of 14.41 h. This result confirms the trade-off relationship between the objectives. NBI approach was able to find uniformly distributed Pareto optimal solutions, which allows to analyze the behavior of the trade-off relationship. Thus, the decision-maker can set extraction conditions to achieve desired characteristics in mucilage.Originality/valueThe novelty of this paper is to confirm the existence of a trade-off relationship between the productivity parameter (yield) and the quality parameter (fiber concentration in the extracted material) during the flaxseed mucilage extraction process. The NBI approach was able to find uniformly distributed Pareto optimal solutions, which allows us to analyze the behavior of the trade-off relationship. This allows the decision-making to the extraction conditions according to the desired characteristics of the final product, thus being able to direct the extraction for the best applicability of the mucilage.

Solving multiobjective optimal control problems using an improved scalarization method

Abstract Detecting the Pareto optimal points on the Pareto frontier is one of the most important topics in multiobjective optimal control problems (MOCPs). This paper presents a scalarization technique to construct an approximate Pareto frontier of MOCPs, using an improved normal boundary intersection (NBI) scalarization strategy. For this purpose, MOCP is first discretized and then using a grid of weights, a sequence of single objective optimal control problems is solved to achieve a uniform distribution of Pareto optimal solutions on the Pareto frontier. The aim is to achieve a more even distribution of Pareto optimal solutions on the Pareto frontier and improve the efficiency of the algorithm. It is shown that in contrast to the NBI method, where Pareto optimality of solutions is not guaranteed, the obtained optimal solution of the scalarized problem is a Pareto optimal solution of the MOCP. Finally, the ability of the proposed method is evaluated and compared with other approaches using several practical MOCPs. The numerical results indicate that the proposed method is more efficient and provides more uniform distribution of solutions on the Pareto frontier than the other methods, such a weighted sum, normalized normal constraint and NBI.

Multi-objective optimization and improved analysis of an organic Rankine cycle coupled with the dynamic turbine efficiency model

Considering the effect of variable turbine efficiency on organic Rankine cycle (ORC) system working fluid selection and parameters determination, a multi-objective optimization model which considered the thermodynamic performance and economic factors simultaneously was coupled with a dynamic turbine efficiency model. The Pareto optimal solutions of optimization with constant turbine efficiency and with variable turbine efficiency were compared, specific quantitative analysis of turbine efficiency on an ORC system is presented and sensitivity analysis of the heat source temperature was conducted. The results reveal that the variation of net power output between constant turbine efficiency and variable turbine efficiency is different, and the distribution of Pareto optimal solutions with constant turbine efficiency and with variable turbine efficiency is also different. With constant turbine efficiency, R245ca is the optimal working fluid. However, with variable turbine efficiency, R365mfc is the optimal working fluid. In specific quantitative analysis, it can be found that the difference in the optimization results is related to the working fluid properties. For R245fa, the variation of turbine efficiency significantly affects the distribution of the Pareto optimal solutions. However, for R365mfc, the variation tendency of turbine efficiency is gentle, which only leads to a small deviation in the optimization results.

A novel multi-criteria self-organising migrating algorithm for engineering problems

Solving engineering design and resources optimisation via multi-objective evolutionary algorithms has attracted much attention in the last few years. In this study, an improved Self-Organising Migrating Algorithm (MOSOMA) is developed and investigated to solve multi-objective engineering design problems. The proposed MOSOMA algorithm uses a migration approach for the search of optima. In order to obtain a uniform distribution of Pareto optimal solutions, the crowding distance method is introduced. Pareto dominance is incorporated into the algorithm in order to allow this heuristic to handle problems with several objective functions. The performance of the MOSOMA algorithm is assessed by applying it to a set of multi-objective standard test functions and constrained engineering design problems. The results show that the proposed approach is competitive and effective compared to other algorithms contemplated in this work and it can also find the result with greater precision.

An Improved NSGA-III Algorithm for Reservoir Flood Control Operation

An improved non-dominated sorting genetic algorithm-III (ENSGA-III) is proposed to solve reservoir flood control operation (RFCO) problem. The highest upstream water level and the largest discharge of the dam are considered as two objective functions for the RFCO problem. In the proposed ENSGA-II, there are three aspects of improvements in the original NSGA-III. First, orthogonal design is adopted to generate initial population for making the population more spread and uniform in the search space. Secondly, e-dominance and constraint violation strategy is designed to find the non-dominated solution set. Thirdly, double populations are updated with three-archive strategy for producing better individuals in evolutionary process. The performance of the proposed ENSGA-III has been tested on the RFCO problem of Three Gorges Reservoir. The simulation results show that the proposed method is able to produce well distributed Pareto optimal solutions for the multi-objective reservoir flood control operation problem in term of solution quality. Compared with the results obtained by other methods, the superiority of the ENSGA-III for solving the RFCO problem is verified.

Normalised normal constraint algorithm applied to multi‐objective security‐constrained optimal generation dispatch of large‐scale power systems with wind farms and pumped‐storage hydroelectric stations

A multi-objective security-constrained optimal generation dispatch (MOSCOGD) model for large-scale power systems with wind farms and pumped-storage hydroelectric (PSH) stations is proposed. In this model, fuel consumption, emission content of atmospheric pollutants, and power purchase costs are taken into account as objective functions, and network security under basic status and N − 1 criterion conditions, and the operational limits of PSH resources are included in the constraints. The difficulties associated with discrete variables for describing the operational characteristics of PSH resources are avoided by introducing two continuous variables, generation power and pumping power, which satisfy complementary constraints. The normalised normal constraint algorithm is used to transform the three-objective optimal dispatch model into a series of single-objective optimisation models, which are solved by the interior point method, resulting in a series of evenly distributed Pareto optimal solutions (POSs) and the complete Pareto frontier surface. The computational efficiency is greatly enhanced by employing a checking–adding–checking again-adding again scheme to address network security constraints. Moreover, parallel computing is employed to improve the computation speed for solving the POSs of the MOSCOGD model. Test results on an actual large-scale power system and the modified IEEE 39-bus system demonstrate the effectiveness of the proposed method.

Multi-objective optimal power flow based on improved strength Pareto evolutionary algorithm

An improved strength Pareto evolutionary algorithm is proposed to solve the multi-objective optimal power flow problem. The fuel cost and emission are considered as two objective functions for the optimal flow problem. In the proposed algorithm, there are three aspects of improvements in the original strength Pareto evolutionary algorithm. First, the external archive population is only composed of the variable size of non-dominated individuals in environmental selection operator. Secondly, the Euclidean distance between the elite individuals and its k-th neighboring individuals is adopted to update the external archive population. Thirdly, the local search strategy is embedded into strength Pareto evolutionary algorithm. The performance of the proposed method has been tested on the IEEE 30-bus and IEEE 57-bus systems. The simulation results show that the proposed method is able to produce well distributed Pareto optimal solutions for the multi-objective optimal power flow problem. Compared with the results obtained by other methods, the superiority of the proposed method is verified.

Multi-objective particle optimization algorithm based on sharing–learning and dynamic crowding distance

A multi-objective particle swarm optimization algorithm, based on share–learning and dynamic crowding distance (MOPSO-SDCD), is proposed to improve the convergence accuracy and keep the diversity of the Pareto optimal solutions. First, the sharing–learning factor is applied to modify the velocity updating formulas, which improves both the global search ability and local search accuracy of the algorithm. Meanwhile, Gaussian mutation and greedy strategy are adopted to update personal best position and external archive, which make the algorithm approximate the Pareto front quickly and avoid premature convergence. Finally, MOPSO-SDCD maintains the external archive based on dynamic crowding distance sorting strategy, whose purpose is boosting the diversity and distribution of Pareto optimal solutions. The ZDT series test functions are used to test the performance of MOPSO-SDCD and compare with other three typical algorithms. Simulation results verify the superiority and effectiveness of the proposed algorithm.

Chaotic improved PSO-based multi-objective optimization for minimization of power losses and L index in power systems

Multi-objective optimal reactive power dispatch (MOORPD) seeks to not only minimize power losses, but also improve the stability of power system simultaneously. In this paper, the static voltage stability enhancement is achieved through incorporating L index in MOORPD problem. Chaotic improved PSO-based multi-objective optimization (MOCIPSO) and improved PSO-based multi-objective optimization (MOIPSO) approaches are proposed for solving complex multi-objective, mixed integer nonlinear problems such as minimization of power losses and L index in power systems simultaneously. In MOCIPSO and MOIPSO based optimization approaches, crossover operator is proposed to enhance PSO diversity and improve their global searching capability, and for MOCIPSO based optimization approach, chaotic sequences based on logistic map instead of random sequences is introduced to PSO for enhancing exploitation capability. In the two approaches, constrain-prior Pareto-dominance method (CPM) is proposed to meet the inequality constraints on state variables, the sorting and crowding distance methods are considered to maintain a well distributed Pareto optimal solutions, and moreover, fuzzy set theory is employed to extract the best compromise solution over the Pareto optimal curve. The proposed approaches have been examined and tested in the IEEE 30 bus and the IEEE 57 bus power systems. The performances of MOCIPSO, MOIPSO, and multi-objective PSO (MOPSO) approaches are compared with respect to multi-objective performance measures. The simulation results are promising and confirm the ability of MOCIPSO and MOIPSO approaches for generating lower power losses and smaller L index than MOPSO method.

Generalized decomposition and cross entropy methods for many-objective optimization

Decomposition-based algorithms for multi-objective optimization problems have increased in popularity in the past decade. Although convergence to the Pareto optimal front (PF) for such algorithms can often be superior to that of Pareto-based alternatives, the problem of effectively distributing Pareto optimal solutions in a high-dimensional space has not been solved. In this work, we introduce a novel concept which we call generalized decomposition. Generalized decomposition provides a framework with which the decision maker (DM) can guide the underlying search algorithm toward specific regions of interest, or the entire Pareto front, with the desired distribution of Pareto optimal solutions. The method simplifies many-objective problems by unifying the three performance objectives of an a posteriori multi-objective optimizer – convergence to the PF, evenly distributed Pareto optimal solutions and coverage of the entire front – to only one, that of convergence. A framework, established on generalized decomposition, and an estimation of distribution algorithm (EDA) based on low-order statistics, namely the cross-entropy method, is created to illustrate the benefits of the proposed concept for many-objective problems. The algorithm – MACE-gD – is shown to be highly competitive with the existing best-in-class decomposition-based algorithm (MOEA/D) and a more elaborate EDA method (RM-MEDA).

An integrated cultural particle swarm algorithm for multi-objective reliability-based design optimization

Uncertainties in design variables and problem parameters are often inevitable in multi-objective optimizations, and they must be considered in an optimization task if reliable Pareto optimal solutions are to be sought. Multi-objective reliability-based design optimization has been raised as a question in design for reliability, but the disadvantages of fixed evolutionary parameters, nonuniformly distributed Pareto optimal solutions and high computational cost hinder engineering applications of reliability-based design. To deal with it, this work proposes an integrated multi-objective cultural-based particle swarm algorithm to solve the double-loop reliability-based design optimization. In the inner optimization loop, the cultural space is composed of the elitism, situational and normative knowledge to adjust the parameters for swarm space, and the crowding distance ranking is introduced to update the global and local optimum and control the maximum number of solutions in elitism knowledge. The hybrid mean value method is improved to perform reliability analysis in the outer loop to suit both concave and convex types of performance functions. In addition, the car side-impact and the injection molding machine are chosen as multi-objective reliability design examples to demonstrate the effectiveness of the proposed approach. Simultaneously, results of car side-impact problem are compared with two traditional multi-objective reliability optimization algorithms, i.e., nondominated sorting genetic algorithm and crowding distance ranking-based multi-objective particle swarm optimizer, to assess the efficiency of the proposed approach. The results denote the proposed cultural-based multi-objective particle swarm optimizer is effective and feasible to solve the reliability-based design optimization problems.

A Novel Pareto-Based Meta-Heuristic Algorithm to Optimize Multi-Facility Location-Allocation Problem

This article proposes a novel Pareto-based multiobjective meta-heuristic algorithm named non-dominated ranking genetic algorithm (NRGA) to solve multi-facility location-allocation problem. In NRGA, a fitness value representing rank is assigned to each individual of the population. Moreover, two features ranked based roulette wheel selection including select the fronts and choose solutions from the fronts, are utilized. The proposed solving methodology is validated using several examples taken from the specialized literature. The performance of our approach shows that NRGA algorithm is able to generate true and well distributed Pareto optimal solutions. Keywords—Non-dominated ranking genetic algorithm, Pareto solutions, Multi-facility location-allocation problem.

Multi-objective optimal design of vertical natural circulation steam generator

In this work, a hybrid non-dominated sorting genetic algorithm was proposed and utilized to perform the multi-objective optimization design of a natural circulation steam generator, which included minimizing of the weight, the volume and the reactor coolant flow-rate. Sensitivity analysis of the design variables was carried out to study the relationships between the optimization variables and the objective functions, which was also helpful for the explanation of the optimization results. The mathematical model of the steam generator was validated by the RELAP5 code. The results show that the mathematical model has a good agreement with the RELAP5 model after modifying the boiling correlation in the secondary side; the proposed hybrid non-dominated sorting genetic algorithm is able to find much better spread of solutions and better convergence near the true Pareto optimal front compared to the non-dominated sorting genetic algorithm; reactor inlet temperature is the most important variable which influences the distribution of Pareto optimal solutions.

Application of Multi-Objective Teaching Learning based Optimization Algorithm to Optimal Power Flow Problem

This paper presents a non-domination based sorting multiobjective teaching-learning-based optimization algorithm, for solving the optimal power flow (OPF) problem. The OPF problem is a nonlinear constrained multi-objective optimization problem where the fuel cost, Transmission losses and L-index are to be minimized. Since the problem is treated as a true multi-objective optimization problem, different trade-off solutions are provided. The decision maker has an option to choose a solution among the different trade-off solutions provided in the pareto-optimal front. The standard IEEE 30-bus test system is used and the results show the effectiveness of MOTLBO and confirm its potential to solve the multi-objective OPF problem. Simulation results clearly show that the proposed method is able to produce true and well distributed Pareto optimal solutions for multiobjective OPF problem and the comparison with the results reported in the literature demonstrates the superiority of the proposed approach and confirms its potential to solve the multi-objective OPF problem.

Multi-objective harmony search algorithm for optimal power flow problem

This paper proposes a multi-objective harmony search (MOHS) algorithm for optimal power flow (OPF) problem. OPF problem is formulated as a non-linear constrained multi-objective optimization problem where different objectives and various constraints have been considered into the formulation. Fast elitist non-dominated sorting and crowding distance have been used to find and manage the Pareto optimal front. Finally, a fuzzy based mechanism has been used to select a compromise solution from the Pareto set. The proposed MOHS algorithm has been tested on IEEE 30 bus system with different objectives. Simulation results are also compared with fast non-dominated sorting genetic algorithm (NSGA-II) method. It is clear from the comparison that the proposed method is able to generate true and well distributed Pareto optimal solutions for OPF problem.

A multi-objective optimal power flow using particle swarm optimization

This paper presents a multi-objective optimal power flow technique using particle swarm optimization. Two conflicting objectives, generation cost, and environmental pollution are minimized simultaneously. A multi-objective particle swarm optimization method is used to solve this highly nonlinear and non-convex optimization problem. A diversity preserving technique is incorporated to generate evenly distributed Pareto optimal solutions. A fuzzy membership function is proposed to choose a compromise solution from the set of Pareto optimal solutions. The algorithm is tested on IEEE 30 and 118 bus systems and its effectiveness is illustrated. Copyright © 2010 John Wiley & Sons, Ltd.