True Pareto-optimal Solutions Research Articles

Evolutionary Multimodal Multiobjective Optimization for Traveling Salesman Problems

Multimodal multiobjective optimization problems (MMOPs) are commonly seen in real-world applications. Many evolutionary algorithms have been proposed to solve continuous MMOPs. However, little effort has been made to solve combinatorial (or discrete) MMOPs. Searching for equivalent Pareto optimal solutions in the discrete decision space is challenging. Moreover, the true Pareto optimal solutions of a combinatorial MMOP are usually difficult to know, which has limited the development of its optimizer. In this paper, we first propose a test problem generator for multimodal multiobjective traveling salesman problems (MMTSPs). It can readily generate MMTSPs with known Pareto optimal solutions. Then we propose a novel evolutionary algorithm to solve MMTSPs. In our proposed algorithm, we develop two new edge assembly crossover operators, which are specialized in searching for superior solutions to MMTSPs. Moreover, the proposed algorithm uses a new environmental selection operator to maintain a good balance between the objective space diversity and decision space diversity. We compare our algorithm with five state-of-the-art designs. Experimental results convincingly show that our algorithm is powerful in solving MMTSPs.

An effective multi-objective bald eagle search algorithm for solving engineering design problems

In this paper, a multi-objective bald eagle search algorithm (MOBES) is proposed. The MOBES introduces an archive mechanism to store the non-dominated solutions obtained by the algorithm. When the archive overflows, remove the most crowded solutions by using the roulette method. The MOBES also adds elite selection strategy to guide other individuals to optimize by selecting elite individuals in the population. The efficiency of MOBES is validated on CEC 2020 benchmark functions, and the results demonstrate that the proposed algorithm is more efficient than its competitors in terms of convergence, diversity and distribution of solutions. The MOBES is also applied to two-objective, tri-objective and four-objective engineering design problems in real world. The results show its superiority in handling challenging multi-objective optimization problems with unknown true Pareto optimal solutions and fronts, and it is more competitive than other algorithms.

A Reference Point-Based Evolutionary Algorithm Solves Multi and Many-Objective Optimization Problems: Method and Validation.

The integration of a decision maker's preferences in evolutionary multi-objective optimization (EMO) has been a common research scope over the last decade. In the published literature, several preference-based evolutionary approaches have been proposed. The reference point-based non-dominated sorting genetic (R-NSGA-II) algorithm represents one of the well-known preference-based evolutionary approaches. This method mainly aims to find a set of the Pareto-optimal solutions in the region of interest (ROI) rather than obtaining the entire Pareto-optimal set. This approach uses Euclidean distance as a metric to calculate the distance between each candidate solution and the reference point. However, this metric may not produce desired solutions because the final minimal Euclidean distance value is unknown. Thus, determining whether the true Pareto-optimal solution is achieved at the end of optimization run becomes difficult. In this study, R-NSGA-II method is modified using the recently proposed simplified Karush-Kuhn-Tucker proximity measure (S-KKTPM) metric instead of the Euclidean distance metric, where S-KKTPM-based distance measure can predict the convergence behavior of a point from the Pareto-optimal front without prior knowledge of the optimum solution. Experimental results show that the algorithm proposed herein is highly competitive compared with several state-of-the-art preference-based EMO methods. Extensive experiments were conducted with 2 to 10 objectives on various standard problems. Results show the effectiveness of our algorithm in obtaining the preferred solutions in the ROI and its ability to control the size of each preferred region separately at the same time.

An effective multi-objective artificial hummingbird algorithm with dynamic elimination-based crowding distance for solving engineering design problems

Artificial hummingbird algorithm (AHA) is a recently developed bio-based metaheuristic and it shows superior performance in handling single-objective optimization problems. Despite the merit, this algorithm can only solve problems with one objective. To solve complex multi-objective optimization problems, including engineering design problems, a multi-objective AHA (MOAHA) is developed in this study. In MOAHA, an external archive is employed to save Pareto optimal solutions, and a dynamic elimination-based crowding distance (DECD) method is developed to maintain this archive to effectively preserve the population diversity. In addition, a non-dominated sorting strategy is merged with MOAHA to construct a solution update mechanism, which effectively refines Pareto optimal solutions for improving the convergence of the algorithm. The superior results over 7 competitors on 28 benchmark functions in terms of convergence, diversity and solution distribution are demonstrated with a suite of comprehensive tests. The MOAHA algorithm is also applied to 5 real-world engineering design problems with multiple objectives, demonstrating its superiority in handling challenging real-world multi-objective problems with unknown true Pareto optimal solutions and fronts. The source code of MOAHA is publicly available at https://ww2.mathworks.cn/matlabcentral/fileexchange/113535-moaha-multi-objective-artificial-hummingbird-algorithm and https://seyedalimirjalili.com/aha.

Handling multimodal multi-objective problems through self-organizing quantum-inspired particle swarm optimization

Multimodal multi-objective optimization problems (MMOPs) involve locating equivalent Pareto optimal solutions in decision space with the same objective values. The key to handling MMOPs is finding all equivalent Pareto optimal solutions and maintaining a promising balance between the convergence and diversity of solutions in both decision space and objective space. To tackle this issue, a self-organizing quantum-inspired particle swarm optimization algorithm (MMO_SO_QPSO) is proposed in this paper for handling MMOPs. In the proposed MMO_SO_QPSO, a self-organizing map is used to find the best neighbor leader of particles. With the aid of neighbor leader particles, a special zone searching method is adopted to update the position of particles and locate equivalent Pareto optimal solutions in decision space. To maintain diversity and convergence of Pareto optimal solutions, a special archive mechanism that relies on the maximum-minimum distance among solutions is introduced into MMO_SO_QPSO. And some outstanding Pareto optimal solutions are maintained in the special archive. In addition, a new performance indicator is developed to estimate properly the similarity between obtained Pareto optimal solutions and true Pareto optimal solutions. The performance of the proposed MMO_SO_QPSO is compared with six state-of-the-art multimodal multi-objective evolutionary algorithms on two well-known benchmark problems. Experimental results demonstrate the superior performance of MMO_SO_QPSO for solving MMOPs. The effectiveness of several strategies is also discussed.

Golden eagle optimizer: A nature-inspired metaheuristic algorithm

This paper proposes a nature-inspired swarm-based metaheuristic for solving global optimization problems called Golden Eagle Optimizer (GEO). The core inspiration of GEO is the intelligence of golden eagles in tuning speed at different stages of their spiral trajectory for hunting. They show more propensity to cruise around and search for prey in the initial stages of hunting and more propensity to attack in the final stages. A golden eagle adjusts these two components to catch the best possible prey in feasible region the shortest possible time. This behavior is mathematically modeled to highlight exploration and exploitation for a global optimization method. The performance of the proposed algorithm is tested and confirmed using 33 benchmark test functions and a scalability test. Results were compared to that of six other well-known algorithms, which revealed GEO’s superiority, which indicates that it can find the global optimum and avoid local optima effectively. The Multi-Objective Golden Eagle Optimizer (MOGEO) is also proposed to solve multi-objective problems. The performance of MOGEO is also tested and verified on ten multi-objective benchmark functions. Results were compared to that of two other multi-objective algorithms, which showed that it can approximate true Pareto optimal solutions better than the other two algorithms. The software (toolbox) and source code for GEO and MOGEO are also provided, which are publicly available.

A Mixture-of-Experts Prediction Framework for Evolutionary Dynamic Multiobjective Optimization.

Dynamic multiobjective optimization requires the robust tracking of varying Pareto-optimal solutions (POS) in a changing environment. When a change is detected in the environment, prediction mechanisms estimate the POS by utilizing information from previous populations to accelerate search toward the true POS. To achieve a robust prediction of POS, a mixture-of-experts-based ensemble framework is proposed. Unlike existing approaches, the framework utilizes multiple prediction mechanisms to improve the overall prediction. A gating network is applied to manage switching among the various predictors based on performance of the predictors at different time intervals of the optimization process. The efficacy of the proposed framework is validated through experimental studies based on 13 dynamic multiobjective benchmark optimization problems. The simulation results show that the proposed framework improves the dynamic optimization performance significantly, particularly for: 1) problems with distinct dynamic POS in decision space over time and 2) problems with highly nonlinear decision variable linkages.

Cooperative multiobjective optimization with bounds on objective functions

When solving large-scale multiobjective optimization problems, solvers can get stuck because of memory and/or time limitations. In such cases, one is left with no information on the distance to the best feasible solution, found before the optimization process has stopped, to the true Pareto optimal solution. In this work, we show how to provide such information. To this aim we make use of the concept of lower shells and upper shells, developed in our earlier works. No specific assumptions about the problems to be solved are made. We illustrate the proposed approach on biobjective multidimensional knapsack problems derived from single-objective multidimensional knapsack problems in the Beasley OR Library. We address cases when a top-class commercial mixed-integer linear solver fails to provide Pareto optimal solutions attempted to be derived by scalarization.

A multi-objective methodology for multi-criteria engineering design

Optimization is more and more significant due to its application in the real engineering problems. The recently proposed imperialist competitive algorithm (ICA) is a successful method in mono-objective optimization. Nevertheless, ICA cannot handle simultaneously the conflicting objectives in multi-objective design problem. In addition, the ICA has the drawback of trapping in local optimum solutions when used for high-dimensional or complex multimodal functions. In order to deal with these situations, in this work, an improved ICA, named modified multi-objective imperialist competitive algorithm (MOMICA) is proposed. In MOMICA, an attraction and repulsion (AR) concept is implemented in the assimilation phase to improve the performances of the algorithm to reach the global optimal position. Moreover, in contrast to ICA, the proposed algorithm integrates the sorting non-dominated strategy (SND) to store the Pareto optimal solutions of multiple conflicting functions. Three performance metrics are used to evaluate the performance of the proposed algorithm: (a) convergence to the true Pareto-optimal set, (b) solutions diversity and (c) robustness, characterized by the variance over 10 runs. The results presented in this paper show that the MOMICA algorithm outperforms the other popular techniques in terms of convergence characteristics and global search ability, for both benchmark functions optimization and multi-objective engineering optimization problems.

A new meta-heuristic optimizer: Pathfinder algorithm

This paper proposes a new meta-heuristic algorithm called Pathfinder Algorithm (PFA) to solve optimization problems with different structure. This method is inspired by collective movement of animal group and mimics the leadership hierarchy of swarms to find best food area or prey. The proposed method is tested on some optimization problems to show and confirm the performance on test beds. It can be observed on benchmark test functions that PFA is able to converge global optimum and avoid the local optima effectively. Also, PFA is designed for multi-objective problems (MOPFA). The results obtained show that it can approximate to true Pareto optimal solutions. The proposed PFA and MPFA are implemented to some design problems and a multi-objective engineering problem which is time consuming and computationally expensive. The results of final case study verify the superiority of the algorithms proposed in solving challenging real-world problems with unknown search spaces. Furthermore, the method provides very competitive solutions compared to well-known meta-heuristics in literature, such as particle swarm optimization, artificial bee colony, firefly and grey wolf optimizer.

Evolutionary multi-objective optimization for bulldozer and its blade in soil cutting

ABSTRACTThe paper targets an optimal soil cutting operation by considering economic and productive aspects. Three realistic objectives and three problem-specific constraints are developed, and the optimization problem is solved using a hybrid evolutionary multi-objective optimization (EMO) technique. In this technique, a set of non-dominated solutions is generated by using an existing EMO technique, and then a few of them are selected for local search using the -constraint method. These selected solutions are used for starting independent local searches using fmincon solver of Matlab. Results demonstrate that the local searches have improved the non-dominated solutions a little, thereby suggesting a closeness of evolved solutions from EMO technique with true Pareto-optimal (PO) solutions. The PO solutions are further validated using experimental data from the literature. Overall, this study offers a platform to choose an appropriate solution from the set of PO solutions. Moreover, the post-optimal analysis demonstrates the commonality principle of few decision variables, which is followed by all PO solutions. The rest of the decision variables decipher important relationships that are responsible for trade-off among the PO solutions. The relationships are later used for preparing guidelines for a practitioner in selecting an appropriate solution for the optimal operation.

A novel multi-objective bacteria foraging optimization algorithm (MOBFOA) for multi-objective scheduling

In grid and cloud computing environment, many users compete for the resources, so the schedule should be generated in the shortest possible time. To address this problem, there have been several research initiatives to use evolutionary and swarm based algorithms to find near-optimal scheduling solutions. The state-of-the-art evolutionary algorithms for handling single/multi-criteria scheduling of m jobs on n resources are still evolving, with efforts aimed at reducing their space/time complexity, maintaining diversity in the population and directing the search towards the true Pareto-optimal solutions. In this paper, we have proposed a multi-objective bacteria foraging optimization algorithm (MOBFOA) to address these objectives. We have attempted to modify the original BFOA to handle the multi-objective scheduling problems using Pareto-optimal front approach. The modification is in terms of selecting bacteria positions from both the dominant as well as non-dominant fronts to obtain diversity in the solutions obtained. The accuracy and speed of the convergence of the BFOA has been improved by introducing adaptive step size in chemotactic step. The proposed MOBFOA uses new fitness assignment method and bacteria selection procedure for simultaneous optimization of multiple objectives, where each solution evaluation is computationally expensive. This paper focuses on the scheduling of independent jobs considering multi-objective trade-offs among the objective functions desired by the users in grid/cloud environment. The performance of the proposed MOBFOA is discussed in terms of convergence towards the Pareto-optimal front and distribution of solutions in the search space. The paper also provides a comparative study of the results obtained by the proposed MOBFOA with other stochastic optimization algorithms, namely, the non-dominated sorting genetic algorithm-II (NSGA-II) and optimised multi-objective particle swarm optimization (OMOPSO).

Reward shaping for knowledge-based multi-objective multi-agent reinforcement learning

AbstractThe majority of multi-agent reinforcement learning (MARL) implementations aim to optimize systems with respect to a single objective, despite the fact that many real-world problems are inherently multi-objective in nature. Research into multi-objective MARL is still in its infancy, and few studies to date have dealt with the issue of credit assignment. Reward shaping has been proposed as a means to address the credit assignment problem in single-objective MARL, however it has been shown to alter the intended goals of a domain if misused, leading to unintended behaviour. Two popular shaping methods are potential-based reward shaping and difference rewards, and both have been repeatedly shown to improve learning speed and the quality of joint policies learned by agents in single-objective MARL domains. This work discusses the theoretical implications of applying these shaping approaches to cooperative multi-objective MARL problems, and evaluates their efficacy using two benchmark domains. Our results constitute the first empirical evidence that agents using these shaping methodologies can sample true Pareto optimal solutions in cooperative multi-objective stochastic games.

Multi-objective differential evolution based on normalization and improved mutation strategy

Developing efficient algorithms for solving multi-objective optimization problems is a challenging and essential task in many applications. This task involves two or more conflicting objectives that need to be simultaneously optimized. Many real-world problems fall into this category. We introduce an improved version of multi-objective differential evolution (DE) algorithm, namely MOnDE that uses a new mutation strategy and a normalization method to select non-dominated solutions. The new mutation strategy “DE/rand-to-nbest” uses the best normalized individual in terms of all the objectives to guide the search towards the true pareto optimal solutions. As a result, the probability of producing superior solutions is increased and a faster convergence is achieved. Summation of normalized objective values method is used instead of non-domination sorting to overcome the high computational complexity and overhead problems of sorting non-dominated solutions. The performance of our approach is tested on a set of benchmark problems that consist of two to five objectives. Different combinations of multi-objective evolutionary programming and multi-objective differential evolution algorithms have been used for comparisons. The results affirm the efficiency and robustness of the proposed approach among other well-known algorithms from the literature.

An Optimality Theory-Based Proximity Measure for Set-Based Multiobjective Optimization

Set-based multiobjective optimization methods, such as evolutionary multiobjective optimization (EMO) methods, attempt to find a set of Pareto-optimal solutions, instead of a single optimal solution. To evaluate these algorithms for their convergence to the efficient set in multiobjective optimization problems, the current performance metrics require the knowledge of the true Pareto-optimal solutions. In this paper, we develop a theoretically motivated Karush–Kuhn–Tucker proximity measure (KKTPM) that can provide an estimate of the proximity of a set of tradeoff solutions from the true Pareto-optimal solutions without any prior knowledge. Besides theoretical development of the proposed metric, the proposed KKTPM is computed for iteration-wise tradeoff solutions obtained from specific EMO algorithms on two-, three-, five-, and ten-objective optimization problems. Results amply indicate the usefulness of the proposed KKTPM as a metric for evaluating different sets of tradeoff solutions and also as a possible termination criterion for an EMO algorithm. Other possible uses of the proposed metric are also highlighted.

An improved imperialist competitive algorithm for multi-objective optimization

This article proposes an improved imperialistic competitive algorithm to solve multi-objective optimization problems. The proposed multi-objective imperialistic competitive algorithm (MOICA) uses the elitist strategy, based on the mutation and crossover as in genetic algorithms, and the Pareto concept to store simultaneously optimal solutions of multiple conflicting functions. Three performance metrics are used to evaluate the performance of the new algorithm: convergence to the true Pareto-optimal set, solution diversity and robustness, characterized by the variance over 10 runs. To validate the efficiency of the proposed algorithm, several multi-objective standard test functions with true solutions are used. The obtained results show that the MOICA outperforms most of the methods available in the literature. The proposed algorithm can also handle multi-objective engineering design problems with high dimensions.

An integer multi-objective optimization model and an enhanced non-dominated sorting genetic algorithm for contraflow scheduling problem

To determine the onset and duration of contraflow evacuation, a multi-objective optimization (MOO) model is proposed to explicitly consider both the total system evacuation time and the operation cost. A solution algorithm that enhances the popular evolutionary algorithm NSGA-II is proposed to solve the model. The algorithm incorporates preliminary results as prior information and includes a meta-model as an alternative to evaluation by simulation. Numerical analysis of a case study suggests that the proposed formulation and solution algorithm are valid, and the enhanced NSGA-II outperforms the original algorithm in both convergence to the true Pareto-optimal set and solution diversity.

A Simple Pareto Adaptive ε-Domination Differential Evolution Algorithm for Multi-Objective Optimization

The two purposes of solving the multi-objective optimization problems are to get solutions close to the true Pa- reto front as much as possible and to obtain promising diversity. To meet these two demands, a new method is proposed in this paper, which has these characteristics: 1) it adopts the orthogonal design method with quantization technology to gen- erate initial population whose individuals are scattered uniformly over the target search space. 2) it is based on an adaptive e concept to obtain a good distribution along the true Pareto-optimal solutions. Experiments on five benchmark problems with different features indicate that the proposed method works well not only in diversity, but also in convergence when compared to other evolutionary algorithms.

Evolutionary algorithm and decisional DNA for multiple travelling salesman problem

In the real world, it is common to face optimization problems that have two or more objectives that must be optimized at the same time, that are typically explained in different units, and are in conflict with one another. This paper presents a hybrid structure that combines set of experience knowledge structures (SOEKS) and evolutionary algorithms, NSGA-II (Non-dominated Sorting Genetic Algorithm II), to solve multiple optimization problems. The proposed structure uses experience that is derived from a former decision event to improve the evolutionary algorithm’s ability to find optimal solutions rapidly and efficiently. It is embedded in a smart experience-based data analysis system (SEDAS) introduced in the paper. Experimental illustrative results of SEDAS application to solve a travelling salesman problem show that our new proposed hybrid model can find optimal or close to true Pareto-optimal solutions in a fast and efficient way.

Hybrid ant optimization system for multiobjective economic emission load dispatch problem under fuzziness

In this paper, a new hybrid optimization system is presented. Our approach integrates the merits of both ant colony optimization and steady state genetic algorithm and it has two characteristic features. Firstly, since there is instabilities in the global market and the rapid fluctuations of prices, a fuzzy representation of the economic emission load dispatch (EELD) problem has been defined, where the input data involve many parameters whose possible values may be assigned by the expert. Secondly, by enhancing ant colony optimization through steady state genetic algorithm, a strong robustness and more effectively algorithm was created. Also, stable Pareto set of solutions has been detected, where in a practical sense only Pareto optimal solutions that are stable are of interest since there are always uncertainties associated with efficiency data. Moreover to help the decision maker DM to extract the best compromise solution from a finite set of alternatives a Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) method is adopted. It is based upon simultaneous minimization of distance from an ideal point (IP) and maximization of distance from a nadir point (NP). The results on the standard IEEE systems demonstrate the capabilities of the proposed approach to generate true and well-distributed Pareto optimal nondominated solutions of the multiobjective EELD.