Local Pareto Optimal Solutions Research Articles

Two-stage evolutionary algorithm with fuzzy preference indicator for multimodal multi-objective optimization

Multimodal multi-objective optimization problems (MMOPs) are multi-objective optimization problems (MOPs) involving multiple equivalent global or local Pareto optimal solution sets (PSs). For decision-makers, not only the global optimal solution sets need to be found, but also the value of local optimal solution sets cannot be ignored. However, most multimodal multi-objective evolutionary algorithms (MMOEAs) tend to select solutions with better convergence, and it is difficult to obtain the global PSs and local PSs at the same time. Therefore, we propose a fuzzy preference indicator-based two-stage evolutionary algorithm (FPITSEA) in this paper. To evaluate more comprehensively the potential of each solution in the population for locating the global and local PS during the evolutionary process, a fuzzy preference indicator is designed in FPITSEA. The fuzzy preference indicator is used to guide the evolution of the population in the first stage to find the global and local Pareto optimal regions. Subsequently, an independent evolution strategy is implemented in the second stage to distinguish different PSs as accurately as possible while also ensuring the convergence quality of the solution set. In addition, an improved distance-based subset selection method is proposed, aiming to simultaneously improve the distribution of the solution set in the decision space and objective space. Experimental results on several test sets of MMOPs show the advantages of FPITSEA over several state-of-the-art MMOEAs.

A multiobjective differential evolution algorithm with subpopulation region solution selection for global and local Pareto optimal sets

Current multimodal multiobjective optimization (MMO) has mostly focused on locating multiple equivalent global Pareto optimal sets (PSs) or local PSs in the decision space, rarely considering finding both simultaneously. To address this issue, this paper explores a crowded individual activation multiobjective differential evolution algorithm based on subpopulation region solution selection, named CAMODE_SR. In CAMODE_SR, a subpopulation region solution selection mechanism is developed to search for more uniformly distributed global and local Pareto optimal solutions. Specifically, two types of specific neighborhood radii are designed depending on the location information of the Pareto optimal solutions discovered during evolution. The two types of radii construct specific neighborhoods for the global and local optimal solutions to maintain the solutions with better diversity. To improve the exploitation capacity of the population, a crowded individual activation mechanism is proposed by developing an activation function associated with iteration. The activation function activates the crowded individuals in local regions to exploit more superior solutions near global optimal solutions. The proposed CAMODE_SR achieves a good tradeoff between the performance of locating both global and local PSs. Extensive experiments on the CEC 2020 MMO benchmark functions demonstrate that the proposed CAMODE_SR is significantly superior to nine state-of-the-art MMEAs in tackling multimodal multiobjective optimization problems with global and local PSs.

Clearing-based multimodal multi-objective evolutionary optimization with layer-to-layer strategy

Equivalent Pareto sets and local Pareto sets in decision space are involved in multimodal multi-objective optimization problems (MMOPs). While popular multimodal multi-objective evolutionary algorithms (MMEAs) devote greater attention to locating equivalent Pareto sets in decision space, resulting in the fact that local Pareto sets are overlooked. To tackle this issue, a clearing-based evolutionary algorithm with layer-to-layer evolution strategy (CEA-LES) is therefore proposed in this paper. A clearing-based niching technique is used in CEA-LES to remove inferior local Pareto optimal solutions, in which the clearing radius dynamically changes with the number of nearest neighbors of each individual. To develop equivalent global Pareto sets and local Pareto sets, a layer-to-layer evolutionary strategy is proposed based on Pareto ranking. Each layer performs mutation selection and reproduction operations with the aid of its closest neighbor layer to produce offspring. Particularly, the whole evolutionary operation is divided into two stages, and the first stage is to select parents using the roulette method and the second stage is using the tournament method. An environmental selection strategy is suggested to maintain diverse solutions and balance the distribution of solutions. The proposed algorithm is performed on CEC 2020 MMOPs benchmark with local Pareto set and Polygon-based MMOPs and is compared with some state-of-the-art MMEAs. Experimental results demonstrate that the proposed algorithm obtains superior performance in comparison to its competing algorithms, and is capable of locating equivalent global Pareto sets and local Pareto sets, simultaneously.

A multi-objective particle swarm optimizer based on reference point for multimodal multi-objective optimization

In real-world applications, there are many multimodal multi-objective optimization problems, which have multiple equivalent global Pareto-optimal solutions or with at least one local Pareto-optimal solution in the decision space. While some evolutionary algorithms have been proposed to find the global solutions recently, they are difficult to handle multimodal multi-objective optimization problems with local solutions. Meanwhile, there have been few studies on searching for local Pareto solutions. However, local solutions are additional alternatives for the decision makers if global solutions are impracticable. This paper proposes a particle swarm optimizer based on reference point, termed RPPSO, which combines a reference point mechanism and a local solution preserving technique. The reference point strategy is utilized to establish multiple evenly distributed neighborhoods and guide particles to evolve independently in their respective neighborhoods, so as to detect more Pareto solutions in the decision space. The local solution preserving technique is employed to estimate the dominant radius of each front, and with this radius to classify the individuals as either non-local or local solutions with the aim of retaining the local solutions. In addition, a set of benchmark test functions with local Pareto solutions are designed. The proposed algorithm is comprehensively evaluated on forty-four benchmark functions and is compared with fourteen state-of-the-art algorithms. The experimental results show that the proposed RPPSO achieves competitive performance than its competitors in terms of the reciprocal of Pareto sets proximity (rPSP). The RPPSO is also applied to solve on one real-world problem (i.e., map-based problem) to further verify the effectiveness and efficiency.

A biobjective chance constrained optimization model to evaluate the economic and environmental impacts of biopower supply chains

Generating electricity by co-combusting biomass and coal, known as biomass cofiring, is shown to be an economically attractive option for coal-fired power plants to comply with emission regulations. However, the total carbon footprint of the associated supply chain still needs to be carefully investigated. In this study we propose a stochastic biobjective optimization model to analyze the economic and environmental impacts of biopower supply chains. We use a life cycle assessment approach to derive the emission factors used in the environmental objective function. We use chance constraints to capture the uncertain nature of energy content of biomass feedstocks. We propose a cutting plane algorithm which uses the sample average approximation method to model the chance constraints and finds high confidence feasible solutions. In order to find Pareto optimal solutions we propose a heuristic approach which integrates the $$\epsilon $$ -constraint method with the cutting plane algorithm. We show that the developed approach provides a set of local Pareto optimal solutions with high confidence and reasonable computational time. We develop a case study using data about biomass and coal plants in North and South Carolina. The results indicate that, cofiring of biomass in these states can reduce emissions by up to 8%. Increasing the amount of biomass cofired will not result in lower emissions due to biomass delivery.

MOEA/D-GLS: a multiobjective memetic algorithm using decomposition and guided local search

This paper proposes an idea of using well studied and documented single-objective optimization methods in multiobjective evolutionary algorithms. It develops a hybrid algorithm which combines the multiobjective evolutionary algorithm based on decomposition (MOEA/D) with guided local search (GLS), called MOEA/D-GLS. It needs to optimize multiple single-objective subproblems in a collaborative way by defining neighborhood relationship among them. The neighborhood information and problem-specific knowledge are explicitly utilized during the search. The proposed GLS alternates among subproblems to help escape local Pareto optimal solutions. The experimental results have demonstrated that MOEA/D-GLS outperforms MOEA/D on multiobjective traveling salesman problems.

Multi-objective differential evolution with diversity enhancement

Multi-objective differential evolution (MODE) is a powerful and efficient population-based stochastic search technique for solving multi-objective optimization problems in many scientific and engineering fields. However, premature convergence is the major drawback of MODE, especially when there are numerous local Pareto optimal solutions. To overcome this problem, we propose a MODE with a diversity enhancement (MODE-DE) mechanism to prevent the algorithm becoming trapped in a locally optimal Pareto front. The proposed algorithm combines the current population with a number of randomly generated parameter vectors to increase the diversity of the differential vectors and thereby the diversity of the newly generated offspring. The performance of the MODE-DE algorithm was evaluated on a set of 19 benchmark problem codes available from http://www3.ntu.edu.sg/home/epnsugan. With the proposed method, the performances were either better than or equal to those of the MODE without the diversity enhancement.

Sensitivity analysis for parametric vector optimization problems using differential equations approach

A new method for obtaining sensitivity information for parametric vector optimization problems (VOP) v is presented, where the parameters in the objective functions and anywhere in the constraints. This method depends on using differential equations technique for solving multiobjective nonlinear programing problems which is very effective in finding many local Pareto optimal solutions. The behavior of the local solutions for slight perturbation of the parameters in the neighborhood of their chosen initial values is presented by using the technique of trajectory continuation. Finally some examples are given to show the efficiency of the proposed method.

Technique for solving multiobjective nonlinear programming using differential equations approach

This paper deals with multiobjective nonlinear programming problems (MONLP). The purpose of this paper is to present an effective approach for solving (MONLP), which is very effective in finding many local pareto-optimal solution, and the method presented here seems to be very efficient for finding the global efficient solution in the case of a bounded feasible region. We apply the technique of differential equation approach, where the (MONLP) is transformed to a nonlinear autonomous system of differential equations, and the relation between the critical points of differential system and local pareto-optimal solutions of the original optimization problem is proved. Finally, the approach presented in this paper is demonstrated with a numerical simple example.

Interactive multiobjective decisionmaking for nonconvex problems based on the penalty scalarizing functions

In this paper, we propose a new interactive multiobjective decisionmaking technique for solving multiobjective nonconvex problems based on the penalty scalarizing functions. Our technique uses the penalty scalarizing problems as a means of generating local Pareto optimal solutions. In our interactive on-line scheme, once the decision maker (DM) specifies initial reference objective levels, the DM is supplied with the corresponding local Pareto optimal solution in a sense close to the DM's requirement. The values of the marginal rates of substitution (MRS) assessed by the DM are used to determine both the optimal direction and the best step-size that maximize the DM's unknown utility function. Then the corresponding local Pareto optimal values of the objective functions become the new reference objective levels for the next iteration. In this way the local preferred solution for the DM can be derived efficiently from among a local Pareto optimal solution set by assessing his MRS. Based on the algorithm of our technique, the time-sharing computer program is written in Fortran to implement man-machine interactive procedures. The interaction processes are demonstrated together with the computer outputs by means of an illustrative numerical example involving nonconvexities.

Neural-genetic techniques for ship control systems

In this paper, artificial neural networks and genetic algorithms for solving optimization problems in ship control systems have been proposed. The gradient Hopfield ANN for linear minimization is considered. Moreover, HANN for finding local Pareto-optimal solutions in multicriteria optimization of designed navigation systems has been considered. Finally, genetic-neural algorithm GNA for improving a quality of solutions have been introduced.