Multiobjective electromagnetic devices design based on improved EDA with sequential optimization of the probability model

Sequential multiobjective optimization method based on radial basis function model is presented in this work to deal with the multiobjective design and optimization problems of engineering electromagnetic devices. Firstly, the initial sample set can be generated by using the sequential optimization method (SOM). SOM can greatly reduce the sample size by integrating the advantages of optimization algorithms and approximate models. Secondly, the optimal Pareto solutions of the multiobjective design problems can be obtained by updating the samples and approximate models sequentially. Thereafter, we calculate the root mean square error for each objective to determine whether it has achieved the default value. Finally, to illustrate the performance of the new method, a classic mathematic test function and a design example of permanent magnet synchronous machine are investigated. It can be found that the proposed method combined the advantages of effectiveness of sequential optimization strategy and lower computation cost of approximate models. The obtained solutions are satisfactory while the computation cost of finite element analysis needed by the new method is less than 1/5 compare with that of direct optimization algorithm.

We present two new sequential optimization strategies, a sequential subspace optimization method (SSOM) and an improved sequential optimization method (SOM) with orthogonal experimental design technique, to deal with optimization design problems of electromagnetic devices in this work. To implement the proposed methods, we first divide the whole design factors into three sets, namely highly-significant, significant, and nonsignificant factors. Then the whole design space can be correspondingly divided into three subspaces with these three sets of factors. Thereafter, SSOM is presented to sequentially optimize those subspaces. In the subspace, we present an improved SOM based on orthogonal experimental design technique to get optimal solutions. Finally, by investigating TEAM benchmark problem 22, we can see that the sampling efficiency can be improved significantly and the computational cost of finite element analysis can be saved remarkably by the proposed methods.

Evolutionary algorithms and elliptical copulas applied to continuous optimization problems

DE/EDA: A new evolutionary algorithm for global optimization

The Pareto front (Pareto set) of a continuous optimization problem with m objectives is a (m-1) dimensional piecewise continuous manifold in the objective space (the decision space) under some mild conditions. Based on this regularity property in the objective space, we have recently developed several multiobjective estimation of distribution algorithms (EDAs). However, this property has not been utilized in the decision space. Using the regularity property in both the objective and decision space, this paper proposes a simple EDA for multiobjective optimization. Since the location information has not efficiently used in EDAs, a combination of EDA and differential evolution (DE) is suggested for improving the algorithmic performance. The hybrid method and the pure EDA method proposed in this paper, and a DE based method are compared on several test instances. Experimental results have shown that the algorithm with the proposed strategy is very promising.

The Estimation of Distribution Algorithms(EDAs) is a new paradigm for Evolutionary Computation. This new class of algorithms generalizes Genetic Algorithms(GAs) by replacing the crossover and mutation operators by learning and sampling the probability distribution of the best individuals of the population at each iteration of the algorithm. In this paper, we review the EDAs for the solution of combinatorial optimization problems and optimization in continuous domains. The paper gives a brief overview of the multiobjective problems(MOP) and estimation of distribution algorithms(EDAs). We introduce a representative algorithm called RMMEDA (Regularity Model Based Multi-objective Estimation of Distribution Algorithm). In order to improve the convergence performance of the algorithm, we improve the traditional RM-MEDA. The improvement we make is using part of the parent population with better performance instead of the entire parent population to establish a more accurate manifold model, and the RM-MEDA based on elitist strategy theory is proposed. Experimental results show that the improved RM-MEDA performs better on the convergence metric and the algorithm runtime than the original one.KeywordsEDARMMEDAElitist Strategy

This paper addresses a practical mean-CVaR portfolio optimization problem, which maximizes mean return and minimizes CVaR. Since the practical constraints are considered, the problem is proved to be NP-hard. To solve this complex problem, we decompose it into an asset selection problem and a proportion allocation problem. For the asset selection problem, an estimation of distribution algorithm (EDA) is developed to determine which assets are included in the portfolio. Once the asset selection is fixed in each generation of the EDA, the proportion of each asset is determined by solving the proportion allocation problem using the linear programming. To guarantee the diversity of the obtained solutions, the probability model (PM) is divided into a set of sub-PMs according to the decomposition of the objective space. A knowledge-based initialization and a cooperation-based local search are designed to improve the solutions obtained in the initialization stage and the search process, respectively. The proposed decomposition-based hybrid EDA (DHEDA) is tested on real-world datasets and compared with an existing algorithm. Numerical results demonstrate the effectiveness and efficiency of the DHEDA.

This paper proposes a new multi-objective optimization problem to find trade-offs in deployment of phasor measurement units (PMUs) and intelligent electronic devices (IEDs) for state estimation in active distribution networks. A new hybrid estimation of distribution algorithm (EDA) has been used to find the optimal number and location of measurement devices, such as PMU and IED, for accurate state estimation. The objective functions to be minimized in this optimization problem are the total cost of PMUs and IEDs, and the root mean square value of state estimation error. As the objectives are conflicting in nature, a multi-objective Pareto-based nondominated sorting (NDS) EDA algorithm is proposed. Moreover, to improve the local searching capability of the traditional EDA algorithm, the interior point method (IPM) is hybridized with EDA to get a near-global optimal solution. The hybridization of EDA with IPM brings a higher degree of balance between the exploration and exploitation capability of the traditional EDA during the search process. Furthermore, the random variation in loads and generations is also considered to check the reliability of the proposed meter placement technique. The viability of the proposed algorithm has been tested on the IEEE 69-bus system and Indian 85-bus system to validate the results. The obtained results have been compared with those of the conventional EDA algorithm, NDS genetic algorithm, and hybrid EDA-simulated annealing algorithm existing in the literature.

Sequential equi-marginal optimization method for ranking strategies for thermal building renovation

Decomposition to smaller sub-problems is a general approach in problem solving. Many of the real-world problems can be decomposed into a number of sub-problems which may be solved easier. Appropriate decomposition is a significant issue specially for optimization problems where the optimal solution is usually obtained by combining the solutions of sub-problems. Estimation of distribution algorithms (EDAs) are a type of evolutionary algorithms that learn a model of problem from the population of candidate solutions. This model is intended to capture the interactions between problem variables, thus facilitating problem decomposition and is used to generate new solutions. In this paper, a novel type of problems is presented that is designed to challenge the model building process in discrete EDAs. The main idea is to propose a set of problems that their candidate solutions can be simultaneously decomposed into different sub-problems. This means that the candidate solution of a problem may be interpreted by two or more different structure where only one is true, resulting in the optimal solution to that problem. Some of these decompositions or structures may be more likely according to the low-order statistics collected from the population of candidate solutions, but may not necessarily lead to the optimal solution. Learning the correct structure/decomposition is a challenge for the model building process in EDA. The experimental results show that the proposed problems are indeed difficult for EDAs even when expressive models such as Bayesian networks are used to capture the interactions in the problem.

Many real-world optimization problems are subjected to uncertainties that may be characterized by the presence of noise in the objective functions. The estimation of distribution algorithm (EDA), which models the global distribution of the population for searching tasks, is one of the evolutionary computation techniques that deals with noisy information. This paper studies the potential of EDAs; particularly an EDA based on restricted Boltzmann machines that handles multi-objective optimization problems in a noisy environment. Noise is introduced to the objective functions in the form of a Gaussian distribution. In order to reduce the detrimental effect of noise, a likelihood correction feature is proposed to tune the marginal probability distribution of each decision variable. The EDA is subsequently hybridized with a particle swarm optimization algorithm in a discrete domain to improve its search ability. The effectiveness of the proposed algorithm is examined via eight benchmark instances with different characteristics and shapes of the Pareto optimal front. The scalability, hybridization, and computational time are rigorously studied. Comparative studies show that the proposed approach outperforms other state of the art algorithms.

Solving slot allocation problem with multiple ATFM measures by using enhanced meta-heuristic algorithm

Although various optimization methods for solving single-objective problems have been developed in the last few decades, these methods have lost their eligibility due to the fact that today’s engineering problems are toward multiple objective optimization problems, in real applications. For single-objective optimization problems, for example, in case of a minimization problem, this value is the decision vector giving the smallest objective that can be achieved within the specified constraints. Hence the minimum decision vector within all possible (feasible) solution vectors is the so-called optimal solution and/or optimal design. However, in multi-objective optimization problems, since a different objective value is generated against each decision vector, the superiority of the solutions over each other is determined by considering the trade-off among the objective values. Therefore, the solution of multi-objective optimization problems, unlike single-objective problems, is a set of vectors rather than a single decision vector. In multi-objective optimization problems, especially if there are intricate objectives, the computational cost of the problem increases. In other words, while synchronously trying to maximize one of the objectives and to minimize another one makes it difficult to find the global optimum design. One of the important techniques used in multi-objective optimization problems is Pareto optimality which enables to select the global optimum solution taking into account the trade-off among all objectives. In this context, using of derivative-based methods has decreased, but the use of metaheuristic methods has increased due to the rapid availability of global optimum solution. This is because the improvements in the field of optimization are progressing in proportion to technology and varying according to the needs. In this chapter, one of the recent metaheuristic optimization methods based on swarm intelligence that is so-called a Pareto-based bat algorithm inspired by the behavior of determining the direction and distance of an object using the echo of the sound called the echolocation of bats is used in order to obtain optimum solutions for multi-objective engineering design problems. In this regard, a four-bar planar truss, a real-sized welded steel beam as well as a multi-layer radar absorber are selected as multi-objective engineering design optimization problems. In case the obtained results (optimal designs) are examined, the potency and the reliability of the proposed multi-objective Pareto-based bat algorithm are demonstrated.

Parameter optimization of a static micro-mixer with a cantilever beam was accomplished for maximizing mixing efficiency using a sequential approximate optimization method. The objective function and design variables were chosen as mixing index, and the length and the angle measured from the horizontal of the cantilever beam, respectively. The Optimization problem of the mixer was considered as a series of sub-problems. Approximation to solve the sub-problems was performed by response surface methodology. To verify the reliability and the accuracy of the approximated objective function, ANOVA table and variable selection method were implemented, respectively. It was verified that the sequential approximate optimization method worked very well, and the mixing efficiency was significantly improved compared with the initial design.

The emigration genetic algorithm, which is a genetic-based algorithm, is proposed to obtain the Pareto optimal solution of vector optimal designs of electromagnetic devices. The proposed algorithm differs from the traditional ones in its design of an emigration operator as well as the inclusion of some useful approaches such as the fitness sharing, clustering, and elitism strategy. Detailed numerical results on three different multiobjective design problems are reported to demonstrate the effectiveness and advantages of the proposed algorithm for solving practical engineering multiobjective optimal design problems.