Archimedean copula-based estimation of distribution algorithm for multi-objective optimisation

Both Estimation of Distribution Algorithms (EDAs) and Copula Theory are hot topics in different research domains. The key of EDAs is modeling and sampling the probability distribution function which need much time in the available algorithms. Moreover, the modeled probability distribution function can not reflect the correct relationship between variables of the optimization target. Copula Theory provides a correlation between univariable marginal distribution functions and the joint probability distribution function. Therefore, Copula Theory could be used in EDAs. Because Archimedean copulas possess many nice properties, an EDA based on Archimedean copulas is presented in this paper. The experimental results show the effectiveness of the proposed algorithm.

Estimation of distribution algorithms are a promising method in evolutionary computation. In the estimation of distribution algorithms, instead of using conventional crossover and mutation operations, probabilistic models are used to sample the genetic information in the next population. Although the use of probabilistic models enables EDAs to have several schema simultaneously, it sometimes causes worse performance due to converse schemata. In this paper, estimation of distribution algorithms with niche separation mechanism is proposed. Coexistence schemata are also newly introduced, where schemata at the same loci but converse information. The proposed method splits population into two subpopulations if such coexistence schemata are found. One of subpopulations is used to constitute the next generation while the probabilistic model of another subpopulation is enqueued. The queued probabilistic model is used if the current population is converged. The experimental results on max-sat problems and Ising spin glass problems show the effectiveness of the proposed method.

Evolutionary algorithms and elliptical copulas applied to continuous optimization problems

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses \probability{proportional{to{size" (PPS) random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.

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.

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.

Estimation of Distribution Algorithms (EDAs) is new kinds of colony evolution algorithms. It produces its new generation by constructing probability distribution model through counting excellent information of individuals of present colony EDAs first, and then sampling the model. To solve the NP-Hard question as EDAs searching optimum network structure, a new Maximum Entropy Estimation of Distribution Algorithm (MEEDA) is proposed; and the paper puts forward the MEEDA to optimize two degree of freedom PID controller parameters. The simulation results show that the optimal 2-DOF PID controller has simultaneously both the characteristics of command tracking and disturbance rejection, so we can see the simulation verifies the effectiveness of the EDA algorithm.

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.

Markov random field (MRF) modelling techniques have been recently proposed as a novel approach to probabilistic modelling for estimation of distribution algorithms (EDAs). An EDA using this technique was called distribution estimation using Markov random fields (DEUM). DEUM was later extended to DEUMd. DEUM and DEUMd use a univariate model of probability distribution, and have been shown to perform better than other univariate EDAs for a range of optimization problems. This paper extends DEUM to use a bivariate model and applies it to the Ising spin glass problems. We propose two variants of DEUM that use different sampling techniques. Our experimental result show a noticeable gain in performance.

Estimation of distribution algorithms (EDAs) are a successful branch of evolutionary algorithms (EAs) that evolve a probabilistic model instead of a population. Analogous to genetic drift in EAs, EDAs also encounter the phenomenon that the random sampling in the model update can move the sampling frequencies to boundary values not justified by the fitness. This can result in a considerable performance loss. This work gives the first tight quantification of this effect for three EDAs and one ant colony optimizer, namely for the univariate marginal distribution algorithm, the compact genetic algorithm, population-based incremental learning, and the max-min ant system with iteration-best update. Our results allow to choose the parameters of these algorithms in such a way that within a desired runtime, no sampling frequency approaches the boundary values without a clear indication from the objective function. This paper for the Hot-off-the-Press track at GECCO 2020 summarizes the work Sharp Bounds for Genetic Drift in Estimation of Distribution Algorithms by B. Doerr and W. Zheng, which has been accepted for publication in the IEEE Transactions on Evolutionary Computation [5].

In this paper, a dynamic immune optimization algorithm with constraints (DCIOA), based on adaptive memory and dynamic recognition functions of artificial systems, was proposed to deal with Knapsack problem with constraints in dynamic environments. A novel dynamic stochastic ranking strategies is used to select excellence antibodies, meanwhile, infeasible antibodies participate in evolution of population; Improving the searching functions utilizes repairing method to remedy infeasible antibodies, and make sure the rate of feasible antibody in current environmental population; Environmental memory pools are constructed to store memory cells, at the same time, environmental recognition operator is designed to examine the environments changing over time, the initial population of similar or same environments are generated via introducing some memory cells into the current population, which accelerates the DCIOA's convergence. In numerical experiments, four well-known dynamic evolutionary algorithms are selected to compare with the DCIOA by three groups of dynamic high dimensional knapsack problems. The results indicate that the DCIOA shows a promising convergence capability, meanwhile, we will also show that our algorithm improves its response over time, a kind of secondary response present in the immune system, and can track more rapidly the optimum in similar environments and requires less time than the other algorithms proposed in literature.

We present an improved estimation of distribution algorithm (EDA) with a new sampling probability model (SPM) to deal with the multiobjective optimization design problems of electromagnetic devices. The new SPM is constructed and updated based on the sequential optimization method. Its sampling efficiency is better than traditional methods and its sample includes more information about the optimization problem. Finally, by investigating a test function and the TEAM Workshop problem 22, we can see that the optimization efficiency of multiobjective EDA can be improved a lot by the new SPM.

This work proposes a linkage-learning niching method that improves the capability of estimation of distribution algorithms (EDAs) on reducing spurious linkages which increase problems difficulty. Concatenated parity function (CPF), a class of allelic pairwise independent problems, causes exponential scalability for hierarchical Bayesian optimization algorithm (hBOA), which is one of powerful EDAs. Empirical results show that restricted tournament replacement (RTR) that hBOA employs results in spurious linkages and increases difficulty on solving CPF. Our research consists of these goals: (1) proposing a mutual information matrix to approximate the implicit linkage-information during EDAs' execution, (2) reducing spurious linkages by utilizing new metric of similarity, and (3) maintaining diversity of population. The results show that hBOA with our proposed niching method reduces the spurious linkages and solves CPF in the polynomial time.

Joint-life insurance pays a sum insured when the first death occurs. This insurance has a case based on the order of exit from the cohort, namely joint life and last survivor. The former means that one of the insured leaves the cohort, while the latter means the last member of the insured has left his or her cohort. For some reasons of simplicity, the insurance premium is usually calculated with the assumption that the husband and wife are mutually independent. However, this assumption is considered unrealistic. Couples are open to the same risks, hence explaining joint survival model should involve dependence structures between the distribution of spouse mortality. In line with this, to understand the dependence structure of multiple random variables, the approach used is Copula. In this context, Copula relates the marginal distribution function of these variables to the joint life distribution. One of the advantages from Copula is that the random variables do not have to come from the same distribution, hence Copula is considered good enough to explain the dependence of the mortality rate between husband and wife. This study aimed to develop a joint survival model for calculating joint life insurance premiums using the concept of Archimedean Copula to discover the minimum premium value by conducting the following steps: first, identifying the marginal distributions of mortality for genders using Indonesian Mortality Table IV (TMI/Tabel Mortalitas Indonesia IV); second, Archimedean copula function-based constructing survival models that captures the relationship between these variables; third, setting dependency parameter θ; fourth, calculating the joint life premium using Archimedean copula based survival modeled for each correlation dependency level; and carrying out optimization to find the minimum premium value. This can be achieved by formulating the problem as an optimization problem, considering an objective function that yields the lowest premium till satisfying the financial requirements of the insurance company.