Covariance matrix adaptation for multi-objective optimization

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most powerful evolutionary algorithms for real-valued single-objective optimization. In this paper, we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist CMA-ES using plus-selection and step size control based on a success rule. This algorithm is compared to the standard CMA-ES. The elitist CMA-ES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in sub-optimal local minima. In the new multi-objective CMAES (MO-CMA-ES) a population of individuals that adapt their search strategy as in the elitist CMA-ES is maintained. These are subject to multi-objective selection. The selection is based on non-dominated sorting using either the crowding-distance or the contributing hypervolume as second sorting criterion. Both the elitist single-objective CMA-ES and the MO-CMA-ES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMA-ES. The benefits of the new MO-CMA-ES in comparison to the well-known NSGA-II and to NSDE, a multi-objective differential evolution algorithm, are experimentally shown.     

Automated parameterization of intermolecular pair potentials using global optimization techniques

In this work, different global optimization techniques are assessed for the automated development of molecular force fields, as used in molecular dynamics and Monte Carlo simulations. The quest of finding suitable force field parameters is treated as a mathematical minimization problem. Intricate problem characteristics such as extremely costly and even abortive simulations, noisy simulation results, and especially multiple local minima naturally lead to the use of sophisticated global optimization algorithms. Five diverse algorithms (pure random search, recursive random search, CMA-ES, differential evolution, and taboo search) are compared to our own tailor-made solution named CoSMoS. CoSMoS is an automated workflow. It models the parameters’ influence on the simulation observables to detect a globally optimal set of parameters. It is shown how and why this approach is superior to other algorithms. Applied to suitable test functions and simulations for phosgene, CoSMoS effectively reduces the number of required simulations and real time for the optimization task.     

基于螺旋模型的仿人机器人步态参数优化算法

步行运动是仿人机器人运动控制的关键环节之一.为了实现快速、稳定的步态,在协方差矩阵自适应进化策略(CMA-ES)的基础上,文中提出仿人机器人螺旋模型算法.在步行优化过程中,将优化任务先划分为3个子任务,按照优化目标分别挑选参数加入相应优化组,同时构建CMA-ES优化器.根据不同的学习目标设计每个CMA-ES优化器,在前一优化组优化结果基础上结合新的需求进行螺旋迭代优化,最终达到既定的学习目标,获得最佳参数值.文中算法应用在HfutEngine仿真3D球队中,机器人的相关步态测试数据显示算法效果较佳.     

A Differential Covariance Matrix Adaptation Evolutionary Algorithm for real parameter optimization

Hybridization in context to Evolutionary Computation (EC) aims at combining the operators and methodologies from different EC paradigms to form a single algorithm that may enjoy a statistically superior performance on a wide variety of optimization problems. In this article we propose an efficient hybrid evolutionary algorithm that embeds the difference vector-based mutation scheme, the crossover and the selection strategy of Differential Evolution (DE) into another recently developed global optimization algorithm known as Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES). CMA-ES is a stochastic method for real parameter (continuous domain) optimization of non-linear, non-convex functions. The algorithm includes adaptation of covariance matrix which is basically an alternative method of traditional Quasi-Newton method for optimization based on gradient method. The hybrid algorithm, referred by us as Differential Covariance Matrix Adaptation Evolutionary Algorithm (DCMA-EA), turns out to possess a better blending of the explorative and exploitative behaviors as compared to the original DE and original CMA-ES, through empirical simulations. Though CMA-ES has emerged itself as a very efficient global optimizer, its performance deteriorates when it comes to dealing with complicated fitness landscapes, especially landscapes associated with noisy, hybrid composition functions and many real world optimization problems. In order to improve the overall performance of CMA-ES, the mutation, crossover and selection operators of DE have been incorporated into CMA-ES to synthesize the hybrid algorithm DCMA-EA. We compare DCMA-EA with original DE and CMA-EA, two best known DE-variants: SaDE and JADE, and two state-of-the-art real optimizers: IPOP-CMA-ES (Restart Covariance Matrix Adaptation Evolution Strategy with increasing population size) and DMS-PSO (Dynamic Multi Swarm Particle Swarm Optimization) over a test-suite of 20 shifted, rotated, and compositional benchmark functions and also two engineering optimization problems. Our comparative study indicates that although the hybridization scheme does not impose any serious burden on DCMA-EA in terms of number of Function Evaluations (FEs), DCMA-EA still enjoys a statistically superior performance over most of the tested benchmarks and especially over the multi-modal, rotated, and compositional ones in comparison to the other algorithms considered here.     

Pseudospectral simulation of turbulent viscoelastic channel flow

The methodology and validation of direct numerical simulations of viscoelastic turbulent channel flow are presented here. Using differential constitutive models derived from kinetic and network theories, numerical simulations have demonstrated drag reduction for various values of the parameters, under conditions where there is a substantial increase in the extensional viscosity compared to the shear viscosity (Sureshkumar, Beris, Handler, Direct numerical simulation of turbulent channel flow of a polymer solution, Phys. Fluids 9 (1997) 743–755 and Dimitropoulos, Sureshkumar, Beris, Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction: effect of the variation of rheological parameters, J. Non-Newtonian Fluid Mech. 79 (1998) 433–468). In this work, new results pertaining to the Reynolds stress and the pressure are presented, and the convergence of the pseudospectral algorithm utilized in the simulations, as well as its parallel implementation, are discussed in detail. It is shown that the lack of mesh refinement, or the use of a larger value for the artificial stress diffusivity used to stabilize the conformation tensor evolution equations, introduce small quantitative errors which qualitatively have the effect of lowering the drag reduction capability of the simulated fluid. However, an insufficient size of the periodic computational domain can also introduce errors in certain cases, which albeit usually small, can qualitatively alter various features of the solution.     

Numerical modelling and passive flow control using porous media

The whole flow over a solid body covered by a porous layer is presented. The three main models used in the literature to compute efficiently the fluid flow are given: the reduction of the porous layer to a boundary condition, the coupling of Darcy equation with Navier-Stokes equations and the Brinkman-Navier-Stokes equations or the penalisation method. Numerical simulations on Cartesian grids using the latest model give easily accurate solutions of the flow around solid bodies with or without porous layers. Adding appropriate porous devices to the solid bodies, an efficient passive control of the two-dimensional incompressible flow is achieved. A strong regularisation of the flow is observed and a significant reduction of the vortex induced vibrations or the drag coefficient is obtained.     

Self-organizing nets for optimization

Given some optimization problem and a series of typically expensive trials of solution candidates sampled from a search space, how can we efficiently select the next candidate? We address this fundamental problem by embedding simple optimization strategies in learning algorithms inspired by Kohonen's self-organizing maps and neural gas networks. Our adaptive nets or grids are used to identify and exploit search space regions that maximize the probability of generating points closer to the optima. Net nodes are attracted by candidates that lead to improved evaluations, thus, quickly biasing the active data selection process toward promising regions, without loss of ability to escape from local optima. On standard benchmark functions, our techniques perform more reliably than the widely used covariance matrix adaptation evolution strategy. The proposed algorithm is also applied to the problem of drag reduction in a flow past an actively controlled circular cylinder, leading to unprecedented drag reduction.     

A Method for Handling Uncertainty in Evolutionary Optimization With an Application to Feedback Control of Combustion

We present a novel method for handling uncertainty in evolutionary optimization. The method entails quantification and treatment of uncertainty and relies on the rank based selection operator of evolutionary algorithms. The proposed uncertainty handling is implemented in the context of the covariance matrix adaptation evolution strategy (CMA-ES) and verified on test functions. The present method is independent of the uncertainty distribution, prevents premature convergence of the evolution strategy and is well suited for online optimization as it requires only a small number of additional function evaluations. The algorithm is applied in an experimental setup to the online optimization of feedback controllers of thermoacoustic instabilities of gas turbine combustors. In order to mitigate these instabilities, gain-delay or model-based ${cal H}_{infty} $ controllers sense the pressure and command secondary fuel injectors. The parameters of these controllers are usually specified via a trial and error procedure. We demonstrate that their online optimization with the proposed methodology enhances, in an automated fashion, the online performance of the controllers, even under highly unsteady operating conditions, and it also compensates for uncertainties in the model-building and design process.      

Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES)

This paper presents a novel evolutionary optimization strategy based on the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). This new approach is intended to reduce the number of generations required for convergence to the optimum. Reducing the number of generations, i.e., the time complexity of the algorithm, is important if a large population size is desired: (1) to reduce the effect of noise; (2) to improve global search properties; and (3) to implement the algorithm on (highly) parallel machines. Our method results in a highly parallel algorithm which scales favorably with large numbers of processors. This is accomplished by efficiently incorporating the available information from a large population, thus significantly reducing the number of generations needed to adapt the covariance matrix. The original version of the CMA-ES was designed to reliably adapt the covariance matrix in small populations but it cannot exploit large populations efficiently. Our modifications scale up the efficiency to population sizes of up to 10n, where n is the problem dimension. This method has been applied to a large number of test problems, demonstrating that in many cases the CMA-ES can be advanced from quadratic to linear time complexity.     

Control of flow using genetic algorithm for a circular cylinder executing rotary oscillation

We propose here a new approach to optimally control incompressible viscous flow past a circular cylinder for drag minimization by rotary oscillation. The flow at Re = 15000 is simulated by solving 2D Navier-Stokes equations in stream function-vorticity formulation. High accuracy compact scheme for space discretization and four stage Runge-Kutta scheme for time integration makes such simulation possible. While numerical solution for this flow field has been reported using a fast viscous-vortex method, to our knowledge, this has not been done at such a high Reynolds number by computing the Navier-Stokes equation before. The importance of scale resolution, aliasing problem and preservation of physical dispersion relation for such vortical flows of the used high accuracy schemes [Sengupta TK. Fundamentals of computational fluid dynamics. Hyderabad, India: University Press; 2004] is highlighted.For the dynamic problem, a novel genetic algorithm (GA) based optimization technique has been adopted, where solutions of Navier-Stokes equations are obtained using small time-horizons at every step of the optimization process, called a GA generation. Then the objective functions is evaluated that is followed by GA determined improvement of the decision variables. This procedure of time advancement can also be adopted to control such flows experimentally, as one obtains time-accurate solution of the Navier-Stokes equation subject to discrete changes of decision variables. The objective function - the time-averaged drag - is optimized using a real-coded genetic algorithm [Deb K. Multi-objective optimization using evolutionary algorithms. Chichester, UK: Wiley; 2001] for the two decision variables, the maximum rotation rate and the forcing frequency of the rotary oscillation. Various approaches to optimal decision variables have been explored for the purpose of drag reduction and the collection of results are self-consistent and furthermore match well with the experimental values reported in [Tokumaru PT, Dimotakis PE. Rotary oscillation control of a cylinder wake. J Fluid Mech 1991;224:77].     

Human evolutionary model: A new approach to optimization

The aim of this paper is to propose the Human Evolutionary Model (HEM) as a novel computational method for solving search and optimization problems with single or multiple objectives. HEM is an intelligent evolutionary optimization method that uses consensus knowledge from experts with the aim of inferring the most suitable parameters to achieve the evolution in an intelligent way. HEM is able to handle experts’ knowledge disagreements by the use of a novel concept called Mediative Fuzzy Logic (MFL). The effectiveness of this computational method is demonstrated through several experiments that were performed using classical test functions as well as composite test functions. We are comparing our results against the results obtained with the Genetic Algorithm of the Matlab’s Toolbox, Evolution Strategy with Covariance Matrix Adaptation (CMA-ES), Particle Swarm Optimizer (PSO), Cooperative PSO (CPSO), G3 model with PCX crossover (G3-PCX), Differential Evolution (DE), and Comprehensive Learning PSO (CLPSO). The results obtained using HEM outperforms the results obtained using the abovementioned optimization methods.     

不同宽高比的V型随行波表面减阻仿真研究

为获得内壁具有不同宽高比的V型随行波的管道模型的微观流场及减阻规律,使用计算机软件对各模型进行了不同速度下的一系列数值仿真计算。在建模过程中为减小计算量,采用了二维管道剖面代替三维管道模型的方法,并选用了更适于旋转流和近壁湍流的湍流RNG K-ε模型进行计算。仿真结果表明：在流动中,管道内壁随行波产生有利于壁面减阻的旋涡,且旋涡疏密与随行波的宽高比有关。相同高度条件下,减阻效果会随着随行波宽高比的变大而增强。结果为减阻技术的工程化应用提供了重要依据。     

A modified Covariance Matrix Adaptation Evolution Strategy with adaptive penalty function and restart for constrained optimization

In the last decades, a number of novel meta-heuristics and hybrid algorithms have been proposed to solve a great variety of optimization problems. Among these, constrained optimization problems are considered of particular interest in applications from many different domains. The presence of multiple constraints can make optimization problems particularly hard to solve, thus imposing the use of specific techniques to handle fitness landscapes which generally show complex properties. In this paper, we introduce a modified Covariance Matrix Adaptation Evolution Strategy (CMA-ES) specifically designed for solving constrained optimization problems. The proposed method makes use of the restart mechanism typical of most modern variants of CMA-ES, and handles constraints by means of an adaptive penalty function. This novel CMA-ES scheme presents competitive results on a broad set of benchmark functions and engineering problems, outperforming most state-of-the-art algorithms as for both efficiency and constraint handling.     

不确定条件下炼钢-连铸生产调度的两阶段鲁棒优化方法

根据炼钢-连铸生产过程的特点,建立一种考虑加工时间和运输时间不确定性的两阶段鲁棒优化模型,即在第1阶段确定排序和指派变量,在第2阶段确定时间变量.针对两阶段鲁棒优化问题的复杂性和非线性难点,运用线性对偶理论将其转换为最差场景下的网络优化问题.针对简化后的网络优化问题,提出一种基于协方差自适应进化策略(covariance matrix adaptation evolution strategy,CMA-ES)的求解算法,并引入基于瓶颈浇次的重启策略以提升其搜索效率.最后,基于不同规模的测试实例进行模型灵敏度分析及算法对比测试.计算和统计结果验证了所提出的调度模型在不确定性条件下的有效性及改进CMA-ES算法的竞争性.     

How to Maximize the Likelihood Function for a DSGE Model

This paper extends two optimization routines to deal with objective functions for DSGE models. The optimization routines are (1) a version of Simulated Annealing developed by Corana A, Marchesi M, Ridella (ACM Trans Math Softw 13(3):262–280, 1987), and (2) the evolutionary algorithm CMA-ES developed by Hansen, Müller, Koumoutsakos (Evol Comput 11(1), 2003). Following these extensions, we examine the ability of the two routines to maximize the likelihood function for a sequence of test economies. Our results show that the CMA-ES routine clearly outperforms Simulated Annealing in its ability to find the global optimum and in efficiency. With ten unknown structural parameters in the likelihood function, the CMA-ES routine finds the global optimum in 95% of our test economies compared to 89% for Simulated Annealing. When the number of unknown structural parameters in the likelihood function increases to 20 and 35, then the CMA-ES routine finds the global optimum in 85 and 71% of our test economies, respectively. The corresponding numbers for Simulated Annealing are 70 and 0%.     

Analysis and control of the near-wake flow over a square-back geometry

A 3D numerical simulation, based on the Lattice Boltzmann method is carried out on the near-wake flow behind a generic square-back blunt body to analyze and establish a method to control the near-wake flow. The flow topology is described by the velocity and the pressure fields. The influence of the wake vortices on the aerodynamic drag is clarified and quantified. In order to reduce this drag, an active open-loop flow control is applied by continuous blowing devices distributed around the base periphery. The blowing effect on the behind body flow is a reduction of the wake section and of the total pressure loss in the wake and an increase of the static pressure on the base of the square body. This control leads to a significant drag reduction of ΔC_x = −29% with a blowing velocity of 1.5V₀. The efficiency is then studied, and we found that the most efficient control is obtained for a blowing velocity of 0.5V₀ and a jet angle of 45°. In this case, a 20% drag reduction is obtained, and the energy needed to control the system is seven times lower than the energy saved by the control.     

基于进化策略有频率禁区的结构优化设计方法

有频率禁区的桁架结构优化设计是在结构保证静态强度的前提下,通过调整构件的截面或节点坐标来改变结构的动力特性,从而避开激振频率带宽。自适应协方差矩阵进化策略（CMA-ES）算法是一种寻优效率高、鲁棒性好的全局优化算法,对处理复杂的非线性多维度的优化问题有很好的适应性。在考虑工艺可行性的基础上,结合有限元分析软件,提出了基于CMA-ES算法的有频率禁区的桁架结构优化设计方法。算例研究表明,该方法是可行的,与传统优化方法、粒子群优化方法相比较,具有全局寻优性能好、效率高的优点。     

Towards inverse modeling of turbidity currents: The inverse lock-exchange problem

A new approach is introduced for turbidite modeling, leveraging the potential of computational fluid dynamics methods to simulate the flow processes that led to turbidite formation. The practical use of numerical flow simulation for the purpose of turbidite modeling so far is hindered by the need to specify parameters and initial flow conditions that are a priori unknown. The present study proposes a method to determine optimal simulation parameters via an automated optimization process. An iterative procedure matches deposit predictions from successive flow simulations against available localized reference data, as in practice may be obtained from well logs, and aims at convergence towards the best-fit scenario. The final result is a prediction of the entire deposit thickness and local grain size distribution. The optimization strategy is based on a derivative-free, surrogate-based technique. Direct numerical simulations are performed to compute the flow dynamics. A proof of concept is successfully conducted for the simple test case of a two-dimensional lock-exchange turbidity current. The optimization approach is demonstrated to accurately retrieve the initial conditions used in a reference calculation.     

A covariance matrix adaptation evolution strategy variant and its engineering application

This paper proposes a novel covariance matrix adaptation evolution strategy (CMA-ES) variant, named AEALSCE, for single-objective numerical optimization problems in the continuous domain. To avoid premature convergence and strengthen the exploration capacity of the basic CMA-ES, AEALSCE is obtained by integrating the CMA-ES with two strategies that can adjust the evolutionary directions and enrich the population diversity. The first strategy is named the anisotropic eigenvalue adaptation (AEA) technique, which adapts the search scope towards the optimal evolutionary directions. It scales the eigenvalues of the covariance matrix anisotropically based on local fitness landscape detection. The other strategy is named the local search (LS) strategy, which is executed under the eigen coordinate system and can be subdivided into two parts. In the first part, the new candidates of superior solutions are sampled around the best solution to perform local exploration. In the other part, the new candidates of inferior solutions are generated using a modified mean point along the fitness descent direction. The proposed AEALSCE algorithm is compared with other top competitors, including the CEC 2014 champion, L-SHADE, and the promising NBIPOP-aCMA-ES, by benchmarking the CEC 2014 testbed. Moreover, AEALSCE is applied in solving three constrained engineering design problems and parameter estimation of photovoltaic (PV) models. According to the statistical results of the experiments, our proposed AEALSCE is competitive with other algorithms in convergence efficiency and accuracy. AEALSCE benefits from a good balance of exploration and exploitation, and it exhibits a potential to address real-world optimization problems.     

MHD flow past a circular cylinder using the immersed boundary method

The immersed boundary method (IB hereafter) is an efficient numerical methodology for treating purely hydrodynamic flows in geometrically complicated flow-domains. Recently Grigoriadis et als. [1] proposed an extension of the IB method that accounts for electromagnetic effects near non-conducting boundaries in magnetohydrodynamic (MHD) flows. The proposed extension (hereafter called MIB method) integrates naturally within the original IB concept and is suitable for magnetohydrodynamic (MHD) simulations of liquid metal flows. It is based on the proper definition of an externally applied current density field in order to satisfy the Maxwell equations in the presence of arbitrarily-shaped, non-conducting immersed boundaries. The efficiency of the proposed method is achieved by fast direct solutions of the two poisson equations for the hydrodynamic pressure and the electrostatic potential.The purpose of the present study is to establish the performance of the new MIB method in challenging configurations for which sufficient details are available in the literature. For this purpose, we have considered the classical MHD problem of a conducting fluid that is exposed to an external magnetic field while flowing across a circular cylinder with electrically insulated boundaries. Two- and three-dimensional, steady and unsteady, flow regimes were examined for Reynolds numbers Re_d ranging up to 200 based on the cylinder’s diameter. The intensity of the external magnetic field, as characterized by the magnetic interaction parameter N, varied from N=0 for the purely hydrodynamic cases up to N=5 for the MHD cases. For each simulation, a sufficiently fine Cartesian computational mesh was selected to ensure adequate resolution of the thin boundary layers developing due to the magnetic field, the so called Hartmann and sidewall layers. Results for a wide range of flow and magnetic field strength parameters show that the MIB method is capable of accurately reproducing integral parameters, such as the lift and drag coefficients, as well as the geometrical details of the recirculation zones. The results of the present study suggest that the proposed MIB methodology provides a powerful numerical tool for accurate MHD simulations, and that it can extend the applicability of existing Cartesian flow solvers as well as the range of computable MHD flows. Moreover, the new MIB method has been used to carrry out a series of accurate simulations allowing the determination of asymptotic laws for the lift and drag coefficients and the extent of the recirculation length as a function of the amplitude of the magnetic field. These results are reported herein.