Evolutionary algorithm assisted by surrogate model in the framework of ordinal optimization and optimal computing budget allocation

Embedding evolutionary strategy in ordinal optimization for hard optimization problems

<abstract><p>The training of artificial neural networks (ANNs) with rectified linear unit (ReLU) activation via gradient descent (GD) type optimization schemes is nowadays a common industrially relevant procedure. GD type optimization schemes can be regarded as temporal discretization methods for the gradient flow (GF) differential equations associated to the considered optimization problem and, in view of this, it seems to be a natural direction of research to <italic>first aim to develop a mathematical convergence theory for time-continuous GF differential equations</italic> and, thereafter, to aim to extend such a time-continuous convergence theory to implementable time-discrete GD type optimization methods. In this article we establish two basic results for GF differential equations in the training of fully-connected feedforward ANNs with one hidden layer and ReLU activation. In the first main result of this article we establish in the training of such ANNs under the assumption that the probability distribution of the input data of the considered supervised learning problem is absolutely continuous with a bounded density function that every GF differential equation admits for every initial value a solution which is also unique among a suitable class of solutions. In the second main result of this article we prove in the training of such ANNs under the assumption that the target function and the density function of the probability distribution of the input data are piecewise polynomial that every non-divergent GF trajectory converges with an appropriate rate of convergence to a critical point and that the risk of the non-divergent GF trajectory converges with rate 1 to the risk of the critical point. We establish this result by proving that the considered risk function is <italic>semialgebraic</italic> and, consequently, satisfies the <italic>Kurdyka-Łojasiewicz inequality</italic>, which allows us to show convergence of every non-divergent GF trajectory.</p></abstract>

Convergence rate analysis for optimal computing budget allocation algorithms

Contributors

Selecting a subset of the best solutions among large-scale problems is an important area of research. When the alternative solutions are stochastic in nature, then it puts more burden on the problem. The objective of this paper is to select a set that is likely to contain the actual best solutions with high probability. If the selected set contains all the best solutions, then the selection is denoted as correct selection. We are interested in maximizing the probability of this selection; P(CS). In many cases, the available computation budget for simulating the solution set in order to maximize P(CS) is limited. Therefore, instead of distributing these computational efforts equally likely among the alternatives, the optimal computing budget allocation (OCBA) procedure came to put more effort on the solutions that have more impact on the selected set. In this paper, we derive formulas of how to distribute the available budget asymptotically to find the approximation of P(CS). We then present a procedure that uses OCBA with the ordinal optimization (OO) in order to select the set of best solutions. The properties and performance of the proposed procedure are illustrated through a numerical example. Overall results indicate that the procedure is able to select a subset of the best systems with high probability of correct selection using small number of simulation samples under different parameter settings.

A hybrid differential evolution(DE) algorithm,namely OHTDE,is proposed to solve the stochastic flow shop scheduling problem(FSSP) with limited buffers between consecutive machines.Two different criteria are considered.The first is to minimize a general earliness/tardiness cost function and the second is to minimize the total completion time.The proposed algorithm is a hybrid of DE and two techniques:the optimal computing budget allocation(OCBA) technique and hypothesis test(HT) .DE is used to execute both global and local search.OCBA is utilized to reasonably allocate a limited sampling budget for every solution,assigning more for better individuals to improve the confidence level of getting good solutions under noise environment.HT is adopted to perform a statistical comparison on solutions performances so that the repeated search on the similar region of the solution space can be avoided to some extent.Moreover,a special crossover,which is performed on some good solutions identified by OCBA and HT,is designed to enhance the capability of local search.Furthermore,the stochastic convergence property of OHTDE is analyzed by using the theory of finite Markov chain.Experimental results and comparison show the effectiveness and robustness of the proposed OHTDE.

This research work deals with aspects concerned with delamination detection in composite structures as revealed by an approach based on vibration measurements. Variations in vibration characteristics generated in composite laminates indicate the existence of delaminations because degradation due to delamination causes reduction in flexural stiffness and strength of the material and as a result vibration parameters like natural frequency responses are changed. Hence, it is possible to monitor the variation in natural frequencies to identify the presence of delamination, and assess its size and location for online structural health monitoring (SHM). The approach to this paper, therefore, typically depends on undertaking the analysis of structural models implemented by finite element analysis (FEA). The numerical solutions using FE models known as the simulator computes the natural frequencies for the delaminated and undelaminated specimens of composite laminates. However, these FE models are computationally expensive, and surrogate (approximation) models are introduced to curtail the computational expense. The simulator is employed to solve the inverse problem using algorithms based on computational intelligence concepts. An artificial neural network (ANN) model is developed to also solve the inverse problem for delamination detection directly and to provide surrogate models integrated with optimization algorithms (the gradient-based local search and non-dominated sorting genetic algorithm-II) to contain the computationally expensive simulations by FEA. This approach is termed as surrogate assisted optimization and it is seen that the engagement of surrogate models in lieu of the FE models in the optimization loop greatly enhances the accuracy of delamination detection results within an affordable computational cost and provides control over handling different variables. Meanwhile, to aid with the building of effective surrogate models using substantial number of training datasets, K-means clustering algorithm is harnessed and this effectively reduces the large training datasets usually required for ANN training. This paper demonstrated that ANN and optimization algorithms with surrogates show immense potentialities for use in delamination damage detection scenarios. Prediction errors of the algorithms were quantified and they were shown to be satisfactory when applied to previously experimental data. The algorithms in their inverse formulations are capable of predicting accurately delamination parameters. Hence, these algorithms should be employed for application in the domain of SHM where their small computational requirements could be exploited for online damage detection.

Choosing suitable buffer setups for network-flow production lines of automated manufacturing systems to augment throughputs is a pragmatic issue. In this work, an approach merging artificial immune system (AIS) and ordinal optimization (OO) is developed to determine an optimal buffer resource allocation of a network-flow production line for maximizing the throughput. The proposed approach consists of two levels. The first level is using AIS assisted by a rough estimation to select a candidate subset of solutions. The second level is to identify an excellent solution among the solutions obtained from the first level using optimal computing budget allocation (OCBA) technique. We have tested the proposed approach on a three-stage ten-node network-flow production line and compared the test results with those obtained by three swarm intelligence methods with accurate estimation. Test results demonstrate that the proposed approach can obtain an excellent solution within a reasonable computing time and outperforms the three swarm intelligence methods.

Ordinal optimization is an efficient technique to choose and rank various engineering designs that require time-consuming discrete-event simulations. Optimal computing budget allocation (OCBA) has been an important tool to enhance its efficiency such that the best design is selected in a timely fashion. It, however, fails to address the issue of selecting the best and worst designs efficiently. The need to select both rapidly given a fixed computing budget has arisen from many applications. This work develops a new OCBA-based approach for selecting both best and worst designs at the same time. Its theoretical foundation is laid. Our numerical results show that it can well outperform all the existing methods in terms of probability of correct selection and computational efficiency.

Many of simulation based learning algorithms have been developed to obtain near optimal policies for Markov decision processes (MDPs) with large state space. However, most of them are for unichain problems. In view that some applications involve multichain processes and it is NP-hard to determine whether a MDP is unichain or not, it is desirable to obtain an algorithm that is applicable to multichain problems as well. This paper presents a rollout algorithm for multichain MDPs with average cost. Preliminary analysis of the estimation error and parameter settings are provided based on the problem structures, i.e., mixing time of transition matrix. Ordinal optimization and Optimal Computing Budget Allocation are also suggested to improve the efficiency of the algorithm.

This paper presents the development and evaluation of Artificial Neural Networks (ANN) based models and optimally selected surrogate models to provide the day-ahead forecast of the hourly-averaged energy load of buildings, by relating it to eight weather parameters as well as the hour of the day. Although ANN and other surrogate models have been used to predict building energy loads in the past, there is a limited understanding of what type of model prescriptions impact their performance as well as how un-recorded impact factors (e.g., human behavior and building repair work) should be accounted for. Here, the recorded energy data of three university buildings, from 9/2013–12/2015, is cleaned and synchronized with the local weather data. The data is then classified into eight classes; weekends and weekdays of Fall/Winter/Spring/Summer semesters. Both Multi-Layer Perceptron (MLP) and Radial Basis Function (RBF) NNs are explored. Differing number of hidden layers and transfer function choices are also explored, leading to the choice of the hyperbolic-tangent-sigmoid transfer function and 60 hidden layers. Similarly, an automated surrogate modeling framework is used to select the best models from among a pool of Kriging, RBF, and SVR models. A baseline concept, that uses energy information from the previous day as an added input to the ANN, helps to account for otherwise unrecorded recent changes in the building behavior, leading to improvement in fidelity of up to 30%.

A hierarchical multiscale off-road mobility model is enhanced through the development of an artificial neural network (ANN) surrogate model that captures the complex material behavior of deformable terrain. By exploiting the learning capability of neural networks, the incremental stress and strain relationship of granular terrain is predicted by the ANN representative volume elements (RVE) at various states of the stress and strain. A systematic training procedure for ANN RVEs is developed with a virtual tire test rig model for producing training data from the discrete-element (DE) RVEs without relying on computationally intensive full vehicle simulations on deformable terrain. The ANN surrogate RVEs are then integrated into the hierarchical multiscale computational framework as a lower-scale model with the scalable parallel computing capability, while the macroscale terrain deformation is described by the finite element (FE) approach. It is demonstrated with several numerical examples that off-road vehicle mobility performances predicted by the proposed FE-ANN multiscale terrain model are in good agreement with those of the FE-DE multiscale model while achieving a substantial computational time reduction. The accuracy and robustness of the ANN RVE for fine-grain sand terrain are discussed for scenarios not considered in training datasets. Furthermore, a drawbar pull test simulation is presented with the ANN RVE developed with data in the cornering scenario and validated against the full-scale vehicle test data. The numerical results confirm the predictive ability of the FE-ANN multiscale terrain model for off-road mobility simulations.

This paper presents a novel technique for the supervised training of feed-forward artificial neural networks (ANN) using the Harmony Search (HS) algorithm. HS is a stochastic meta-heuristic that is inspired from the improvisation process of musicians. Unlike Backpropagation, HS is non-trajectory driven. By modifying an existing improved version of HS & adopting a suitable ANN data representation, we propose a training technique where two of HS probabilistic parameters are determined dynamically based on the best-to-worst (BtW) harmony ratio in the current harmony memory instead of the improvisation count. This would be more suitable for ANN training since parameters and termination would depend on the quality of the attained solution. We have empirically tested and verified our technique by training an ANN with a benchmarking problem. In terms of overall training time and recognition, our results have revealed that our method is superior to both the original improved HS and standard Backpropagation.

Stochastic design optimization: Application to reacting flows

Although particle swarm optimization (PSO) is a powerful evolutionary algorithm for solving nonlinear optimization problems in deterministic environments, many practical problems have some stochastic noise. The optimal computing budget allocation (OCBA) has been integrated into PSO in various ways to cope with this. The OCBA can mitigate the effect of noise on PSO by selecting the best solution under a limited evaluation budget. Recently, with the increasing complexity of PSO applications, the evaluation costs are also increasing rapidly, which has sparked the need for more efficient PSO in stochastic environments. This article proposes a simple yet effective adjustment to the integration of OCBA to further improve the efficiency of PSO. The proposed adjustment allows OCBA to expand its search space to find the global best position more correctly such that the entire swarm can move on a better direction under stochastic noise. The experimental results on various benchmarks demonstrate the improved performance of PSO by the proposed adjustment under a limited budget compared with the latest studies. In addition, the results regarding fighters' evasion flight optimization emphasize the practical need for the proposed adjustment.