Expensive Multiobjective Optimization Research Articles

An Ensemble Surrogate-Based Framework for Expensive Multiobjective Evolutionary Optimization

Surrogate-assisted evolutionary algorithms (SAEAs) have become very popular for tackling computationally expensive multiobjective optimization problems (EMOPs), as the surrogate models in SAEAs can approximate EMOPs well, thereby reducing the time cost of the optimization process. However, with the increased number of decision variables in EMOPs, the prediction accuracy of surrogate models will deteriorate, which inevitably worsens the performance of SAEAs. To deal with this issue, this article suggests an ensemble surrogate-based framework for tackling EMOPs. In this framework, a global surrogate model is trained under the entire search space to explore the global area, while a number of surrogate submodels are trained under different search subspaces to exploit the subarea, so as to enhance the prediction accuracy and reliability. Moreover, a new infill sampling criterion is designed based on a set of reference vectors to select promising samples for training the models. To validate the generality and effectiveness of our framework, three state-of-the-art evolutionary algorithms [nondominated sorting genetic algorithm III (NSGA-III), multiobjective evolutionary algorithm based on decomposition with differential evolution (MOEA/D-DE) and reference vector-guided evolutionary algorithm (RVEA)] are embedded, which significantly improve their performance for solving most of the test EMOPs adopted in this article. When compared to some competitive SAEAs for solving EMOPs with up to 30 decision variables, the experimental results also validate the advantages of our approach in most cases.

A surrogate-assisted evolutionary algorithm with hypervolume triggered fidelity adjustment for noisy multiobjective integer programming

Although surrogate-assisted evolutionary algorithms (SAEAs) have been widely developed to address computationally expensive multi-objective optimization problems (MOPs), they still encounter difficulties in solving the expensive and noisy combinatorial MOPs. To this end, we propose a novel SAEA to handle this kind of problem. In the proposed algorithm, the averaging method is used to denoise. To balance the conflict between the time cost and the effect of noises, multi-fidelity surrogate models are constructed according to the averaged evaluation results. The number of independent repeated evaluations represents the fidelity level of surrogate models. In the optimization process, the hypervolume indicator is employed as a trigger to determine whether the fidelity level should be increased. In addition, a lightweight local search method, the semi-variable neighborhood search, is proposed to improve the global search efficiency of the proposed algorithm in discrete decision spaces. Experimental results show that our proposed algorithm achieves competitive performance on most benchmark problems.

High-dimensional expensive multi-objective optimization via additive structure

• A high-dimensional expensive multi-objective optimization method is proposed. • A multi-point sampling strategy to recommend multiple candidate solutions is introduced. • Additive Gaussian Structure in expensive multi-objective optimization in highdimensional cases is addressed. Expensive multi-objective problems (MOPs) are extremely challenging due to the high evaluation cost to find satisfying solutions with adequate precision, especially in high-dimensional cases. However, most of the current EGO-based algorithms for expensive MOPs are limited to low decision dimensions because of the exponential difficulty in high dimensional circumstances. This paper presents High-Dimensional Expensive Multi-objective Optimization with Additive structure (ADD-HDEMO) to solve high-dimensional expensive MOPs via additive structural kernel and identifies two key challenges in this endeavor. First, we integrate multiple sub-objectives in high-dimensional expensive MOPs into a single objective with the decision space unchanged. Then, we infer the dependence correlation between the decision and objective space of the augmented EMOP via an additive GP kernel structure where Gibbs sampling is used to learn the latent additive structure. Furthermore, we parallel the proposed algorithm by introducing a multi-point sampling mechanism when recommending infill points. The effectiveness of the proposed method is evaluated on ZDT and DTLZ benchmarks compared with three other EGO-based multi-objective optimization approaches, ParEGO, SMS-EGO and MOEA/D-EGO. Our analyses demonstrate that ADD-HDEMO is effective in solving high-dimensional expensive MOPs.

A diversity preservation method for expensive multi-objective combinatorial optimization problems using Novel-First Tabu Search and MOEA/D

Expensive multi-objective combinatorial optimization problems have constraints in the number of objective function evaluations due to time, financial, or resource restrictions. As most combinatorial problems, they are subject to a high number of duplicated solutions. Given the fact that expensive environments limit the number of objective function evaluations, the existence of duplicated solutions heavily impacts the optimization process due to poor diversity and low convergence speed. This paper proposes the Novel-First Tabu Search, a greedy-strategy mechanism that uses Knowledge-Assisted Local Search methods to preserve the population diversity and increase the exploration and exploitation ability of MOEA/D. Experiments are conducted on constrained, unconstrained, multimodal, deceptive, linear, convex, and non-convex Pareto Front multi-objective combinatorial optimization benchmark problems. This paper also conducts an experiment on the real-world, expensive problem of Well Placement Optimization using a benchmark case based on the Namorado oil field, located in the Campos Basin, Brazil. The experimental results and performance comparison with state-of-the-art algorithms demonstrate that the proposed design significantly preserves diversity and increases convergence without violating the constraint in the number of objective function evaluations.

An adaptive model switch-based surrogate-assisted evolutionary algorithm for noisy expensive multi-objective optimization

To solve noisy and expensive multi-objective optimization problems, there are only a few function evaluations can be used due to the limitation of time and/or money. Because of the influence of noises, the evaluations are inaccurate. It is challenging for the existing surrogate-assisted evolutionary algorithms. Due to the influence of noises, the performance of the surrogate model constructed by these algorithms is degraded. At the same time, noises would mislead the evolution direction. More importantly, because of the limitations of function evaluations, noise treatment methods consuming many function evaluations cannot be applied. An adaptive model switch-based surrogate-assisted evolutionary algorithm is proposed to solve such problems in this paper. The algorithm establishes radial basis function networks for denoising. An adaptive model switch strategy is adopted to select suited surrogate model from Gaussian regression and radial basis function network. It adaptively selects the sampling strategies based on the maximum improvement in the convergence, diversity, and approximation uncertainty to make full use of the limited number of function evaluations. The experimental results on a set of test problems show that the proposed algorithm is more competitive than the five most advanced surrogate-assisted evolutionary algorithms.

A bagging-based surrogate-assisted evolutionary algorithm for expensive multi-objective optimization

A bagging-based surrogate-assisted evolutionary algorithm for expensive multi-objective optimization

Expensive Multiobjective Evolutionary Optimization Assisted by Dominance Prediction

We propose a new surrogate-assisted evolutionary algorithm for expensive multiobjective optimization. Two classification-based surrogate models are used, which can predict the Pareto dominance relation and <inline-formula> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula>-dominance relation between two solutions, respectively. To make such surrogates as accurate as possible, we formulate dominance prediction as an imbalanced classification problem and address this problem using deep learning techniques. Furthermore, to integrate the surrogates based on dominance prediction with multiobjective evolutionary optimization, we develop a two-stage preselection strategy. This strategy aims to select a promising solution to be evaluated among those produced by genetic operations, taking proper account of the balance between convergence and diversity. We conduct an empirical study on a number of well-known multiobjective and many-objective benchmark problems, over a relatively small number of function evaluations. Our experimental results demonstrate the superiority of the proposed algorithm compared with several representative surrogate-assisted algorithms.

Expensive Multiobjective Optimization Algorithm Based on Equivariate Component Analysis

In order to reduce the cost of candidate solution evaluation in the process of solving expensive optimization problems, an expensive multi-objective optimization algorithm based on equivalence component analysis was proposed to study the influence of decision space equivalence components on the prediction accuracy of agent models. Based on the analysis of the equivalence of decision space attributes, a limit learning network based on the equivalence components was constructed for Pareto dominance prediction among candidate solutions. A multi-objective test problem with equivalent components was selected in Pareto dominance prediction experiments, the results of which showed that the algorithm can effectively improve the accuracy of Pareto dominance prediction among candidate solutions. Successively the candidate solutions were scored with multiple ELM (Extreme Learning Machine) models, selected for evaluation and updated, and integrated into the Pareto-based multi-objective evolutionary algorithm. Through comparative experiments on the test problem, the method could achieve a better Pareto approximation solution under the limitation of a limited number of evaluations, and the goal of reducing the cost of expensive multi-objective optimization calculations.

Adjusting normalization bounds to improve hypervolume based search for expensive multi-objective optimization

When solving expensive multi-objective optimization problems, surrogate models are often used to reduce the number of true evaluations. Based on predictions from the surrogate models, promising candidate solutions, also referred to as infill solutions, can be identified for evaluation to expedite the search towards the optimum. This infill process in turn involves optimization of certain criteria derived from the surrogate models. In this study, predicted hypervolume maximization is considered as the infill criterion for expensive multi/many-objective optimization. In particular, we examine the effect of normalization bounds on the performance of the algorithm building on our previous study on bi-objective optimization. We propose a more scalable approach based on “surrogate corner” search that shows improved performance where some of the conventional techniques face challenges. Numerical experiments on a range of benchmark problems with up to 5 objectives demonstrate the efficacy and reliability of the proposed approach.

An Expensive Multi-Objective Optimization Algorithm Based on Decision Space Compression

Numerous surrogate-assisted expensive multi-objective optimization algorithms were proposed to deal with expensive multi-objective optimization problems in the past few years. The accuracy of the surrogate models degrades as the number of decision variables increases. In this paper, we propose a surrogate-assisted expensive multi-objective optimization algorithm based on decision space compression. Several surrogate models are built in the lower dimensional compressed space. The promising points are generated and selected in the lower compressed decision space and decoded to the original decision space for evaluation. Experimental studies show that the proposed algorithm achieves a good performance in handling expensive multi-objective optimization problems with high-dimensional decision space.

Adaptive dropout for high-dimensional expensive multiobjective optimization

Various works have been proposed to solve expensive multiobjective optimization problems (EMOPs) using surrogate-assisted evolutionary algorithms (SAEAs) in recent decades. However, most existing methods focus on EMOPs with less than 30 decision variables, since a large number of training samples are required to build an accurate surrogate model for high-dimensional EMOPs, which is unrealistic for expensive multiobjective optimization. To address this issue, we propose an SAEA with an adaptive dropout mechanism. Specifically, this mechanism takes advantage of the statistical differences between different solution sets in the decision space to guide the selection of some crucial decision variables. A new infill criterion is then proposed to optimize the selected decision variables with the assistance of surrogate models. Moreover, the optimized decision variables are extended to new full-length solutions, and then the new candidate solutions are evaluated using expensive functions to update the archive. The proposed algorithm is tested on different benchmark problems with up to 200 decision variables compared to some state-of-the-art SAEAs. The experimental results have demonstrated the promising performance and computational efficiency of the proposed algorithm in high-dimensional expensive multiobjective optimization.

Expensive multi-objective optimization of electromagnetic mixing in a liquid metal

This paper presents a novel trust-region method for the optimization of multiple expensive functions. We apply this method to a biobjective optimization problem in fluid mechanics, the optimal mixing of particles in a flow in a closed container. The three-dimensional time-dependent flows are driven by Lorentz forces that are generated by an oscillating permanent magnet located underneath the rectangular vessel. The rectangular magnet provides a spatially non-uniform magnetic field that is known analytically. The magnet oscillation creates a steady mean flow (steady streaming) similar to those observed from oscillating rigid bodies. In the optimization problem, randomly distributed mass-less particles are advected by the flow to achieve a homogeneous distribution (objective function 1) while keeping the work done to move the permanent magnet minimal (objective function 2). A single evaluation of these two objective functions may take more than two hours. For that reason, to save computational time, the proposed method uses interpolation models on trust-regions for finding descent directions. We show that, even for our significantly simplified model problem, the mixing patterns vary significantly with the control parameters, which justifies the use of improved optimization techniques and their further development.

A Kriging Model-Based Expensive Multiobjective Optimization Algorithm Using R2 Indicator of Expectation Improvement

Most of the multiobjective optimization problems in engineering involve the evaluation of expensive objectives and constraint functions, for which an approximate model-based multiobjective optimization algorithm is usually employed, but requires a large amount of function evaluation. Aiming at effectively reducing the computation cost, a novel infilling point criterion EIR2 is proposed, whose basic idea is mapping a point in objective space into a set in expectation improvement space and utilizing the R2 indicator of the set to quantify the fitness of the point being selected as an infilling point. This criterion has an analytic form regardless of the number of objectives and demands lower calculation resources. Combining the Kriging model, optimal Latin hypercube sampling, and particle swarm optimization, an algorithm, EIR2-MOEA, is developed for solving expensive multiobjective optimization problems and applied to three sets of standard test functions of varying difficulty and comparing with two other competitive infill point criteria. Results show that EIR2 has higher resource utilization efficiency, and the resulting nondominated solution set possesses good convergence and diversity. By coupling with the average probability of feasibility, the EIR2 criterion is capable of dealing with expensive constrained multiobjective optimization problems and its efficiency is successfully validated in the optimal design of energy storage flywheel.

An adaptive Bayesian approach to surrogate-assisted evolutionary multi-objective optimization

Surrogate models have been widely used for solving computationally expensive multi-objective optimization problems (MOPs). The efficient global optimization (EGO) algorithm, a Bayesian approach to surrogate-assisted optimization, has become very popular in surrogate-assisted evolutionary optimization. In this paper, we propose an adaptive Bayesian approach to surrogate-assisted evolutionary algorithm to solve expensive MOPs. The main idea is to tune the hyperparameter in the acquisition function according to the search dynamics to determine which candidate solutions are to be evaluated using the expensive real objective functions. In addition, the sampling selection criterion switches between an angle based distance and an angle-penalized distance over the course of optimization to achieve a better balance between exploration and exploitation. The performance of the proposed algorithm is examined on a set of benchmark problems and an airfoil design optimization problem using a maximum of 300 real fitness evaluations. Our experimental results show that the proposed algorithm is competitive compared to four popular multi-objective evolutionary algorithms.

Deep neural networks compression learning based on multiobjective evolutionary algorithms

This work addresses the problem of deep neural network compression, which is a promising technique to reduce the number of network parameters and/or speed up the network evaluation process significantly. Most of the existing methods rely on domain experts’ experience for the selection of hyperparameters such as the rank and sparsity ratio of the weight matrix in order to get an appropriate compression result without serious performance downgrade. However, they usually suffer from heavy computational loads due to the large number of tests in revealing the best hyperparameters. In this work, we propose an efficient approach to network compression from the perspective of multiobjective evolution. The contributions in the paper are twofolds: (1) We build a multiobjective compression learning model that considers the model classification error rate and compression rate as two objectives in the optimization, which can provide a compromise of the tradeoffs between these two objectives. (2) A mechanism for approximate compressed model generation is devised in the framework of expensive multiobjective optimization, which is able to reduce the high model training costs involved in the optimization process. Experiments are carried out to confirm the superiority of the proposed algorithm.

Surrogate-assisted particle swarm optimization algorithm with Pareto active learning for expensive multi-objective optimization

For multi-objective optimization problems, particle swarm optimization &#x0028 PSO &#x0029 algorithm generally needs a large number of fitness evaluations to obtain the Pareto optimal solutions. However, it will become substantially time-consuming when handling computationally expensive fitness functions. In order to save the computational cost, a surrogate-assisted PSO with Pareto active learning is proposed. In real physical space &#x0028 the objective functions are computationally expensive &#x0029, PSO is used as an optimizer, and its optimization results are used to construct the surrogate models. In virtual space, objective functions are replaced by the cheaper surrogate models, PSO is viewed as a sampler to produce the candidate solutions. To enhance the quality of candidate solutions, a hybrid mutation sampling method based on the simulated evolution is proposed, which combines the advantage of fast convergence of PSO and implements mutation to increase diversity. Furthermore, &#x03B5 -Pareto active learning &#x0028 &#x03B5 -PAL &#x0029 method is employed to pre-select candidate solutions to guide PSO in the real physical space. However, little work has considered the method of determining parameter &#x03B5. Therefore, a greedy search method is presented to determine the value of where the number of active sampling is employed as the evaluation criteria of classification cost. Experimental studies involving application on a number of benchmark test problems and parameter determination for multi-input multi-output least squares support vector machines &#x0028 MLSSVM &#x0029 are given, in which the results demonstrate promising performance of the proposed algorithm compared with other representative multi-objective particle swarm optimization &#x0028 MOPSO &#x0029 algorithms.

Handling expensive multi-objective optimization problems with a cluster-based neighborhood regression model

This paper gives attention to multi-objective optimization in scenarios where objective function evaluation is expensive, that is, expensive multi-objective optimization. We firstly propose a cluster-based neighborhood regression model, which incorporates the linear regression technique to predict the descent direction and generate new potential offspring. Combining this model with the classical decomposition-based multi-objective optimization framework, we propose an efficient and effective algorithm for tackling computationally expensive multi-objective optimization problems. As opposed to the conventional approach of replacing the original time-consuming objective functions with the approximated ones obtained by surrogate model, the proposed algorithm incorporates the proposed regression model to serve as an operator producing higher-quality offspring so that the algorithm requires fewer iterations to reach a given solution quality. The proposed algorithm is compared with several state-of-the-art surrogate-assisted algorithms on a variety of well-known benchmark problems. Empirical results demonstrate that the proposed algorithm outperforms or is competitive with other peer algorithms, and has the ability to keep a good trade-off between solution quality and running time within a fairly small number of function evaluations. In particular, our proposed algorithm shows obvious superiority in terms of the computational time used for the algorithm components, and can obtain acceptable solutions for expensive problems with high efficiency.

Prediction of Pareto Dominance Using an Attribute Tendency Model for Expensive Multi-Objective Optimization

This paper proposes a novel surrogate-assisted multi-objective evolutionary algorithm, MOEA-ATCM, to solve expensive or black-box multi-objective problems with small evaluation budgets. The proposed approach encompasses a state-of-the-art MOEA based on a nondominated sorting genetic algorithm assisted by multi-fidelity optimization methods. A high-fidelity attribute tendency (AT) surrogate model was used to construct a linear decision space by introducing the knowledge of the objective space. A coarse model (CM) based on the AT model and correlation analyses of the objective functions and decision attributes were used to predict the Pareto dominance for candidates in the new decision space constructed by the AT model. Two major roles of MOEA-ATCM were identified: (1) the development of a new multi-fidelity surrogate-model-based method to predict Pareto dominance in a decision space that was then applied to MOEA, which does not need to dynamically update surrogate models in the optimization process and (2) the development of a Pareto dominance prediction method to obtain good nondominated solutions of expensive or black box problems with relatively few objective function evaluations. The advantages of MOEA-ATCM were verified by mathematical benchmark problems and a real-world multi-objective parameter optimization problem.

Wavelet multi-resolution approximation for multiobjective optimal control.

A new sequential method based on multi-resolution approximation is proposed for solving computationally expensive multi-objective optimization problems. A traditional strategy is to decompose a multi-objective optimization problem into a number of single-objective optimization problems, whereby the PF can be regarded as a function of weights. Therefore, it is very natural to use wavelet multi-resolution approximation techniques for setting weight vectors. In our framework, the sequential approach starts with sampling aggressive functions on the initial coarsest grid with a few collocation points; once a rough PF is obtained, new points are automatically added on the basis of an adaptive wavelet collocation method. Therefore, the PF can be approximated with a relatively small number of weights. The efficiency of our method is demonstrated on two examples: a typical multi-objective optimization problem and an expensive multi-objective control optimal problem.

A survey on handling computationally expensive multiobjective optimization problems with evolutionary algorithms

Evolutionary algorithms are widely used for solving multiobjective optimization problems but are often criticized because of a large number of function evaluations needed. Approximations, especially function approximations, also referred to as surrogates or metamodels are commonly used in the literature to reduce the computation time. This paper presents a survey of 45 different recent algorithms proposed in the literature between 2008 and 2016 to handle computationally expensive multiobjective optimization problems. Several algorithms are discussed based on what kind of an approximation such as problem, function or fitness approximation they use. Most emphasis is given to function approximation-based algorithms. We also compare these algorithms based on different criteria such as metamodeling technique and evolutionary algorithm used, type and dimensions of the problem solved, handling constraints, training time and the type of evolution control. Furthermore, we identify and discuss some promising elements and major issues among algorithms in the literature related to using an approximation and numerical settings used. In addition, we discuss selecting an algorithm to solve a given computationally expensive multiobjective optimization problem based on the dimensions in both objective and decision spaces and the computation budget available.