Are You Concerned about Limited Function Evaluations: Data-Augmented Pareto Set Learning for 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.

Expensive multiobjective optimization problems (EMOPs) are complex optimization problems exacted from real-world applications, where each objective function evaluation (FE) involves expensive computations or physical experiments. Many surrogate-assisted evolutionary algorithms (SAEAs) have been designed to solve EMOPs. Nevertheless, EMOPs with large-scale decision variables remain challenging for existing SAEAs, leading to difficulties in maintaining convergence and diversity. To address this deficiency, we proposed a variable reconstruction-based SAEA (VREA) to balance convergence enhancement and diversity maintenance. Generally, a cluster-based variable reconstruction strategy reconstructs the original large-scale decision variables into low-dimensional weight variables. Thus, the population can be rapidly pushed towards the Pareto set (PS) by optimizing low-dimensional weight variables with the assistance of surrogate models. Population diversity is improved due to the cluster-based variable reconstruction strategy. An adaptive search step size strategy is proposed to balance exploration and exploitation further. Experimental comparisons with four state-of-the-art SAEAs are conducted on benchmark EMOPs with up to 1000 decision variables and an aerodynamic design task. Experimental results demonstrate that VREA obtains well-converged and diverse solutions with limited real FEs.

A dual surrogate assisted evolutionary algorithm based on parallel search for expensive multi/many-objective optimization

Accelerating surrogate assisted evolutionary algorithms for expensive multi-objective optimization via explainable machine learning

Computationally expensive multiobjective optimization problems are difficult to solve using solely evolutionary algorithms (EAs) and require surrogate models, such as the Kriging model. To solve such problems efficiently, we propose infill criteria for appropriately selecting multiple additional sample points for updating the Kriging model. These criteria correspond to the expected improvement of the penalty-based boundary intersection (PBI) and the inverted PBI. These PBI-based measures are increasingly applied to EAs due to their ability to explore better nondominated solutions than those that are obtained by the Tchebycheff function. In order to add sample points uniformly in the multiobjective space, we assign territories and niche counts to uniformly distributed weight vectors for evaluating the proposed criteria. We investigate these criteria in various test problems and compare them with established infill criteria for multiobjective surrogate-based optimization. Both proposed criteria yield better diversity and convergence than those obtained with other criteria for most of the test problems.

The development of multi-objective evolutionary algorithms (MOEAs) assisted by surrogate models has significantly increased in the last few years. However, in realworld applications, the high modality and dimensionality that functions normally have, often causes problems to such models. Therefore, if the Pareto optimal set of a multi-objective optimization problem is located in a search space in which the surrogate model is not able to shape the corresponding region, the search could be misinformed and thus converge to wrong regions. This has motivated the idea of incorporating refinement mechanisms to such approaches, in order to improve the search. In this paper, we present a local search mechanism which improves the search of a MOEA assisted by surrogate models. Our preliminary results indicate that our proposed approach can produce good quality results when it is restricted to performing only between 1,000 and 5,000 fitness function evaluations. Our proposed approach is validated using a set of standard test problems and an airfoil design problem.

Surrogate-assisted evolutionary algorithms (SAEAs) have been widely developed to solve complex and computationally expensive multiobjective optimization problems (EMOPs) in recent years. However, when dealing with high-dimensional optimization problems in decision space, the performance of these surrogate-assisted multiobjective evolutionary algorithms (MOEAs) deteriorates drastically. In this work, a novel classifier-assisted rank-based learning and local model-based multiobjective evolutionary algorithm (CLMEA) is proposed for high-dimensional EMOPs. CLMEA makes full use of the uncertainty of solutions in the decision space and objective space to explore the uncertain but informative space toward high-dimensional problems. Specifically, the offspring in different ranks uses rank-based learning strategy to generate more promising and informative candidates for function evaluations (FEs). To reduce the search region of high-dimensional problems and maintain the diversity of solutions, the most uncertain sample point from the nondominated solutions measured by the crowding distance is selected as the center to conduct local search. The experimental results of benchmark problems and a real-world application on geothermal reservoir heat extraction optimization demonstrate superior performance of CLMEA compared with the state-of-the-art surrogate-assisted MOEAs. The source code for this work is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/JellyChen7/CLMEA</uri>

In this paper, we focus on expensive multiobjective optimization problems and propose a method to predict an approximation of the Pareto optimal set using classification of sampled decision vectors as dominated or nondominated. The performance of our method, called EPIC, is demonstrated on a set of benchmark problems used in the multiobjective optimization literature and compared with state-of the-art methods, ParEGO and PAL. The initial results are promising and encourage further research in this direction.

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.

Most of the current expensive multiobjective optimization (MOO) algorithms focus on identifying Pareto optimal solutions. However, in some applications such as multiobjective modular design, decision-makers often prefer a set of optimal solutions that share common components in the decision space, which may conflict with Pareto optimality. Existing expensive MOO algorithms are not specifically designed to address this preference. To bridge this gap, we propose modeling the component-sharing preference in MOO as a special bi-level multiobjective optimization problem. Specifically, the upper-level is a single-objective optimization problem that seeks the optimal shared variables, while the lower-level is a multiobjective optimization problem aimed at identifying trade-off solutions for given shared variable values. Moreover, the lower-level objective is expensive-to-evaluate and can only be evaluated for a limited number of times. To efficiently solve this problem, we introduce a data-efficient algorithm called Bayesian Bi-level Search (BBS). The effectiveness of BBS is validated through six new benchmark problems and a real-world application involving the planform shape design of Blended-Wing-Body underwater glider. The results show that our method effectively identifies solutions with shared components within limited computational budgets.

Practitioners often encounter computationally expensive multiobjective optimization problems to be solved in a variety of real-world applications. On the purpose of challenging these problems, we propose a new surrogate-based multiobjective optimization algorithm that does not require a large evaluation budget. It is called Multiobjective Tree-structured Parzen Estimator (MOTPE) and is an extension of the tree-structured Parzen estimator widely used to solve expensive single-objective optimization problems. Our empirical evidences reveal that MOTPE can approximate Pareto fronts of many benchmark problems better than existing methods with a limited budget. In this paper, we discuss furthermore the influence of MOTPE configurations to understand its behavior.

Due to the adversarial limitations between the output solutions in the multi-objective optimization stage, the energy efficiency optimization of large pumping stations has the disadvantage of difficult coordination between power consumption and efficiency. Therefore, a multi-objective Bayesian hyperparameter optimization algorithm design for energy efficiency optimization of large pumping stations has been carried out. A multi-objective optimization model for energy efficiency of large pumping stations was constructed by combining the input power of the main motor, transmission energy loss, transformer loss, and energy consumption of auxiliary equipment, with efficiency as the constraint. We extract sequence features that affect pump water demand, and use feature labels as hyperparameters for the multi-objective optimization model of energy efficiency in large pumping stations. We transform the multi-objective optimization problem into a hyperparameter optimization problem, avoiding adversarial output solutions. We iterate the query points collected by EI, so the output model reaches a convergent optimal solution. The test results show that the designed algorithm can improve the pumping efficiency of the pump station while controlling the power consumption of the pump station, achieving collaborative optimization of efficiency.

In multi-objective Bayesian optimization, an infill criterion is an important part, as it is the indicator to evaluate how much good a new set of solutions is, compared to a Pareto-front approximation set. This paper presents a deterministic algorithm for computing the Expected R2 Indicator for bi-objective problems and studies its use as an infill criterion in Bayesian Global Optimization. The R2-Indicator was introduced in 1998 by M. Hansen and A. Jaszkiewicz for performance assessment in multi-objective optimization and is more recently also used in indicator-based multi-criterion evolutionary algorithms (IBEAs). In Bayesian Global Optimization, we propose the Expected R2-indicator Improvement (ER2I) as an infill criterion. It is defined as the expected decrease of the R2 indicator by a point that is sampled from a predictive Gaussian distribution. The ER2I can also be used as a pre-selection criterion in surrogate-assisted IBEAs. It provides an alternative to the Expected Hypervolume-Indicator Improvement (EHVI) that requires a reference point, bounding the Pareto front from above. In contrast, the ER2I works with a utopian reference point that bounds the Pareto front from below. In addition, the ER2I supports preference modelling with utility functions and its computation time grows only linearly with the number of considered weight combinations. It is straightforward to approximate the ER2I by Monte Carlo Integration, but so far a deterministic algorithm to solve the non-linear integral remained unknown. We outline a deterministic algorithm for the computation of the bi-objective ER2I with Chebychev utility functions. Moreover, we study monotonicity properties of the ER2I w.r.t. parameters of the predictive distribution and numerical simulations demonstrate fast convergence to Pareto fronts of different shapes and the ability of the ER2I Bayesian optimization to fill gaps in the Pareto front approximation.

In some offline data-driven optimization problems, only small data can be gathered from real applications, which may decrease the reliability of surrogate models. To overcome the issues mentioned above, a data augmentation strategy based on generative adversarial networks (GANs) is adopted to improve the performance of data-driven multiobjective particle swarm optimization (DDMOPSO). Besides the fitting information of the original data, the distribution information is also considered to create synthetic data. The synthetic data is utilized to increase the accuracy of surrogate models. Therefore, a novel offline data-driven multiobjective particle swarm optimization based on data augmentation strategy (DDMOPSO-A) is proposed in this paper. The experiment results show that the proposed algorithm is superior to four competitive algorithms on seven benchmark problems.

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