Expensive Multiobjective Optimization Problems Research Articles

A New Multitask Joint Learning Framework for Expensive Multi-Objective Optimization Problems

A New Multitask Joint Learning Framework for Expensive Multi-Objective Optimization Problems

Hypervolume-Guided Decomposition for Parallel Expensive Multiobjective Optimization

The hypervolume metric is widely used to guide the search in multiobjective optimization. However, in parallel expensive multiobjective optimization, the hypervolume-based multi-point expected improvement (EI) suffers from high computational overhead and scales poorly with the batch size. To address this issue, we integrate hypervolume-based EI with the MOEA/D framework and propose a novel EI, named the expected direction-based hypervolume improvement (DirHV-EI). The DirHV-EI only measures the hypervolume improvement within each axis-parallel box induced by the modified Tchebycheff scalarization. Thus, it has a simple analytical expression that can be easily computed. Theoretical analysis indicates that the maximization of our proposed improvement function can help to maximize both the weighted hypervolume and the Tchebycheff improvement metrics. Using DirHV-EI, we design a decomposition-based Bayesian optimization algorithm, called DirHV-EGO, for solving expensive multiobjective optimization problems. At each iteration, the MOEA/D is used to maximize the DirHV-EI values with respect to a number of direction vectors in a collaborative manner, and a number of candidate solutions can be obtained. Then, a submodularity-based greedy selection strategy is used to select multiple query points from the candidates. Experimental results on both benchmark instances and real-world problems show that our proposed algorithm is an efficient and effective method for parallel expensive multiobjective optimization.

Are You Concerned about Limited Function Evaluations: Data-Augmented Pareto Set Learning for Expensive Multi-Objective Optimization

Optimizing multiple conflicting black-box objectives simultaneously is a prevalent occurrence in many real-world applications, such as neural architecture search, and machine learning. These problems are known as expensive multi-objective optimization problems (EMOPs) when the function evaluations are computationally or financially costly. Multi-objective Bayesian optimization (MOBO) offers an efficient approach to discovering a set of Pareto optimal solutions. However, the data deficiency issue caused by limited function evaluations has posed a great challenge to current optimization methods. Moreover, most current methods tend to prioritize the quality of candidate solutions, while ignoring the quantity of promising samples. In order to tackle these issues, our paper proposes a novel multi-objective Bayesian optimization algorithm with a data augmentation strategy that provides ample high-quality samples for Pareto set learning (PSL). Specifically, it utilizes Generative Adversarial Networks (GANs) to enrich data and a dominance prediction model to screen out high-quality samples, mitigating the predicament of limited function evaluations in EMOPs. Additionally, we adopt the regularity model to expensive multi-objective Bayesian optimization for PSL. Experimental results on both synthetic and real-world problems demonstrate that our algorithm outperforms several state-of-the-art and classical algorithms.

An XGBoost-assisted evolutionary algorithm for expensive multiobjective optimization problems

Many expensive optimization problems exist in various real-world applications. However traditional evolutionary algorithms are inadequate for solving these problems directly. Surrogate-assisted evolutionary algorithm (SAEA) can effectively solve expensive optimization problems using computationally inexpensive surrogate models. However, both the Kriging and ensemble models most SAEAs adopted have limited uncertainty of prediction, especially for expensive multiobjective optimization problems (EMOPs). To enhance the optimization performance of SAEA for EMOPs, this paper proposes a new XGBoost-assisted evolutionary algorithm, calling XGBEA. Specifically, XGBoost is used as the surrogate model, and a neighborhood density selection strategy based on a mixed population and archive space (NDS-MPA) is proposed to measure the uncertainties of individuals. XGBoost helps to best fit objective functions with different fitness landscapes. NDS-MPA selects non-dominated individuals with minimal density for re-evaluation, incorporating considerations of convergence, diversity and uncertainty. Experimental results on two well-studied benchmarks demonstrated the superiority of XGBEA over seven state-of-the-art SAEAs.

Evolutionary multiobjective optimization assisted by scalarization function approximation for high-dimensional expensive problems

Surrogate-assisted evolutionary algorithms (SAEAs) are a promising approach for solving expensive multiobjective optimization problems, but they often cannot address high-dimensional problems. Although one common approach to overcoming this challenge is to construct reliable surrogates, their accuracy inevitably deteriorates in a high-dimensional search space. Thus, this paper presents a novel SAEA based on scalarization function approximation, which is designed to strengthen its robustness against this deterioration. The proposed algorithm constructs an approximation model for each scalarization function defined in a decomposition-based framework. Each decomposed problem is then solved using multiple independent models trained for its neighbor problems. The intent is to decrease the risk of search performance degradations caused by unreliable approximations and retain the redundancy of the surrogate-assisted search to hedge the risk of over-fitting. Furthermore, each approximation model is adapted to a promising region of its corresponding decomposed problem to reduce the complexity of model fitting given a limited number of training samples. Experimental results show that the proposed algorithm is competitive with state-of-the-art SAEAs adapted for high-dimensional problems.

Evolutionary Optimization with Simplified Helper Task for High-dimensional Expensive Multiobjective Problems

In recent years, surrogate-assisted evolutionary algorithms (SAEAs) have been sufficiently studied for tackling computationally expensive multiobjective optimization problems (EMOPs), as they can quickly estimate the qualities of solutions by using surrogate models to substitute for expensive evaluations. However, most existing SAEAs only show promising performance for solving EMOPs with no more than 10 dimensions, and become less efficient for tackling EMOPs with higher dimensionality. Thus, this article proposes a new SAEA with a simplified helper task for tackling high-dimensional EMOPs. In each generation, one simplified task will be generated artificially by using random dimension reduction on the target task (i.e., the target EMOPs). Then, two surrogate models are trained for the helper task and the target task, respectively. Based on the trained surrogate models, evolutionary multitasking optimization is run to solve these two tasks, so that the experiences of solving the helper task can be transferred to speed up the convergence of tackling the target task. Moreover, an effective model management strategy is designed to select new promising samples for training the surrogate models. When compared to five competitive SAEAs on four well-known benchmark suites, the experiments validate the advantages of the proposed algorithm on most test cases.

Solving High-Dimensional Expensive Multiobjective Optimization Problems by Adaptive Decision Variable Grouping

Solving High-Dimensional Expensive Multiobjective Optimization Problems by Adaptive Decision Variable Grouping

Inverse distance weighting and radial basis function based surrogate model for high-dimensional expensive multi-objective optimization

Radial basis function (RBF) models have attracted a lot of attention in assisting evolutionary algorithms for solving computationally expensive optimization problems. However, most RBFs cannot directly provide the uncertainty information of their predictions, making it difficult to adopt principled infill sampling criteria for model management. To overcome this limitation, an inverse distance weighting (IDW) and RBF based surrogate assisted evolutionary algorithm, named IR-SAEA, is proposed to address high-dimensional expensive multi-objective optimization problems. First, an RBF-IDW model is developed, which can provide both the predicted objective values and the uncertainty of the predictions. Moreover, a modified lower confidence bound infill criterion is proposed based on the RBF-IDW for the balance of exploration and exploitation. Extensive experiments have been conducted on widely used benchmark problems with up to 100 dimensions. The empirical results have validated that the proposed algorithm is able to achieve a competitive performance compared with state-of-the-art SAEAs.

Surrogate-assisted evolutionary algorithm with decomposition-based local learning for high-dimensional multi-objective optimization

When the evolutionary algorithm is applied to handle high-dimensional expensive multi-objective optimization problems (MOPs), population evolution is crucial since it controls exploration and exploitation and decides if promising candidate solutions could be generated. However, little attention has been paid to this issue, evolution operators based on Genetic Algorithm (GA) and Differential Evolution (DE) are still the two most common approaches, whose convergence on high-dimensional MOPs with a limited number of fitness evaluations remains challenging. In this paper, we propose a decomposition-based local learning strategy to accelerate convergence in the high-dimensional search space of MOPs. Specifically, an individual is updated by learning from one of the best solutions of its corresponding local area based on the multi-objective decomposition approach. Accordingly, a surrogate-assisted evolutionary algorithm is proposed for better solving expensive high-dimensional MOPs. Experimental studies on MOPs with up to 100 decision variables and with 300 fitness evaluations demonstrate the effectiveness of the proposed method. Furthermore, we use the proposed method to solve a 2-objective shape design problem of the blended wing-body underwater glider (BWBUG) with 39 decision variables, and an impressive solution set is obtained.

A trust-region-like algorithm for expensive multi-objective optimization

In solving expensive multi-objective optimization problems, surrogate models have been widely investigated. The existing multi-objective algorithms adopting surrogate models can be classified into two categories: surrogate-assisted evolutionary algorithms (SAEAs) and surrogate-based optimization algorithms (SBAs). However, their efficiency and convergence remain considerably inadequate. In this work, we propose a trust-region-like algorithm for dealing with expensive multi-objective problems, where only a small number of expensive objective function evaluations are allowed. The optimization process of the proposed algorithm mainly includes two stages. The first stage is the surrogate-assisted stage, in which a promising population with the consideration of convergence and diversity is obtained by reference vector-guided evolutionary algorithms (RVEA). The promising population is considered the preliminary trust-region. Moreover, a new adaptive sampling selection criterion switching between two different sampling strategies is used to further narrow the trust-region. The second stage is the surrogate-based stage, wherein the selection of the most promising individuals is carried out from the ultimate trust-region using the pseudo hypervolume-based expected improvement matrix (PEIM) criterion. Comparison results on the benchmark functions demonstrate that the proposed algorithm is competitive with five state-of-art algorithms in a limited computational budget. Finally, the proposed algorithm is used in the design optimization of a variable stiffness composite cylinder.

A Steady-State Algorithm for Solving Expensive Multiobjective Optimization Problems With Nonparallelizable Evaluations

Expensive multi-objective optimization problems (EMOPs) refer to those wherein evaluation of each candidate solution incurs a significant cost. To solve such problems within a limited number of solution evaluations, surrogate-assisted evolutionary algorithms (SAEAs) are often used. However, existing SAEAs typically operate in a generational framework wherein multiple solutions are identified for evaluation in each generation. There exist relatively few proposals in steady-state framework, wherein only a single solution is evaluated in each iteration. The development of such algorithms is crucial to efficiently solve EMOPs for which the evaluation of candidate designs cannot be parallelized. Furthermore, regardless of the framework used, the performance of current SAEAs tends to degrade when the Pareto front (PF) of the problem has irregularities, such as extremely concave/convex segments, even for 2/3-objective problems. To contextualize the motivation of this study, the performance of a few state-of-the-art SAEAs is first demonstrated on some such selected problems. Then, to address the above research gaps, we propose a surrogate-assisted steady-state EA (SASSEA), which incorporates a number of novel elements including (a) effective use of model uncertainty information to aid the search, including the use of probabilistic dominance and Mahalanobis distance, (b) two-step infill identification using non-dominance (ND) and distance based selection, and (c) a shadow ND mechanism to avoid repeated selection and evaluation of dominated solutions. The efficacy of the proposed approach is demonstrated through extensive benchmarking on a range of test problems. It shows competitive performance relative to many state-of-the-art SAEAs, including both steady-state and generational approaches.

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

Numerous optimization problems in the real world involve multi-objective and computationally expensive simulations (i.e., expensive multi-objective optimization problems). This paper purposes a dual surrogates-assisted evolutionary algorithm (SAEA) based on parallel search, termed DSAEA-PS, for this issue. Approximation and classification are two main implementation forms of surrogate models, but the existing methods of expensive multi-objective optimization only apply one kind of them, and scarce works have paid attention to combining approximation and classification to improve the optimization performance. In the proposed algorithm, to enhance the prediction accuracy and reliability, both the approximation model and classification model are applied to cooperate to provide the quality and uncertainty information of candidate solutions. Meanwhile, the parallel search based on heterogeneous multi-objective evolutionary algorithms is introduced for better exploration of the decision space. In addition, combined with the strengthened dominance relation (SDR), a sampling strategy that comprehensively considers the quality of candidate solutions and their uncertainty information is proposed. Experimental results with five peer competitors on a set of widely-used benchmark problems demonstrate the ability of DSAEA-PS. Furthermore, DSAEA-PS is adopted for a five-objective blended-wing-body underwater glider design problem that involves time-consuming simulations of fluid dynamics and structural strength. A series of high-performance solutions obtained from DSAEA-PS verifies its effectiveness on engineering applications.

Surrogate-assisted expensive constrained Bi-objective optimization with highly heterogeneous evaluations

Expensive multiobjective optimization problem has been thoroughly studied in the past few years. However, one type of problem is often overlooked. Still, it is common in practice, whose partial target functions (objectives and constraints) are expensive that require approximations by surrogate models, while others have explicit formulas that can calculate immediately. To address this problem, a general representation is first defined as a highly heterogeneous optimization problem (HHOP) in this work. In terms of bi-objective constrained HHOP, an algorithm is proposed based on the Kriging surrogate model with a new exact 2-D expected hypervolume improvement (EHVI) calculation method and a new constraint handling approach considering both the expensive and inexpensive target functions. In EHVI calculation, the integral is only performed in the expensive objective dimension due to the deterministic of the inexpensive objective, resulting in the reduction of the time complexity and increase of the accuracy. Combining a multiobjective and a dominance-based constraint handling approaches, a new constraint handling strategy is proposed to deal with the constrained HHOP specifically. The effectiveness of the proposed EHVI calculation method and constraint handling strategy is verified by the benchmark comparison problems. whose results also indicate the superiority of the proposed algorithm compared with the other state-of-art algorithms. Furthermore, the proposed algorithm is implemented for a head sheave optimization problem to demonstrate its practicability in real-world problems.

A two-stage dominance-based surrogate-assisted evolution algorithm for high-dimensional expensive multi-objective optimization

In the past decades, surrogate-assisted evolutionary algorithms (SAEAs) have become one of the most popular methods to solve expensive multi-objective optimization problems (EMOPs). However, most existing methods focus on low-dimensional EMOPs because a large number of training samples are required to build accurate surrogate models, which is unrealistic for high-dimensional EMOPs. Therefore, this paper develops a two-stage dominance-based surrogate-assisted evolution algorithm (TSDEA) for high-dimensional EMOPs which utilizes the RBF model to approximate each objective function. First, a two-stage selection strategy is applied to select individuals for re-evaluation. Then considering the training time of the model, proposing a novel archive updating strategy to limit the number of individuals for updating. Experimental results show that the proposed algorithm has promising performance and computational efficiency compared to the state-of-the-art five SAEAs.

Machine Learning Assisted Evolutionary Multi-Objective Optimization [Guest Editorial

Optimization and learning are two main paradigms of artificial intelligence in addressing complex real-world problems, with their respective focuses but frequently enhanced by each other. Evolutionary multi-objective optimization (EMO) algorithms are a family of nature-inspired algorithms widely used for solving multi-objective optimization problems (MOPs). Despite the great success achieved by the existing EMO algorithms, most of these algorithms have also encountered many challenges in terms of optimization performance and efficiency for solving complex MOPs such as large-scale MOPs, expensive MOPs, dynamic MOPs, and real-world MOPs. To cope with these challenges, there has been increasing interest in applying machine learning (ML) techniques to enhance the EMO algorithms. Specifically, ML techniques can be adopted to extract useful knowledge hidden in the data generated by EMO algorithms in the search process, which can be leveraged to assist different components in EMO algorithms in different ways, e.g., problem formulation, offspring generation, fitness evaluation, and/or environmental selection. These machine learning assisted components have substantially enhanced the ability of EMO algorithms in handling complex MOPs.

Expensive Multiobjective Optimization Based on Information Transfer Surrogate

Objective value estimation based on computationally efficient surrogate models is widely used to reduce the computational cost in solving expensive multiobjective optimization problems (MOPs). However, due to the scarcity of training data and the lack of data sharing between training tasks in a surrogate-based system, the estimation effectiveness of the surrogate models might not be satisfactory. In this study, we present a novel surrogate methodology based on information transfer to deal with this problem. Particularly, in the proposed framework, the objectives of an MOP that may have little apparent similarity or correlation are linearly mapped to a number of related tasks. Afterward, the related tasks are used to train a multitask Gaussian process (MTGP). MTGP expands the training data leading to more confident learning of the parameters of the model. The predicted values of the objective functions can be obtained by a reverse mapping from the learned MTGP model. In this way, the computational burden of the expensive objective functions of an MOP can be substantially reduced while maintaining good estimation accuracy. MTGP facilitates mutual information transfer across tasks, avoids learning from scratch for new tasks, and captures the underlying structural information between tasks. The proposed surrogate approach is merged into MOEA/D to address MOPs. Experimental tests under various scenarios indicate that the resultant algorithm outperforms other state-of-the-art surrogate-based multiobjective optimization algorithms.

Walsh-based surrogate-assisted multi-objective combinatorial optimization: A fine-grained analysis for pseudo-boolean functions

The aim of this paper is to study surrogate-assisted algorithms for expensive multiobjective combinatorial optimization problems. Targeting pseudo-boolean domains, we provide a fine-grained analysis of an optimization framework using the Walsh basis as a core surrogate model. The considered framework uses decomposition in the objective space, and integrates three different components, namely, (i) an inner optimizer for searching promising solutions with respect to the so-constructed surrogate, (ii) a selection strategy to decide which solution is to be evaluated by the expensive objectives, and (iii) the strategy used to setup the Walsh order hyper-parameter. Based on extensive experiments using two benchmark problems, namely bi-objective NK-landscapes and unconstrained binary quadratic programming problems (UBQP), we conduct a comprehensive in-depth analysis of the combined effects of the considered components on search performance, and provide evidence on the effectiveness of the proposed search strategies. As a by-product, our work shed more light on the key challenges for designing a successful surrogate-assisted multi-objective combinatorial search process.

Choose Appropriate Subproblems for Collaborative Modeling in Expensive Multiobjective Optimization.

In dealing with the expensive multiobjective optimization problem, some algorithms convert it into a number of single-objective subproblems for optimization. At each iteration, these algorithms conduct surrogate-assisted optimization on one or multiple subproblems. However, these subproblems may be unnecessary or resolved. Operating on such subproblems can cause server inefficiencies, especially in the case of expensive optimization. To overcome this shortcoming, we propose an adaptive subproblem selection (ASS) strategy to identify the most promising subproblems for further modeling. To better leverage the cross information between the subproblems, we use the collaborative multioutput Gaussian process surrogate to model them jointly. Moreover, the commonly used acquisition functions (also known as infill criteria) are investigated in this article. Our analysis reveals that these acquisition functions may cause severe imbalances between exploitation and exploration in multiobjective optimization scenarios. Consequently, we develop a new acquisition function, namely, adaptive lower confidence bound (ALCB), to cope with it. The experimental results on three different sets of benchmark problems indicate that our proposed algorithm is competitive. Beyond that, we also quantitatively validate the effectiveness of the ASS strategy, the CoMOGP model, and the ALCB acquisition function.

An approach for computationally expensive multi-objective optimization problems with independently evaluable objectives

Multi-objective optimization problems involve simultaneous optimization of two or more objectives in conflict. For example, in automotive design, one might be interested in simultaneously minimizing the aerodynamic drag and maximizing the torsional rigidity/collision strength of the vehicle. For a number of problems encountered in engineering design, the objectives can be independently evaluated and such evaluations are often computationally expensive. While surrogate-assisted optimization (SAO) methods are typically used to deal with such problems, they evaluate all objectives for the chosen infill solution(s). If however the objectives can be independently evaluated, there is an opportunity to improve the computational efficiency by evaluating the selected objective(s) only. In this study, we introduce a SAO approach capable of selectively evaluating the objective(s) of the infill solution(s). The approach exploits principles of non-dominance and sparse subset selection to facilitate decomposition and identifies the infill solutions through maximization of probabilistic dominance measure. Thereafter, for each of these infill solutions, one or more objectives are evaluated, taking into account the evaluation status of its closest neighbor and the probability of improvement along each objective. The performance of the approach is benchmarked against state-of-the-art methods on a range of mathematical problems to highlight the efficacy of the approach. Thereafter, we present the performance on two engineering design problems namely a vehicle crashworthiness design problem and an airfoil design problem. We hope that this study would motivate further algorithmic developments to cater to such classes of problems.

A data augmentation based Kriging-assisted reference vector guided evolutionary algorithm for expensive dynamic multi-objective optimization

Expensive dynamic multi-objective optimization problems (EXDMOPs) involve multiple objective functions changing over time steps. In this kind of problems, only a small number of function evaluations can be allowed in each time step. The challenge of EXDMOP is how to quickly and accurately track the changing optimal solutions with only a small number of function evaluations. Although EXDMOP can be solved as a series of independent expensive multi-objective optimization problems or as an ordinary dynamic multi-objective optimization problems, these two methods either ignore the connection between different time steps or degrade the performance significantly due to the requirement for a large number of function evaluations. In this paper, we incorporate two proposed strategies named training data augmentation and sampling pool expansion to the Kriging-assisted reference vector guided evolutionary algorithm to construct a novel algorithm for solving EXDMOPs. The training data augmentation strategy tries to look for the previous similar time step, and adds the solutions evaluated by the objective functions at that time step to current time step. The accuracies of the Kriging models are improved in this way. The sampling pool expansion strategy sends the predicted optimal solutions to the competition of the function evaluations assignment at each time step, where the predicted optimal solutions are generated by the improved population prediction strategy. We set the ablation experiment for the proposed algorithm and compare it with four dynamic multi-objective evolutionary algorithms on various of problems. The experiment results validate the effectiveness of the proposed two strategies. It also indicates that the proposed algorithm is promising for solving EXDMOPs.