Balancing optimality and robustness is the key to solving expensive robust multiobjective optimization problems (ExRMOPs) by evolutionary algorithms. However, existing studies usually design algorithms based on either the average perspective or the worst perspective, overlooking the complementarity of these two perspectives-the former prefers optimality, whereas the latter prefers robustness. Therefore, this article proposes a Kriging-assisted evolutionary algorithm with dual perspectives and dual indicators (called KPI) to solve ExRMOPs. In KPI, we develop a dual-perspective aggregation function (DPAF) as the replaced objective to guide the evolutionary search. Specifically, in terms of each original objective, DPAF of each solution is defined as the weighted sum of the performance evaluated from the average perspective and the worst perspective. The weight used in DPAF is related to the stability level of the current population, enabling DPAF to adaptively balance optimality and robustness. In addition, we design a dual-indicator candidate selection strategy to identify high-quality candidates from the final population of the evolutionary search for expensive function evaluations. In this strategy, we first eliminate solutions with poor robust optimality by the proposed robust optimality indicator. Subsequently, based on the robust optimality indicator and a common diversity indicator, several solutions with good robust optimality and diversity are selected as candidates from the remaining solutions. Extensive experiments on two test suites and a real-world application verify the superiority of KPI.

This article proposes a novel constrained multiobjective evolutionary Bayesian optimization algorithm based on decomposition (named CMOEBO/D) for expensive constrained multiobjective optimization problems (CMOPs). In CMOEBO/D, an expensive CMOP is decomposed into some approximate constrained subproblems by Gaussian process models, reference vectors, and the augmented Tchebycheff function. Then, we devise a new infill criterion (named CPoB) to evaluate the performance of solutions. Specifically, in CPoB, on each approximate constrained subproblem, any two solutions are compared based on the product of two probabilities. For each solution, the first probability (denoted as PoB) is its likelihood of outperforming the other in terms of the predicted augmented Tchebycheff function value, and the second probability (denoted as PoF) is its likelihood of satisfying all constraints. It is obvious that PoB and PoF measure the convergence and feasibility of a solution, respectively. Based on CPoB, the approximate constrained subproblems are solved via collaborative evolutionary optimization to obtain their near-optimal solutions. Furthermore, by combining the information of the database, we design a bilevel candidate selection strategy to select some of these near-optimal solutions for expensive fitness evaluations, which can make good diversity, convergence, and feasibility contributions simultaneously to the database. Extensive experiments verify the competitiveness of CMOEBO/D.

This paper proposes a surrogate-assisted evolutionary framework (called SELF) to solve expensive multitask optimization problems (ExMTOPs). SELF consists of two main phases: global knowledge transfer phase and local knowledge transfer phase. In the former, a multitask Gaussian process model (MTGP) is established by fusing previously evaluated solutions of multiple optimization tasks. MTGP can capture task-relevant information and the knowledge of landscapes. Then, differential evolution assisted with MTGP is proposed to preselect high-quality candidates. During the preselection, the knowledge of landscapes is transferred among multiple optimization tasks for locating promising regions quickly. In the latter, for each optimization task, Bayesian optimization is adopted to improve the quality of the best individual in the population. Moreover, the improved best individuals in the populations of multiple optimization tasks are adaptively transferred based on a transfer probability, which is computed through the task-relevant information provided by MTGP. By combining these two phases, SELF not only achieves the tradeoff between exploration and exploitation, but also utilizes the global and local knowledge transfer to improve the efficiency for solving ExMTOPs. We test SELF on seven benchmark test problems in the IEEE CEC2017 evolutionary multitask optimization competition. The results demonstrate that the performance of SELF is better than that of other seven advanced methods. In addition, we also apply SELF to deal with two real-world ExMTOPs. The designs provided by SELF exhibit the best performance among all the compared methods, verifying the potential of SELF in practical engineering applications.

This article proposes a distribution information-based Kriging-assisted evolutionary algorithm (named DISK) to tackle expensive many-objective optimization problems (EMaOPs). In DISK, we design a new Pareto dominance relationship (called DIPD) to guide the evolutionary search and candidate selection. DIPD works based on the Kriging models and incorporates the decision-space distribution information of the nondominated solutions in the database. Such distribution information can be used to assess the possibility of an unknown solution being located in/close to the decision-space promising region. Thanks to this property, DIPD is capable of preserving the predicted elitist solutions located in/close to the decision-space promising region. These solutions are very likely to possess good original Pareto optimality and are beneficial for improving the convergence of the nondominated-solution set in the database. In addition, to further ensure the diversity of the nondominated-solution set in the database, we also design an adaptive exploration strategy, which explores the objective-space unknown region farthest away from the nondominated solutions in the database once the optimization process stagnates. Furthermore, through a feasibility-first mechanism, we extend DISK to deal with constrained EMaOPs, obtaining$\textrm {DISK}^{+}$. Finally, we verify the competitiveness of DISK and$\textrm {DISK}^{+}$via extensive experiments.

This paper proposes a new parameterless constraint-handling technique, named constrained probabilistic Pareto dominance (CPPD), for expensive constrained multiobjective optimization problems (CMOPs). In CPPD, when comparing two solutions, in terms of each original objective, we design a new objective for each solution, which is the negative product of two probabilities calculated based on the predicted fitness mean values and the uncertainty information provided by Kriging models: 1) the probability that this solution satisfies all constraints, denoted as PoF, and 2) the probability that this solution is better than the other on the original objective, denoted as PoB. It is evident that for each solution, PoF and PoB indicate its feasibility and its optimality on the corresponding original objective, respectively. Then, Pareto dominance based on new objectives is executed. As a result, both competitive feasible solutions and promising infeasible solutions with good diversity can be preserved by CPPD. These two kinds of solutions can help the population to exploit the located feasible parts and to explore new feasible parts, respectively. Further, based on CPPD, we develop a Pareto-based Kriging-assisted constrained multiobjective evolutionary algorithm (called PEA) to deal with expensive CMOPs with two or three objectives. Finally, PEA is generalized to solve expensive constrained many-objective optimization problems, named PEA+. The effectiveness of CPPD, PEA, and PEA+ is verified by comprehensive experiments.

This article proposes a constrained evolutionary Bayesian optimization (CEBO) algorithm to cope with expensive constrained optimization problems with inequality constraints. The uniqueness of CEBO lies in its capability of balancing feasibility and objective improvement under a limited function evaluation budget, which is achieved by designing two strategies to obtain promising solutions. The first strategy prefers feasibility. It tends to obtain a feasible solution by utilizing the predicted value and uncertainty provided by Gaussian process (GP). The second strategy prefers objective improvement. It maintains and evolves the population of evolutionary algorithms, and selects a solution with a good objective function value and violating the constraints not too much based on the predicted value and uncertainty provided by GP at each iteration. The sequential implementation of these two strategies allows CEBO to balance feasibility and objective improvement. The effectiveness of CEBO is verified by 26 test instances and a practical application. The results demonstrate that CEBO is able to find high-quality solutions with 100 FEs.

This article devises a two-phase Kriging-assisted evolutionary algorithm (named TEA) to tackle expensive constrained multiobjective optimization problems (CMOPs). In the first phase, only objectives are considered, which can help the population to cross infeasible obstacles and to evolve toward the unconstrained Pareto front. Since the unconstrained Pareto front is in front of the feasible region in the objective space, the first phase can find some feasible solutions during the evolution. In the second phase, both objectives and constraints are considered. In this article, we also propose two transition conditions to judge whether the search should be switched from the first phase to the second phase, by making use of the candidates evaluated by the original objectives and constraints in the first phase. These two transition conditions aim at maintaining some high-quality feasible solutions when the first phase ends, which is able to motivate the population to converge toward the constrained Pareto front with good diversity in the second phase. Furthermore, in both phases, we design a new Pareto dominance relationship (called PDPD) by incorporating the probability distribution information derived from the Kriging models. PDPD is further generalized to handle constraints in expensive CMOPs, Constrained PDPD (CPDPD), which provides high credibility for the comparison between two individuals with respect to both objectives and constraints. Finally, three benchmark test suites and a real-world application confirm the superiority of TEA.

Although many methods have been proposed to solve expensive optimization problems (EOPs), they often consume hundreds of function evaluations (FEs) to find the optimal solution, which is unacceptable when facing highly EOPs. To reduce the number of FEs, we incorporate the population distribution into the well-known expected improvement (EI); thus, a new infill criterion called evolutionary EI (EEI) is proposed. In EEI, the covariance matrix adaptation evolution strategy is used to provide the population distribution. Compared with the original EI, EEI focuses more on promising regions provided by the population distribution, thus, reducing the FEs wasted in unpromising regions. By employing EEI as the infill criterion of Bayesian optimization, a new algorithm called EEI-BO is designed. Moreover, we also introduce an extended version of EEI-BO, called EEI-BO+, to handle multitask EOPs. To verify the effectiveness of EEI-BO, it is used to solve 10−, 20−, and 30-D test problems by using only 40, 50, and 60 FEs, respectively. The results show that EEI-BO is able to obtain high-quality solutions by consuming limited FEs. In addition, we apply EEI-BO to deal with the lightweight and crashworthiness design of the side body of an automobile. The results demonstrate that EEI-BO performs well on solving it. Furthermore, the performance of EEI-BO+is investigated by nine test problems. The results show that it has the capability to solve multitask EOPs with fast convergence speed.

As an effective optimization tool for expensive optimization problems (EOPs), surrogate-assisted evolutionary algorithms (SAEAs) have been widely studied in recent years. However, most current SAEAs are designed for continuous/ combinatorial EOPs, which are not suitable for mixed-variable EOPs. This article focuses on one kind of mixed-variable EOP: EOPs with continuous and categorical variables (EOPCCVs). A multisurrogate-assisted ant colony optimization algorithm (MiSACO) is proposed to solve EOPCCVs. MiSACO contains two main strategies: 1) multisurrogate-assisted selection and 2) surrogate-assisted local search. In the former, the radial basis function (RBF) and least-squares boosting tree (LSBT) are employed as the surrogate models. Afterward, three selection operators (i.e., RBF-based selection, LSBT-based selection, and random selection) are devised to select three solutions from the offspring solutions generated by ACO, with the aim of coping with different types of EOPCCVs robustly and preventing the algorithm from being misled by inaccurate surrogate models. In the latter, sequence quadratic optimization, coupled with RBF, is utilized to refine the continuous variables of the best solution found so far. By combining these two strategies, MiSACO can solve EOPCCVs with limited function evaluations. Three sets of test problems and two real-world cases are used to verify the effectiveness of MiSACO. The results demonstrate that MiSACO performs well in solving EOPCCVs.

A considerable number of surrogate-assisted evolutionary algorithms (SAEAs) have been developed to solve expensive optimization problems (EOPs) with continuous objective functions. However, in the real-world applications, we may face EOPs with discontinuous objective functions, which are also called EOPs with discontinuous responses (EOPDRs). Indeed, EOPDRs pose a great challenge to current SAEAs. In this article, a surrogate-assisted differential evolution (DE) algorithm with region division is proposed, named ReDSADE. ReDSADE includes three main strategies: 1) the region division strategy; 2) the Kriging-based search; and 3) the radial basis function (RBF)-based local search. In the region division strategy, we define a new distance measure, called the objective-decision distance. Based on this distance, the evaluated solutions are partitioned into several clusters, and several support vector machine (SVM) classifiers are trained to classify them. These SVM classifiers divide the decision space into several subregions, with the aim of making the objective function continuous in them. In the Kriging-based search, a Kriging model is established in each subregion and combined with DE to search for the optimal solution. In the RBF-based local search, DE is coupled with RBF to search around the best solution found so far, thus accelerating the convergence. By combining these three strategies, ReDSADE is able to solve EOPDRs with limited function evaluations. Three sets of test problems and a real-world application are utilized to verify the effectiveness of ReDSADE. The results demonstrate that ReDSADE exhibits good convergence accuracy and convergence speed.

For expensive constrained optimization problems (ECOPs), the computation of objective function and constraints is very time-consuming. This paper proposes a novel global and local surrogate-assisted differential evolution (DE) for solving ECOPs with inequality constraints. The proposed method consists of two main phases: 1) global surrogate-assisted phase and 2) local surrogate-assisted phase. In the global surrogate-assisted phase, DE serves as the search engine to produce multiple trial vectors. Afterward, the generalized regression neural network is used to evaluate these trial vectors. In order to select the best candidate from these trial vectors, two rules are combined. The first is the feasibility rule, which at first guides the population toward the feasible region, and then toward the optimal solution. In addition, the second rule puts more emphasis on the solution with the highest predicted uncertainty, and thus alleviates the inaccuracy of the surrogates. In the local surrogate-assisted phase, the interior point method coupled with radial basis function is utilized to refine each individual in the population. During the evolution, the global surrogate-assisted phase has the capability to promptly locate the promising region and the local surrogate-assisted phase is able to speed up the convergence. Therefore, by combining these two important elements, the number of fitness evaluations can be reduced remarkably. The proposed method has been tested on numerous benchmark test functions from three test suites and two real-world cases. The experimental results demonstrate that the performance of the proposed method is better than that of other state-of-the-art methods.

In many real-world engineering applications, a uniform design needs to be conducted in a constrained experimental domain that includes linear/nonlinear and inequality/equality constraints. In general, these constraints make the constrained experimental domain small and irregular in the decision space. Therefore, it is difficult for current methods to produce a predefined number of samples and make the samples distribute uniformly in the constrained experimental domain. This paper presents a two-phase differential evolution for uniform designs in constrained experimental domains. In the first phase, considering the constraint violation as the fitness function, a clustering DE is proposed to guide the population toward the constrained experimental domain from different directions promptly. As a result, a predefined number of samples can be obtained in the constrained experimental domain. In the second phase, maximizing the minimum Euclidean distance among samples is treated as another fitness function. By optimizing this fitness function, the samples produced in the first phase can be scattered uniformly in the constrained experimental domain. The performance of the proposed method has been tested and compared with another state-of-the-art method. Experimental results suggest that our method is significantly better than the compared method in the uniform designs of a new type of automotive crash box and five benchmark test problems. Moreover, the proposed method could be considered as a general and promising framework for other uniform designs in constrained experimental domains.