Multi-objective Efficient Global Optimization Research Articles

Adaptive nested Monte Carlo approach for multi-objective efficient global optimization

Adaptive nested Monte Carlo approach for multi-objective efficient global optimization

Many-to-Few Decomposition: Linking R2-Based and Decomposition-Based Multiobjective Efficient Global Optimization Algorithms

Many-to-Few Decomposition: Linking R2-Based and Decomposition-Based Multiobjective Efficient Global Optimization Algorithms

GPU-Accelerated Infill Criterion for Multi-Objective Efficient Global Optimization Algorithm and Its Applications

In this work, a novel multi-objective efficient global optimization (EGO) algorithm, namely GMOEGO, is presented by proposing an approach of available threads’ multi-objective infill criterion. The work applies the outstanding hypervolume-based expected improvement criterion to enhance the Pareto solutions in view of the accuracy and their distribution on the Pareto front, and the values of sophisticated hypervolume improvement (HVI) are technically approximated by counting the Monte Carlo sampling points under the modern GPU (graphics processing unit) architecture. As compared with traditional methods, such as slice-based hypervolume integration, the programing complexity of the present approach is greatly reduced due to such counting-like simple operations. That is, the calculation of the sophisticated HVI, which has proven to be the most time-consuming part with many objectives, can be light in programed implementation. Meanwhile, the time consumption of massive computing associated with such Monte Carlo-based HVI approximation (MCHVI) is greatly alleviated by parallelizing in the GPU. A set of mathematical function cases and a real engineering airfoil shape optimization problem that appeared in the literature are taken to validate the proposed approach. All the results show that, less time-consuming, up to around 13.734 times the speedup is achieved when appropriate Pareto solutions are captured.

An exploitation-enhanced multi-objective efficient global optimization algorithm for expensive aerodynamic shape optimizations

The traditional multi-objective efficient global optimization (EGO) algorithms have been hybridized and adapted to solving the expensive aerodynamic shape optimization problems based on high-fidelity numerical simulations. Although the traditional EGO algorithms are highly efficient in solving some of the optimization problems with very complex landscape, it is not preferred to solve most of the aerodynamic shape optimization problems with relatively low-degree multi-modal design spaces. A new infill criterion encouraging more local exploitation has been proposed by hybridizing two traditional multi-objective expected improvements (EIs), namely, statistical multi-objective EI and expected hypervolume improvement, in order to improve their robustness and efficiency in aerodynamic shape optimization. Different analytical test problems and aerodynamic shape optimization problems have been investigated. In comparison with traditional multi-objective EI algorithms and a standard evolutionary multi-objective optimization algorithm, the proposed method is shown to be more robust and efficient in the tests due to its hybrid characteristics, easier handling of sub-optimization problems, and enhanced exploitation capability.

Variable-fidelity hypervolume-based expected improvement criteria for multi-objective efficient global optimization of expensive functions

Variable-fidelity surrogate-based efficient global optimization (EGO) method with the ability to adaptively select low-fidelity (LF) and high-fidelity (HF) infill point has emerged as an alternative to further save the computational cost of the single-fidelity EGO method. However, in terms of the variable-fidelity surrogate-assisted multi-objective optimization methods, existing methods rely on empirical parameters or are unable to adaptively select LF/HF sample in the optimal search process. In this paper, two variable-fidelity hypervolume-based expected improvement criteria with analytic expressions for variable-fidelity multi-objective EGO method are proposed. The first criterion relies on the concept of variable-fidelity expected improvement matrix (VFEIM) and is obtained by aggregating the VFEIM using a simplified hypervolume-based aggregation scheme. The second criterion termed as VFEMHVI is derived analytically based on a modified hypervolume definition. Both criteria can adaptively select new LF/HF samples in the iterative optimal search process to update the variable-fidelity models towards the HF Pareto front, distinguishing the proposed methods to the rests in the open literature. The constrained versions of the two criteria are also derived for problems with constraints. The effectiveness and efficiency of the proposed methods are verified and validated over analytic problems and demonstrated by two engineering problems including aerodynamic shape optimizations of the NACA0012 and RAE2822 airfoils. The results show that the VFEMHVI combined with the normalization-based strategy to define the reference point is the most efficient one over the compared methods.

Multi-objective efficient global optimization of expensive simulation-based problem in presence of simulation failures

The multi-objective efficient global optimization (MOEGO), an extension of the single-objective efficient global optimization algorithm with the intention to handle multiple objectives, is one of the most frequently studied surrogate model-based optimization algorithms. However, the evaluation of the infill point obtained in each MOEGO update iteration using simulation tool may fail. Such evaluation failures are critical to the sequential MOEGO method as it leads to a premature halt of the optimization process due to the impossibility of updating the Kriging models approximating objectives. In this paper, a novel strategy to prevent the premature halt of the sequential MOEGO method is proposed. The key point is to introduce an additional Kriging model to predict the success possibility of the simulation at an unvisited point. Multi-objective expected improvement-based criteria incorporating the success possibility of the simulation are proposed. Experiments are performed on a set of six analytic problems, five low-fidelity airfoil shape optimization problems, and a high-fidelity axial flow compressor tandem cascade optimization problem. Results suggest that the proposed MOEGO-Kriging method is the only method that consistently performs well on analytic and practical problems. The methods using the least-square support vector machine (LSSVM) or weighted LSSVM as the predictor of success possibility perform competitively or worse compared with MOEGO-Kriging. The penalty-based method, assigning high objective values to the failed evaluations in minimization problem, yields the worst performance.

Aerodynamic optimization of forward-curved blade centrifugal fan characterized by inclining bionic volute tongue

The profile and axial designs of a volute tongue have great influences on the internal flow conditions and aerodynamic performance of a forward-curved blade centrifugal fan. In order to achieve a high quality of internal flow fields and fan aerodynamic performance, we propose an inclined bionic volute tongue (IBVT) described with rotation angle and scale coefficient at the front and back walls of the volute, which varies the profile of the volute tongue and the distance between the impeller and the volute tongue in the axial direction. Meanwhile, an adaptive pseudo expected improvement matrix (APEIM) parallel multi-objective efficient global optimization (EGO) algorithm is developed by introducing a flexible number of added points for each iteration of optimization on account of the optimization problem itself. On this basis, IBVT is optimized to maximize the fan aerodynamic performance in terms of total pressure efficiency and static pressure rise by virtue of numerical simulation. The results show that the developed APEIM parallel multi-objective EGO algorithm is capable of accurately and efficiently finding the Pareto front solutions. IBVT is considered to have the effectiveness of aerodynamic performance improvement for a forward-curved blade centrifugal fan. Within the fan with IBVT, the flow fields in the axial direction become more uniform with the dimension reduction of the inactive zone around the front span. The impeller-volute tongue interaction is greatly improved with the decreases of reversed and recirculated flow coefficients. IBVT also benefits the reduction of flow losses in the volute. By adopting the analysis of variance (ANOVA), the aerodynamic performance of the fan is highly affected by the rotation angle and scale coefficient of IBVT at the front wall of the volute. This work is helpful to guide the design of the volute tongue for a forward-curved blade centrifugal fan.

Convergence rates of the efficient global optimization algorithm for improving the design of analog circuits

Optimal sizing of analog circuits is a hard and time-consuming challenge. Nowadays, analog designers are more than ever interested in developing solutions for automating such a task. In order to overcome well-known drawbacks of the conventional equation-based and simulation-based sizing techniques, analog designers are being attracted by the so-called metamodeling techniques and recently have used them for establishing accurate models of circuits’ performances. Metamodels have been associated to optimization routines to maximize circuits’ performances. In this work we deal with the newly proposed efficient global optimization (EGO) algorithm that intrinsically offers both the metamodel generation and the optimization routine. Furthermore, it performs the requested task by using a relatively very small number of performance evaluations. Firstly, we focus on the convergence rates of the EGO technique via twenty benchmark test problems. Then, we use EGO for the optimal design of a couple of analog CMOS circuits. Comparison between EGO performances and those obtained using two surrogate-assisted metaheuristics is provided to show potentialities of the proposed approach. Finally, The case of muti-objective problems is also considered. The multi-objective efficient global optimization algorithm is used for generating Pareto fronts of conflicting perormances of two analog circuits. Obtained results are compared to those of the conventional in-loop optimization technique.

Multi-Objective Multidisciplinary Design Optimization Approach for Partially Reusable Launch Vehicle Design

Reusability of the first stage of launch vehicles may offer new perspectives to lower the cost of payload injection into orbit if sufficient reliability and efficient refurbishment can be achieved. One possible option that may be explored is to design the vehicle first stage for both reusable and expendable uses, in order to increase the flexibility and adaptability to different target missions. This paper proposes a multilevel multidisciplinary design optimization (MDO) approach to design aerospace vehicles addressing multimission problems. The proposed approach is focused on the design of a family of launchers for different missions sharing commonalities using multi-objective MDO to account for the computational cost associated with the discipline simulations. The multimission problem addressed considers two missions: 1) a reusable configuration for a sun synchronous orbit with a medium payload range and recovery of the first stage using a gliding-back strategy; 2) an expendable configuration for a medium payload injected into a geostationary transfer orbit. A dedicated MDO formulation introducing couplings between the missions is proposed in order to efficiently solve such a coupled problem while limiting the number of calls to the exact multidisciplinary analysis thanks to the use of Gaussian processes and multi-objective efficient global optimization.

Parallelized multiobjective efficient global optimization algorithm and its applications

In engineering practice, most optimization problems have multiple objectives, which are usually in a form of expensive black-box functions. The multiobjective efficient global optimization (MOEGO) algorithms have been proposed recently to sequentially sample the design space, aiming to seek for optima with a minimum number of sampling points. With the advance in computing resources, it is wise to make optimization parallelizable to shorten the total design cycle further. In this study, two different parallelized multiobjective efficient global optimization algorithms were proposed on the basis of the Kriging modeling technique. With use of the multiobjective expectation improvement, the proposed algorithm is able to balance local exploitation and global exploration. To implement parallel computing, the “Kriging Believer” and “multiple good local optima” strategies were adopted here to develop new sample infill criteria for multiobjective optimization problems. The proposed algorithms were applied to five mathematical benchmark examples first, which demonstrated faster convergence and better accuracy with more uniform distribution of Pareto points, in comparison with the two other conventional algorithms. The best performed “Kriging Believer” strategy approach was then applied to two more sophisticated real-life engineering case studies on the tailor-rolled blank (TRB) structures for crashworthiness design. After optimization, the TRB hat-shaped tube achieved a 3% increase in energy absorption and a 10.7% reduction in mass, and the TRB B-pillar attained a 10.1% reduction in mass and a 12.8% decrease in intrusion, simultaneously. These benchmark and engineering examples demonstrated that the proposed methods are fairly promising for being an effective tool for a range of design problems.

A modified hypervolume based expected improvement for multi-objective efficient global optimization method

The hypervolume indicator has been proved as an outstanding metric for the distribution of Pareto points, and the derived hypervolume based expected improvement (HVEI) has received a particular attention in the multi-objective efficient global optimization (EGO) method. However, the high computational cost has become the bottle neck which limits the application of HVEI on many objective optimization. Aiming at this problem, a modified version of HVEI (MHVEI) is proposed in this paper, which is easier to implement, maintains all the desired properties, and has a much lower computational cost. The theoretical study shows that the new criterion can be considered as a weighted integral form of HVEI, and it prefers the new point with a higher uncertainty compared with HVEI. The numerical tests show that the MHVEI performs similar as HVEI on the lower dimensional problem, and the advantage of MHVEI becomes more obvious as the dimension grows.

Multi-Fidelity Multi-Objective Efficient Global Optimization Applied to Airfoil Design Problems

In this study, efficient global optimization (EGO) with a multi-fidelity hybrid surrogate model for multi-objective optimization is proposed to solve multi-objective real-world design problems. In the proposed approach, a design exploration is carried out assisted by surrogate models, which are constructed by adding a local deviation estimated by the kriging method and a global model approximated by a radial basis function. An expected hypervolume improvement is then computed on the basis of the model uncertainty to determine additional samples that could improve the model accuracy. In the investigation, the proposed approach is applied to two-objective and three-objective optimization test functions. Then, it is applied to aerodynamic airfoil design optimization with two objective functions, namely minimization of aerodynamic drag and maximization of airfoil thickness at the trailing edge. Finally, the proposed method is applied to aerodynamic airfoil design optimization with three objective functions, namely minimization of aerodynamic drag at cruising speed, maximization of airfoil thickness at the trialing edge and maximization of lift at low speed assuming a landing attitude. XFOILis used to investigate the low-fidelity aerodynamic force, and a Reynolds-averaged Navier–Stokes simulation is applied for high-fidelity aerodynamics in conjunction with a high-cost approach. For comparison, multi-objective optimization is carried out using a kriging model only with a high-fidelity solver (single fidelity). The design results indicate that the non-dominated solutions of the proposed method achieve greater data diversity than the optimal solutions of the kriging method. Moreover, the proposed method gives a smaller error than the kriging method.

Gaussian Process Surrogate Model for Levering Similar Trends Across Concepts

The computational budget for designing and optimizing engineering systems has been tightened by the growth in the number of technology alternatives to satisfy the increasingly demanding goals in modern systems and the use of sophisticated, but also computationally burdensome, simulation tools to quantitatively choose the best technology alternative more accurately. This paper develops a new Gaussian process metamodel, which leverages trends in engineering objectives that are similar across derivative concepts, as opposed to current approximation methods that handle concepts independently. This metamodel is based on a multifidelity approach that enhances a new concept prediction with observations of a previous concept with similar trends. The influence of the sizes of the new and the old concept training sets in the performance of the proposed metamodel is explored in this paper. The proposed multifidelity metamodel is tested against traditional independent surrogates in approximating the engine shaft horsepower of a UH-60A modification with a Fenestron tail, where observations from the UH-60A with a conventional tail are available. The proposed metamodels are up to two times as accurate as independent concept modeling for small new concept training sets; then, a multiobjective efficient global optimization algorithm is applied to the proposed metamodel to search for minimal UH-60A power consumption in hover and cruise. This results in Pareto-optimal solutions that are 10% more efficient than the UH-60A baseline.

A Parallel Multiobjective Efficient Global Optimization: The Finite Element Method in Optimal Design and Model Development

Conceiving electromagnetic devices using finite element modeling tools is a complex task and also time-costly. The Efficient Global Optimization method, based on the progressive construction of surrogate models, is studied. The method uses Kriging models and allows for multiobjective optimization. An original infill criterion, combining the surrogate models and an estimate of their error is proposed. Moreover, two techniques for the calculi distribution, adapted to the algorithm, are tested on an eight-core machine. An advantage of the method consists in its capability of providing sufficiently accurate models for each objective and constraint function around the obtained Pareto front. The SMES device of the TEAM problem 22 is used as benchmark.