Expensive Problems Research Articles

Multifidelity Bayesian Optimization: A Review

Resided at the intersection of multifidelity optimization (MFO) and Bayesian optimization (BO), MF BO has found a niche in solving expensive engineering design optimization problems, thanks to its advantages in incorporating physical and mathematical understandings of the problems, saving resources, addressing exploitation–exploration trade-off, considering uncertainty, and processing parallel computing. The increasing number of works dedicated to MF BO suggests the need for a comprehensive review of this advanced optimization technique. This paper surveys recent developments of two essential ingredients of MF BO: Gaussian process (GP) based MF surrogates and acquisition functions. First the existing MF modeling methods and MFO strategies are categorized to locate MF BO in a large family of surrogate-based optimization and MFO algorithms. Then, the common properties shared between the methods from each ingredient of MF BO are exploited to describe important GP-based MF surrogate models and to review various acquisition functions. This presentation aims to provide a structured understanding of MF BO. Finally, important aspects are examined that require further research for applications of MF BO in solving intricate yet important design optimization problems, including constrained optimization, high-dimensional optimization, optimization under uncertainty, and multiobjective optimization.

A surrogate-assisted differential evolution for high-dimensional expensive constrained optimization problems with mixed-integer variables

A surrogate-assisted differential evolution for high-dimensional expensive constrained optimization problems with mixed-integer variables

Hierarchical Online Air Combat Maneuver Decision Making and Control Based on Surrogate-Assisted Differential Evolution Algorithm

One-to-one within-visual-range air combat of unmanned combat aerial vehicles (UCAVs) requires fast, continuous, and accurate decision-making to achieve air combat victory. In order to solve the current problems of insufficient real-time performance of traditional intelligent optimization algorithms for solving decision-making problems and the mismatch between the planning trajectory and the actual flight trajectory caused by the difference between the decision-making model and the actual aircraft model, this paper proposes a hierarchical on-line air combat maneuvering decision-making and control framework. Considering the real-time constraints, the maneuver decision problem is transformed into an expensive optimization problem at the decision planning layer. The surrogate-assisted differential evolution algorithm is proposed on the basis of the original differential evolution algorithm, and the planning trajectory is obtained through the 5 degrees of freedom (DOF) model. In the control execution layer, the planning trajectory is tracked through the nonlinear dynamic inverse tracking control method to realize the high-precision control of the 6DOF model. The simulation is carried out under four different initial situation scenarios, including head-on neutral, dominant, parallel neutral, and disadvantaged situations. The Monte Carlo simulation results show that the Surrogate-assisted differential evolution algorithm (SADE) can achieve a win rate of over 53% in all four initial scenarios. The proposed maneuver decision and control framework in this article achieves smooth flight trajectories and stable aircraft control, with each decision average taking 0.08 s, effectively solving the real-time problem of intelligent optimization algorithms in maneuver decision problems.

Optimization of high-dimensional expensive multi-objective problems using multi-mode radial basis functions

Numerous surrogate-assisted evolutionary algorithms are developed for multi-objective expensive problems with low dimensions, but scarce works have paid attention to that with high dimensions, i.e., generally more than 30 decision variables. In this paper, we propose a multi-mode radial basis functions-assisted evolutionary algorithm (MMRAEA) for solving high-dimensional expensive multi-objective optimization problems. To improve the reliability, the proposed algorithm uses radial basis functions based on three modes to cooperate to provide the qualities and uncertainty information of candidate solutions. Meanwhile, bi-population based on competitive swarm optimizer and genetic algorithm are applied for better exploration and exploitation in high-dimensional search space. Accordingly, an infill criterion based on multi-mode of radial basis functions that comprehensively considers the quality and uncertainty of candidate solutions is proposed. Experimental results on widely-used benchmark problems with up to 100 decision variables demonstrate the effectiveness of our proposal. Furthermore, the proposed method is applied to the structure optimization of the blended-wing-body underwater glider (BWBUG) and gets impressive solutions.

Surrogate-assisted global and distributed local collaborative optimization algorithm for expensive constrained optimization problems

This paper presents a surrogate-assisted global and distributed local collaborative optimization (SGDLCO) algorithm for expensive constrained optimization problems where two surrogate optimization phases are executed collaboratively at each generation. As the complexity of optimization problems and the cost of solutions increase in practical applications, how to efficiently solve expensive constrained optimization problems with limited computational resources has become an important area of research. Traditional optimization algorithms often struggle to balance the efficiency of global and local searches, especially when dealing with high-dimensional and complex constraint conditions. For global surrogate-assisted collaborative evolution phase, the global candidate set is generated through classification collaborative mutation operations to alleviate the pre-screening pressure of the surrogate model. For local surrogate-assisted phase, a distributed central region local exploration is designed to achieve intensively search for promising distributed local areas which are located by affinity propagation clustering and mathematical modeling. More importantly, a three-layer adaptive selection strategy where the feasibility, diversity and convergence are balanced effectively is designed to identify promising solutions in global and local candidate sets. Therefore, the SGDLCO efficiently balances global and local search during the whole optimization process. Experimental studies on five classical test suites demonstrate that the SGDLCO provides excellent performance in solving expensive constrained optimization problems.

A survey on expensive optimization problems using differential evolution

A survey on expensive optimization problems using differential evolution

Deep optimal experimental design for parameter estimation problems

Abstract Optimal experimental design is a well studied field in applied science and engineering. Techniques for estimating such a design are commonly used within the framework of parameter estimation. Nonetheless, in recent years parameter estimation techniques are changing rapidly with the introduction of deep learning techniques to replace traditional estimation methods. This in turn requires the adaptation of optimal experimental design that is associated with these new techniques. In this paper we investigate a new experimental design methodology that uses deep learning. We show that the training of a network as a Likelihood Free Estimator can be used to significantly simplify the design process and circumvent the need for the computationally expensive bi-level optimization problem that is inherent in optimal experimental design for non-linear systems. Furthermore, deep design improves the quality of the recovery process for parameter estimation problems. As proof of concept we apply our methodology to two different systems of Ordinary Differential equations.

Integrating metaheuristic and machine learning for optimization of a full-scale transmission line tower

In long transmission lines (TLs), the design of a transmission line tower (TLT) can be replicated multiple times, a contrast to most structures that feature a unique design. Various optimization methods have been developed to reduce the overall mass of these structures. Historically, most research has relied on metaheuristic algorithms to identify optimal solutions. More recently, machine learning (ML) techniques have begun to be integrated with metaheuristics to speed up the optimization process, although ML has primarily been applied to smaller TLTs, in academic settings, or to simplify structural analysis, potentially compromising accuracy. This paper introduces a new approach that combines the Backtracking Search Algorithm (BSA), known for its efficacy in similar real-world TLT optimization challenges, with Kriging-based Efficient Global Optimization (EGO). This methodology starts by optimizing the size using BSA and subsequently employs EGO to refine the shape. This dual-step optimization effectively significantly reduces the mass of the TLT with only a few hundred additional objective function evaluations (OFEs). In comparison, simultaneous size and shape optimization using BSA alone requires over one hundred thousand additional OFEs to achieve comparable results, showing the potential of the proposed approach in dealing with expensive engineering design optimization problems.

Solving Expensive Optimization Problems in Dynamic Environments With Meta-Learning.

Dynamic environments pose great challenges for expensive optimization problems, as the objective functions of these problems change over time and thus require remarkable computational resources to track the optimal solutions. Although data-driven evolutionary optimization and Bayesian optimization (BO) approaches have shown promise in solving expensive optimization problems in static environments, the attempts to develop such approaches in dynamic environments remain rarely explored. In this article, we propose a simple yet effective meta-learning-based optimization framework for solving the expensive dynamic optimization problems. This framework is flexible, allowing any off-the-shelf continuously differentiable surrogate model to be used in a plug-in manner, either in data-driven evolutionary optimization or BO approaches. In particular, the framework consists of two unique components: 1) the meta-learning component, in which a gradient-based meta-learning approach is adopted to learn experience (effective model parameters) across different dynamics along the optimization process and 2) the adaptation component, where the learned experience (model parameters) is used as the initial parameters for fast adaptation in the dynamic environment based on few shot samples. By doing so, the optimization process is able to quickly initiate the search in a new environment within a strictly restricted computational budget. Experiments demonstrate the effectiveness of the proposed algorithm framework compared to several state-of-the-art algorithms on common benchmark test problems under different dynamic characteristics.

Radiofrequency Coil Tuning Using Matching Circuit Co-Simulation with Magnetic Resonance Integral Equations

Abstract The proposed extension to the conventional circuit co-simulation (CCS) method reliably and efficiently introduces arbitrary matching networks into CCS routines and is here dubbed as matching circuit co-simulation (MCCS). It further achieves coil tuning for resonance and an optimum scattering parameters condition that resembles a fully matched coil. Combined with magnetic resonance integral equations (MARIE), the proposed MCCS further accurately produces a full EM simulation of the coils. We first validated MARIE with the MCCS routine to use it to model simple single-channel coils and parallel transmit (pTx) head arrays coupled with Duke, a digital human body model. Having successfully validated the MCCS routine, we swept through both tuning and matching network parameters to optimize the coil. While optimizing for all parameters simultaneously is a viably efficient option for coil models of low complexity, these models are not realistic, and we found that it becomes a highly computationally expensive problem to solve for more complex coil geometries, such as birdcage coils and pTx coils with higher number of channels, especially when shielding is included. Consequently, we separated tuning and matching into two steps, by first tuning the coil to the right frequency, and then introducing matching networks to enhance the power transmission into the body. We observed lower on-diagonal S-parameters after matching in all pTx coils, but 1 coil still had strong cross-coupling. Finally, electromagnetic fields obtained via MARIE for separate port excitation were successfully combined using MCCS.

System architecture optimization strategies: dealing with expensive hierarchical problems

Abstract Choosing the right system architecture for the problem at hand is challenging due to the large design space and high uncertainty in the early stage of the design process. Formulating the architecting process as an optimization problem may mitigate some of these challenges. This work investigates strategies for solving system architecture optimization (SAO) problems: expensive, black-box, hierarchical, mixed-discrete, constrained, multi-objective problems that may be subject to hidden constraints. Imputation ratio, correction ratio, correction fraction, and max rate diversity metrics are defined for characterizing hierarchical design spaces. This work considers two classes of optimization algorithms for SAO: multi-objective evolutionary algorithms such as NSGA-II, and Bayesian optimization (BO) algorithms. A new Gaussian process kernel is presented that enables modeling hierarchical categorical variables, extending previous work on modeling continuous and integer hierarchical variables. Next, a hierarchical sampling algorithm that uses design space hierarchy to group design vectors by active design variables is developed. Then, it is demonstrated that integrating more hierarchy information in the optimization algorithms yields better optimization results for BO algorithms. Several realistic single-objective and multi-objective test problems are used for investigations. Finally, the BO algorithm is applied to a jet engine architecture optimization problem. This work shows that the developed BO algorithm can effectively solve the problem with one order of magnitude less function evaluations than NSGA-II. The algorithms and problems used in this work are implemented in the open-source Python library SBArchOpt.

Surrogate-assisted fully-informed particle swarm optimization for high-dimensional expensive optimization

Surrogate-Assisted Evolutionary Algorithms (SAEAs) have been proven to be powerful optimization tools for tackling Expensive Optimization Problems (EOPs) where a limited number of function evaluations are available. However, many SAEAs are only designed for low- or medium-dimensional EOPs. Existing SAEAs are challenging to address High-dimensional EOPs (HEOPs) owing to the curse of dimensionality and lack of powerful exploitation capacity. To tackle HEOPs efficiently, a Surrogate-Assisted Fully-informed Particle Swarm Optimization (SA-FPSO) algorithm is proposed in this paper. Firstly, a generation-based Social Learning-based PSO (SLPSO) is adopted to explore the whole decision space with the help of the global surrogate model. Secondly, the fully-informed search scheme is incorporated into the framework of SLPSO to improve its exploitation capacity in the surrogate-assisted search environment. Thirdly, a local space identification strategy is proposed to determine the search range for the local surrogate-assisted search. Seven commonly used expensive benchmark functions with dimensions ranging from 30D to 300D are used to verify the performance of SA-FPSO for HEOPs. Experiment results indicate that SA-FPSO obtains superior performance over several state-of-the-art SAEAs both in terms of convergence speed and solution accuracy.

Evolutionary Optimization with a 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.

Optimization of expensive black-box problems with penalized expected improvement

This paper proposes a new infill criterion for the optimization of expensive black-box design problems. The method complements the classical Efficient Global Optimization algorithm by considering the distribution of improvement instead of merely the expectation. During the optimization process, we maximize a penalized expected improvement acquisition function from a specially collected infill candidate set. Specifically, the acquisition function is formulated by penalizing the expected improvement with the variation of improvement, and the infill candidate set is composed of some global and local maxima of the expected improvement function which are identified to be “mutually non-dominated”. Some conditions necessary for setting the penalty coefficient of the acquisition function are investigated, and the definition of “mutually non-dominated infill candidates” is presented. The proposed method is demonstrated with a 1-D analytical function and benchmarked using six 10-D analytical functions and an underwater vehicle structural optimization problem. The results show that the proposed method is efficient for the optimization of expensive black-box design problems.

A novel surrogate-assisted differential evolution for mixed-integer expensive constrained optimization problems

A novel surrogate-assisted differential evolution for mixed-integer expensive constrained optimization problems

Solving higher dimensional expensive black box global optimization problems using sparse directional search on surrogates

We present a new algorithm Directional Optimization Search with Surrogate (DOSS), for optimizing problems with box constraints and a computationally expensive black-box objective function. DOSS is a radial basis function (RBF) based method that mainly focuses on higher dimensional and computationally expensive objective functions that can be multimodal. DOSS introduces three new techniques not previously used in earlier RBF algorithms, including using a combination of the coordinates knowledge level, fewer initial sampling points, and the surrogate’s gradient information. Numerical results on a test set including 14 test problems with 36, 48, and 60 dimensions show that DOSS outperforms two recently published algorithms RBFOpt and TuRBO, and earlier RBF algorithms such as DYCORS. TuRBO is a Gaussian process based optimization algorithm, which outperformed earlier state-of-the-art methods. DOSS algorithm also has a good performance on real-world optimization problems, including robot pushing and rover trajectory planning problems. Almost sure convergence for the DOSS algorithm is also proven in this paper. An implementation of DOSS is available at https://doi.org/10.5281/zenodo.13731558.

Multi-region hierarchical surrogate-assisted quantum-behaved particle swarm optimization for expensive optimization problems

Surrogate-assisted evolutionary algorithms (SAEAs) have been successfully applied to solve computationally expensive optimization problems. However, most SAEAs struggle to achieve good results in solving complex multimodal problems, especially high-dimensional ones. Moreover, for problems with complex landscapes, SAEAs typically require constructing complex global surrogates to model the landscape and performing many iterations to identify the surrogate’s optimum, thereby reducing the efficiency of SAEAs. To deal with these issues, this paper proposes a multi-region hierarchical surrogate-assisted quantum-behaved particle swarm optimization (MHS-QPSO) algorithm for expensive optimization problems. To better balance exploration and exploitation, a search behavior selection strategy is proposed, enabling MHS-QPSO to appropriately switch between global and local searches. For the global search, the search space is divided into multiple regions that can adaptively adjust the size of the areas. A surrogate is constructed in each region, requiring only a small number of QPSO iterations to find the optimum of each surrogate. Furthermore, a novel reliability-based criterion is proposed to screen candidate solutions in different regions for exact evaluations, which can save the number of exact function evaluations and can rapidly improve the fitting accuracy of the surrogates in regions with superior fitness. During local searches, a dynamic boundary adjustment strategy is introduced to guide the QPSO to faster approach the potential optimal region. Experimental results on seven benchmark functions with dimensions from 10 to 100, and on a complex real application, demonstrate that MHS-QPSO significantly outperforms several state-of-the-art algorithms within a limited computational budget. Code for MHS-QPSO is available at https://github.com/quanshuzhang/MHS-QPSO.git.

Expected coordinate improvement for high-dimensional Bayesian optimization

Bayesian optimization (BO) algorithm is very popular for solving low-dimensional expensive optimization problems. Extending Bayesian optimization to high dimension is a meaningful but challenging task. One of the major challenges is that it is difficult to find good infill solutions as the acquisition functions are also high-dimensional. In this work, we propose the expected coordinate improvement (ECI) criterion for high-dimensional Bayesian optimization. The proposed ECI criterion measures the potential improvement we can get by moving the current best solution along one coordinate. The proposed approach selects the coordinate with the highest ECI value to refine in each iteration and covers all the coordinates gradually by iterating over the coordinates. The greatest advantage of the proposed ECI-BO (expected coordinate improvement based Bayesian optimization) algorithm over the standard BO algorithm is that the infill selection problem of the proposed algorithm is always a one-dimensional problem thus can be easily solved. Numerical experiments show that the proposed algorithm can achieve significantly better results than the standard BO algorithm and competitive results when compared with five high-dimensional BOs and six surrogate-assisted evolutionary algorithms. This work provides a simple but efficient approach for high-dimensional Bayesian optimization. A Matlab implementation of our ECI-BO is available at https://github.com/zhandawei/Expected_Coordinate_Improvement.

Two-layer surrogate-assisted collaborative framework for expensive constrained optimization problems involving mixed integer variables

Many surrogate-assisted evolutionary algorithms (SAEAs) have been shown exceptional abilities in solving expensive constrained optimization problems (ECOPs). However, few of them are designed for ECOPs involving mixed integer variables (ECOPs-MI). Therefore, a two-layer surrogate-assisted collaborative framework called TLSACF is designed, in which the radial basis function (RBF) and Gaussian Process (GP)-based mixed-integer collaborative frameworks are combined effectively. In the RBF-based collaborative framework, the combination between RBF-assisted prescreening and RBF-based local search achieves effective population update by balancing the exploration and exploitation on high potential parental information. For the GP-based collaborative framework, a variable type-based cooperative mutation is utilized to obtain potential candidate individuals by assigning targeted mutation for different variables, and a hypervolume-based complete expected improvement matrix function is derived to prescreen these candidate solutions from the perspective of balancing the optimization on constraints and the objective for the effective search on disconnected sub-feasible regions caused by integer variables. Experimental results on ten classical problems show that TLSACF performs excellent in solving ECOPs-MI.

A Data-Driven Evolutionary Transfer Optimization for Expensive Problems in Dynamic Environments

A Data-Driven Evolutionary Transfer Optimization for Expensive Problems in Dynamic Environments