Number Of Function Evaluations Research Articles

Application of Classic and Novel Metaheuristic Algorithms in a BIM-Based Resource Tradeoff in Dam Projects

In recent years, dam construction has become more complex, requiring an effective project management method. Building Information Modeling (BIM) affects how construction projects are planned, designed, executed, and operated. Therefore, reducing execution time, cost, and risk and increasing quality are the primary goals of organizations. In this paper, first, the time and cost of the project were obtained via the BIM process. Subsequently, optimization between the components of the survival pyramid (time, cost, quality, and risk) in construction projects was completed in a case study of the Ghocham storage dam in five different modes, including contractor’s offers, BIM, actual, and two other modes based on the expert’s opinions. For this aim, five different meta-heuristic optimization algorithms were utilized, including two classical algorithms (Genetic and Simulated Annealing) and three novel algorithms (Black Widow Optimization, Battle Royale Optimization, and Black Hole Mechanics Optimization). In four cases, once each element of the survival pyramid was optimized separately, all four cases were traded off simultaneously. Moreover, the results were obtained from all the mentioned algorithms in five scenarios based on the number of function evaluation (Nfe), Standard Deviation (SD), Computation Time (CT), and Best Cost (BC). MATLAB software completed the coding related to the objective functions and optimization algorithms. The results indicated the appropriate performance of GA and BHMO algorithms in some scenarios. However, only the GAs should be considered effective algorithms in a dam construction projects’ time–cost–quality–risk (TCQR) tradeoff.

A subset-selection-based derivative-free optimization algorithm for dynamic operation optimization in a steel-making process

ABSTRACT Steel-making production is a dynamic process that has the characteristics of high temperature and heat, and a complex reaction mechanism that causes the mechanism model possibly to be unavailable and the system to be a black box. In this article, a dynamic operation optimization (DOO) problem is refined from the basic oxygen furnace (BOF) steel-making process, and the system model is formulated by a data analytics method. This prevents to solve the optimization problem with derivative-based optimization methods. To circumvent these difficulties, a surrogate-model-based derivative-free optimization algorithm is proposed for solving the DOO problem. In order to establish the surrogate model with the least number of function evaluations, a subset selection strategy is designed to find a sparse structure for the optimization problem, based on which a set of simple bases is determined to establish the surrogate model. Moreover, this also reduces the scale of the parameter optimization problem. Numerical experiments on actual production data verify the applicability and effectiveness of the proposed method.

Dual adaption based evolutionary algorithm for optimized the smart healthcare communication service of the Internet of Things in smart city

The Internet of Things (IoT) is a revolutionary technique of sharing data for smart devices that generates huge amounts of data from smart healthcare systems. Therefore, healthcare systems utilize the convergence power and traffic analysis of sensors that cannot be satisfactorily handled by the IoT. In this article, a novel mutation operator is devised and incorporated with the differential evolution (DE) algorithm. Two tests have been conducted in the validation process. Firstly, the newly dual adaption-based operators incorporated with the differential evolution algorithm are being proposed. The proposed approach provides sufficient diversity and enhances the search speed of nature’s local and global search environments in the problem. The proposed method incorporates the application of IoT-based smart healthcare. Second, an application-based test has been conducted, in which the proposed approach is applied to the application in the smart healthcare system. Therefore, IoT sensor deployment is an optimization problem to minimize service time, delay, and energy loss by considering the communication constraint between sensors(objects). The proposed algorithm is applied in this article to solve this optimization problem. Further, in the experimentation and comparative study, the proposed method is superior to the standard evolutionary algorithms in IoT applications concerning the minimum number of function evaluations and minimization of traffic services. The proposed approach also achieves efficiency in the minimum loss of energy in each service and reduces load and delay.

A Multi-Fidelity Approach for Reliability Assessment Based on the Probability of Classification Inconsistency

Abstract Most multi-fidelity schemes for optimization or reliability assessment rely on regression surrogates, such as Gaussian processes. Contrary to these approaches, we propose a classification-based multi-fidelity scheme for reliability assessment. This technique leverages multi-fidelity information to locally construct failure boundaries using support vector machine (SVM) classifiers. SVMs are subsequently used to estimate the probability of failure using Monte Carlo simulations. The use of classification has several advantages: It can handle discontinuous responses and reduce the number of function evaluations in the case of a large number of failure modes. In addition, in the context of multi-fidelity techniques, classification enables the identification of regions where the predictions (e.g., failure or safe) from the various fidelities are identical. At the core of the proposed scheme is an adaptive sampling routine driven by the probability of classification inconsistency between the models. This sampling routine explores sparsely sampled regions of inconsistency between the models of various fidelity to iteratively refine the approximation of the failure domain boundaries. A lookahead scheme, which looks one step into the future without any model evaluations, is used to selectively filter adaptive samples that do not induce substantial changes in the failure domain boundary approximation. The model management strategy is based on a framework that adaptively identifies a neighborhood of no confidence between the models. The proposed scheme is tested on analytical examples of dimensions ranging from 2 to 10, and finally applied to assess the reliability of a miniature shell and tube heat exchanger.

Using a new coefficient conjugate gradient method for solving unconstrained optimization problems

The conjugate gradient technique is a numerical solution strategy for finding minimization in mathematics. We present a simple, straightforward, efficient, and resilient conjugate gradient technique in this study. To address the convergence difficulty and descent property, the new technique is built on the quadratic model. Under some assumptions, the new improved approach meets the convergence characteristics and the adequate descent criterion. The suggested unique strategy is substantially more efficient than the classic FR method, according to our numerical analysis. The number of function evaluations, iterations and restarts are all included in the numerical results. The computational efficiency of the proposed approach is proved by comparative results.

An Improved Blind Kriging Surrogate Model for Design Optimization Problems

Surrogate modeling techniques are widely employed in solving constrained expensive black-box optimization problems. Therein, Kriging is among the most popular surrogates in which the trend function is considered as a constant mean. However, it also encounters several challenges related to capturing the overall trend with a relatively limited number of function evaluations as well as searching feasible points with complex or discontinuous feasible regions. To address this above issue, this paper presents an improved surrogate blind Kriging (IBK) and a combined infill strategy to find the optimal solution. According to enhancing the prediction accuracy of metamodels of objective and constraints, the high-order effects of regression function in the blind Kriging are identified by promising a variable selection technique. In addition, an infill strategy is developed based on the probability of feasibility, penalization, and constrained expected improvement for updating blind Kriging metamodels of the objective and constraints. At each iteration, two infill sample points are allocated at the positions to achieve improvement in optimality and feasibility. The IBK metamodels are updated by the newly-added infill sample points, which leads the proposed framework search to rapidly converge to the optimal solution. The performance and applicability of the proposed model are tested on several numerical benchmark problems via comparing with other metamodel-based constrained optimization methods. The obtained results indicate that IBK generally has a greater efficiency performance and outperforms the competitors in terms of a limited number of function evaluations. Finally, IBK is successfully applied to structural design optimization. The optimization results show that IBK is able to find the best feasible design with fewer evaluation functions compared with other studies, and this demonstrates the effectiveness and practicality of the proposed model for solving the constrained expensive black-box engineering design optimization problems.

Evaluation of QAOA based on the approximation ratio of individual samples

The quantum approximate optimization algorithm (QAOA) is a hybrid quantum–classical algorithm to solve binary-variable optimization problems. Due to the short circuit depth and its expected robustness to systematic errors it is a promising candidate likely to run on near-term quantum devices. We simulate the performance of QAOA applied to the Max-Cut problem and compare it with some of the best classical alternatives. When comparing solvers, their performance is characterized by the computational time taken to achieve a given quality of solution. Since QAOA is based on sampling, we utilize performance metrics based on the probability of observing a sample above a certain quality. In addition, we show that the QAOA performance varies significantly with the graph type. In particular for three-regular random graphs, QAOA performance shows improvement by up to two orders of magnitude compared to previous estimates, strongly reducing the performance gap with classical alternatives. This was possible by reducing the number of function evaluations per iteration and optimizing the variational parameters on small graph instances and transferring to large via training. Because QAOA’s performance guarantees are only known for limited applications and contexts, we utilize a framework for the search for quantum advantage which incorporates a large number of problem instances and all three classical solver modalities: exact, approximate, and heuristic.

Local buckling of omega-stringer-stiffened composite panels under compression–shear interaction

In the preliminary design of thin-walled structures the computational efficiency is especially important, as a high number of function evaluations of any computational analysis model is necessary. To further identify possible ways to reduce computational effort, the convergence behaviour and quality of different trial functions is investigated in the context of a computational model based on the Ritz-method. In the analysis framework, the local buckling behaviour of an omega-stringer-stiffened composite panel is investigated, where continuity conditions are ensured by the Lagrange multiplier method. The quality of the different trial functions assembled using trigonometric functions and Legendre polynomials is assessed in convergence and parametric studies.The load cases considered are uniaxial compression, biaxial compression, shear loading and their combinations. The most promising improvement on the results is identified by using a combination of a sine series and Legendre polynomials with simply supported boundary conditions.

A Two-stage Surrogate-Assisted Evolutionary Algorithm (TS-SAEA) for Expensive Multi/Many-objective Optimization

In this paper, a two-stage surrogate-assisted evolutionary algorithm (TS-SAEA) is presented for computationally expensive multi/many-objective optimization, which consists of a convergence stage and a diversity stage. In the convergence stage, the objective space is partitioned into several sub-regions by reference vectors, where the individuals compete with each other. In the diversity stage, the converged individuals and the current non-dominated solutions are combined to form a potential sample set, on which a secondary selection is conducted to further improve the diversity. Specifically, the proposed diversity strategy firstly defines the initial boundary individuals and a candidate pool. The individuals with “max-min angles” will continuously be selected from the pool to supplement the boundary individuals until the number of the boundary individuals equals the number of the current non-dominated solutions. At last, the points with the better space-filling features are picked out from the updated boundary individuals to evaluate the true objectives. The above-mentioned process keeps running until the maximal number of function evaluations is satisfied. To evaluate the performance of TS-SAEA on both low and high-dimensional multi/many-objective problems, it is compared with four state-of-art algorithms on 52 benchmark problems and one engineering application. The experimental results show that TS-SAEA has significant advantages on computationally expensive multi/many-objective optimization problems.

Resource allocation problems with expensive function evaluations

The resource allocation problem is among the classical problems in operations research, and has been studied extensively for decades. However, current solution approaches are not able to efficiently handle problems with expensive function evaluations, which can occur in a variety of applications. We study the integer resource allocation problem with expensive function evaluations, for both convex and non-convex separable cost functions. We present several solution methods, both heuristics and exact methods, that aim to limit the number of function evaluations. The methods are compared in numerical experiments using both randomly generated instances and instances from two resource allocation problems occurring in radiation therapy planning. Results show that the presented solution methods compare favorably against existing derivative free optimization solvers.

Numerical Integration of Stiff Differential Systems Using Non-Fixed Step-Size Strategy

Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to be attained. This is because such systems have solution components that decay rapidly and/or slowly than others over a given integration interval. In order to curb this challenge, there is a need to propose a method that can vary the step size within a defined integration interval. This challenge motivated the development of Non-Fixed Step-Size Algorithm (NFSSA) using the Lagrange interpolation polynomial as a basis function via integration at selected limits. The NFSSA is capable of integrating highly stiff differential systems in both small and large intervals and is also efficient in terms of economy of computer time. The validation of properties of the proposed algorithm which include order, consistence, zero-stability, convergence, and region of absolute stability were further carried out. The algorithm was then applied to solve some samples mildly and highly stiff differential systems and the results generated were compared with those of some existing methods in terms of the total number of steps taken, number of function evaluation, number of failure/rejected steps, maximum errors, absolute errors, approximate solutions and execution time. The results obtained clearly showed that the NFSSA performed better than the existing ones with which we compared our results including the inbuilt MATLAB stiff solver, ode 15s. The results were also computationally reliable over long intervals and accurate on the abscissae points which they step on.

Efficient and relevant stepwise covariate model building for pharmacometrics.

Covariate modeling is an important opportunity for pharmacometrics to influence decision making in drug development. The stepwise covariate model (SCM) building procedure is the most common method for covariate model development. Despite its advantages, the traditional SCM method is known to have long runtimes and the suboptimal ability to select relevant covariates, especially in more complex phase III settings. In this work, two alternative approaches are presented: SCM+, which introduces the “adaptive scope reduction” and changes to general estimation settings, and “stage‐wise filtering,” which groups covariates into categories based on their importance (mechanistic, structural, and exploratory). The three methods (SCM, SCM+, and SCM+ with stage‐wise filtering) are applied to data from a simulated phase III population pharmacokinetic study and are compared in terms of efficiency and relevance. The two SCM+ methods were considerably more efficient than the traditional SCM: the number of function evaluations was reduced by 70% for SCM+ and by 76% for SCM+ with stage‐wise filtering compared to SCM; the corresponding number of executed models was reduced by 44% for SCM+ and 70% for SCM+ with stage‐wise filtering. In addition, among the three methods, SCM+ with stage‐wise filtering selected the highest number of relevant covariates. Given the improved efficiency and ability to select relevant covariates shown in this work, the use of SCM+ and stage‐wise filtering can greatly increase the efficiency of covariate modeling in drug development, which will ultimately facilitate more timely support for decision making.

A Benchmark Study on Steepest Descent and Conjugate Gradient Methods-Line Search Conditions Combinations in Unconstrained Optimization

In this paper, it is aimed to computationally conduct a performance benchmarking for the steepest descent and the three well-known conjugate gradient methods (i.e., Fletcher-Reeves, Polak-Ribiere and Hestenes-Stiefel) along with six different step length calculation techniques/conditions, namely Backtracking, Armijo-Backtracking, Goldstein, weakWolfe, strongWolfe, Exact local minimizer in the unconstrained optimization. To this end, a series of computational experiments on a test function set is completed using the combinations of those optimization methods and line search conditions. During these experiments, the number of function evaluations for every iteration are monitored and recorded for all the optimization method-line search condition combinations. The total number of function evaluations are then set a performance measure when the combination in question converges to the functions minimums within the given convergence tolerance. Through those data, the performance and data profiles are created for all the optimization method-line search condition combinations with the purpose of a reliable and an efficient benchmarking. It has been determined that, for this test function set, the steepest descent-Goldstein combination is the fastest one whereas the steepest descent-exact local minimizer is the most robust one with a high convergence accuracy. By making a trade-off between convergence speed and robustness, it has been identified that the steepest descent-weak Wolfe combination is the optimal choice for this test function set.

Rapid aerodynamic shape optimization under uncertainty using a stochastic gradient approach

A common approach in aerodynamic design is to optimize a performance function—provided some constraints—defined by a choice of an aerodynamic model at nominal operating conditions. Practical experience indicates that such a deterministic approach may result in considerably sub-optimal designs when the adopted aerodynamic model does not lead to accurate predictions, or when the actual operating conditions differ from those considered in the design. One approach to address this shortcoming is to consider an average or robust design, wherein the statistical moments of the performance function, given the uncertainty in the operating conditions and the aerodynamic model, is optimized. However, when the number of uncertain inputs is large or the performance function exhibits significant variability, an accurate evaluation of these moments may require a large number of function evaluations at each optimization iteration, rendering the problem significantly expensive. To tackle this difficulty, we consider a variant of the stochastic gradient descent method where in each iteration, a stochastic approximation of the objective, constraints, and their gradients is generated. This is done via a small number of forward/adjoint solutions corresponding to random selections of the uncertainties. The methodology is applied to the robust optimization of the NACA-0012 airfoil subject to operating condition and turbulence model uncertainty. With a cost that is only a small factor larger than that of the deterministic methodology, the stochastic gradient approach significantly improves the performance of the aerodynamic design for a wide range of operating conditions and turbulence models.

Bayesian Optimization over Permutation Spaces

Optimizing expensive to evaluate black-box functions over an input space consisting of all permutations of d objects is an important problem with many real-world applications. For example, placement of functional blocks in hardware design to optimize performance via simulations. The overall goal is to minimize the number of function evaluations to find high-performing permutations. The key challenge in solving this problem using the Bayesian optimization (BO) framework is to trade-off the complexity of statistical model and tractability of acquisition function optimization. In this paper, we propose and evaluate two algorithms for BO over Permutation Spaces (BOPS). First, BOPS-T employs Gaussian process (GP) surrogate model with Kendall kernels and a Tractable acquisition function optimization approach to select the sequence of permutations for evaluation. Second, BOPS-H employs GP surrogate model with Mallow kernels and a Heuristic search approach to optimize the acquisition function. We theoretically analyze the performance of BOPS-T to show that their regret grows sub-linearly. Our experiments on multiple synthetic and real-world benchmarks show that both BOPS-T and BOPS-H perform better than the state-of-the-art BO algorithm for combinatorial spaces. To drive future research on this important problem, we make new resources and real-world benchmarks available to the community.

Toward investigating the application of reservoir opportunity index in facilitating well placement optimization under geological uncertainty

There have been several attempts to reduce the computational costs associated with well placement optimization under geological uncertainty by selecting a representative subset of realizations or by reducing the number of function evaluations to speed up the optimization. Although previous researchers have studied this issue, they have used static or dynamic properties of the reservoir to select a reduced subset of realizations or shrank the search space of the optimizer to reduce the computational burden of the task. Using static parameters is fast but lacks the accuracy of the dynamic approach. On the other hand, dynamic parameters require a computationally expensive full flow simulation. In this paper, a workflow based on the Reservoir Opportunity Index (ROI) maps was proposed to address the two issues mentioned above. The ROI maps were constructed using both the static and dynamic reservoir properties. The dynamic feature was obtained by the computationally efficient Fast Marching Method instead of full reservoir simulation. In this workflow, ROI density (DROI) maps were constructed by averaging the ROI values of adjacent cells to incorporate the effect of neighboring cells in finding potential regions to constrain the search space of the optimizer. In addition, these DROI maps were used to select a representative subset of realizations. Furthermore, the effect of re-selection of the representative realizations was investigated in this study. The proposed workflow was applied to optimize the location of production wells under geological uncertainty and was compared with the Full Case (using all realizations in optimization) in terms of three criteria, namely the expected value, the standard deviation, and the cumulative distribution function (CDF) of the objective function. Results of optimization showed that the constrained search with a fixed number of realizations and the statistical clustering technique outperformed other approaches with minimum differences of 1.5% and 8.9% from the objective function and its standard deviation of the values obtained from the Full Case, respectively. Also, it had 28.64% less difference between its CDF and the Full Case in comparison to the second-best technique. In addition, it was shown that the constrained search required fewer function evaluations to achieve similar results as the global search with approximately three times fewer function evaluations. The results of this paper can be used as a guidance for reservoir engineers to select a proper approach for optimization (constrained or unconstrained), clustering the geological realizations (statistical or K-means), and proper implementation of ROI maps for shrinking the search space of the optimizer and selecting a representative subset of realizations.

A classification-assisted environmental selection strategy for multiobjective optimization

Environmental selection of multiobjective evolutionary algorithms (MOEAs) is a key component that chooses promising solutions from a candidate set for later usage. Most environmental selection strategies choose solutions based on their function values. However, in real-world optimization problems, function evaluations can be time consuming. The necessity of a large number of function evaluations leads to the low efficiency of MOEAs. How to decrease the number of function evaluations is one of the main issues of MOEAs. The environmental selection can be regarded as a classification process. The selected solutions are the promising class, and the discarded solutions are the unpromising class. Based on this consideration, we propose a classification-assisted environmental selection (CAES) strategy in this paper to decrease the number of function evaluations in MOEAs. In the proposed method, solutions are divided into two classes. One is non-dominated solutions (i.e. promising class) and the other is dominated solutions (i.e. unpromising class). The classifier is built to classify the offspring solutions into these two classes. Only promising offspring are evaluated (unpromising ones are removed with no function evaluations). Therefore, the number of function evaluations is reduced. We integrate the proposed CAES strategy into six MOEAs. The effectiveness of the proposed CAES strategy is examined through computational experiments on various test suites and three real-world application problems. Our experimental results show that the proposed CAES strategy clearly reduces the number of function evaluations without severely degrading the search ability of the original MOEAs.

Geometri Uydurma için En İyi Eşlenik Gradyan Yönteminin Tespit Edilmesi

In this study, it is aimed to determine the optimal conjugate gradient (CG) method for the geometry fitting of 2D measured profiles. To this end, the three well-known CG methods such as the Fletcher-Reeves, Polak-Ribiere and Hestenes-Stiefel were employed. For testing those methods performances, the five primitive geometries accommodating circle, square, triangle, ellipse and rectangle were first built with a 3D printer, and then they were scanned with a coordinate measuring machine (CMM) to achieve their 2D profiles. The nonlinear least squares procedure was implemented to minimize the error between those measured data and modeled ones. An iterative line search was utilized for this task. The search direction was calculated using the above-mentioned CG methods. During the geometry fitting process, the number of function evaluations at each iteration were computed and the total number of function evaluations were set to be a performance measure of the CG method in question when it converged. By using these performance measures, the performance and data profiles were created to efficiently determine the optimal CG method. Based on performance profiles, it can be stated that the Fletcher-Reeves and Polak-Ribiere methods are the fastest ones on three test geometries out of five. In addition to that, all the CG methods were able to complete the geometry fitting of 80% of test geometries. On the other hand, by examining the data profiles, it was determined that the Polak-Ribiere and Hestenes-Stiefel methods achieve their maximum capabilities of the completing geometry fitting (i.e., 80%) with much lower number of function evaluations than the Fletcher-Reeves method. Besides, in most geometries, the Polak-Ribiere method outperformed the others, thereby it was determined to be the optimal one for the geometry fitting. As a conclusion, the reported results in this work might help the end-users who study on the CMM data processing to conduct an efficient geometry fitting.

Mixed-precision explicit stabilized Runge–Kutta methods for single- and multi-scale differential equations

Mixed-precision algorithms combine low- and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge–Kutta–Chebyshev (RKC) methods, where high precision is used for accuracy, and low precision for stability. Generally speaking, RKC methods are low-order explicit schemes with a stability domain growing quadratically with the number of function evaluations. For this reason, most of the computational effort is spent on stability rather than accuracy purposes. In this paper, we show that a naïve mixed-precision implementation of any Runge–Kutta scheme can harm the convergence order of the method and limit its accuracy, and we introduce a new class of mixed-precision RKC schemes that are instead unaffected by this limiting behavior. We present three mixed-precision schemes: a first- and a second-order RKC method, and a first-order multirate RKC scheme for multiscale problems. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first- and second-order methods respectively) in high precision, while the rest are performed in low precision. We prove that while these methods are essentially as cheap as their fully low-precision equivalent, they retain the stability and convergence order of their high-precision counterpart. Indeed, numerical experiments confirm that these schemes are as accurate as the corresponding high-precision method.

Hybrid Surrogate-Based Constrained Optimization With a New Constraint-Handling Method.

Surrogate-based-constrained optimization for some optimization problems involving computationally expensive objective functions and constraints is still a great challenge in the optimization field. Its difficulties are of two primary types. One is how to handle the constraints, especially, equality constraints; another is how to sample a good point to improve the prediction of the surrogates in the feasible region. Overcoming these difficulties requires a reliable constraint-handling method and an efficient infill-sampling strategy. To perform inequality- and equality-constrained optimization of expensive black-box systems, this work proposes a hybrid surrogate-based-constrained optimization method (HSBCO), and the main innovation is that a new constraint-handling method is proposed to map the feasible region into the origin of the Euclidean subspace. Thus, if the constraint violation of an infeasible solution is large, then it is far from the origin in the Euclidean subspace. Therefore, all constraints of the problem can be transformed into an equivalent equality constraint, and the distance between an infeasible point and the origin in the Euclidean subspace represents the constraint violation of the infeasible solution. Based on the distance, the objective function of the problem can be penalized by a Gaussian penalty function, and the original constrained optimization problem becomes an unconstrained optimization problem. Thus, the feasible solutions of the original minimization problem always have a lower objective function value than any infeasible solution in the penalized objective space. To improve the optimization performance, kriging-based efficient global optimization (EGO) is used to find a locally optimal solution in the first phase of HSBCO, and starting from this locally optimal solution, RBF-model-based global search and local search strategies are introduced to seek global optimal solutions. Such a hybrid optimization strategy can help the optimization process converge to the global optimal solution within a given maximum number of function evaluations, as demonstrated in the experimental results on 23 test problems. The method is shown to achieve the global optimum more closely and efficiently than other leading methods.