Convex Estimators for Optimization of Kriging Model Problems

2012 ◽  
Vol 134 (11) ◽  
Author(s):  
Karim Hamza ◽  
Mohammed Shalaby
Keyword(s):  
Quadratic Programming ◽  
Sequential Quadratic Programming ◽  
Original Problem ◽  
Kriging Model ◽  
Global Optimum ◽  
Water Desalination ◽  
Test Functions ◽  
Convex Underestimators ◽  
Vehicle Crashworthiness ◽  
Relaxed Problem

This paper presents a framework for identification of the global optimum of Kriging models that have been tuned to approximate the response of some generic objective function and constraints. The framework is based on a branch and bound scheme for subdivision of the search space into hypercubes while constructing convex underestimators of the Kriging models. The convex underestimators, which are the key development in this paper, provide a relaxation of the original problem. The relaxed problem has two main features: (i) convex optimization algorithms such as sequential quadratic programming (SQP) are guaranteed to find the global optimum of the relaxed problem and (ii) objective value of the relaxed problem is a lower bound within a hypercube for the original (Kriging model) problem. As accuracy of the convex estimators improves with subdivision of a hypercube, termination of a branch happens when either: (i) solution of the relaxed problem within the hypercube is no better than current best solution of the original problem or (ii) best solution of the original problem and that of the relaxed problem are within tolerance limits. To assess the significance of the proposed framework, comparison studies against genetic algorithm (GA), particle swarm optimization (PSO), random multistart sequential quadratic programming (mSQP), and DIRECT are conducted. The studies include four standard nonlinear test functions and two design application problems of water desalination and vehicle crashworthiness. The studies show the proposed framework deterministically finding the optimum for all the test problems. Among the tested stochastic search techniques (GA, PSO, mSQP), mSQP had the best performance as it consistently found the optimum in less computational time than the proposed approach except on the water desalination problem. DIRECT deterministically found the optima for the nonlinear test functions, but completely failed to find it for the water desalination and vehicle crashworthiness problems.

Convex Estimators for Optimization of Kriging Model Problems

2011 ◽  
Author(s):  
Karim Hamza ◽  
Mohammed Shalaby
Keyword(s):  
Original Problem ◽  
Search Space ◽  
Kriging Model ◽  
Global Optimum ◽  
Water Desalination ◽  
Tolerance Limits ◽  
Design Variables ◽  
Search Termination ◽  
Relaxed Problem ◽  
Better Than

This paper presents a framework for identification of the global optimum of Kriging models. The framework is based on a branch and bound scheme for sub-division of the search space into hypercubes while constructing convex under-estimators of the Kriging models. The convex under-estimators, which are a key development in this paper, provide a relaxation of the original problem. The relaxed problem has two key features: i) convex optimization algorithms such as sequential quadratic programming (SQP) are guaranteed to find the global optimum of the relaxed problem, and ii) objective value of the relaxed problem is a lower bound on the best attainable solution within a hypercube for the original (Kriging model) problem. The convex under-estimators improve in accuracy as the size of a hypercube gets smaller via the branching search. Termination of a hypercube branch is done when either: i) solution of the relaxed problem within the hypercube is no better than current best solution of the original problem, or ii) best solution of the original problem and that of the relaxed problem are within tolerance limits. To assess the significance of the proposed framework, comparison studies against genetic algorithm (GA) are conducted using Kriging models that approximate standard nonlinear test functions, as well as application problems of water desalination and vehicle crashworthiness. Results of the studies show the proposed framework deterministically providing a solution within tolerance limits from the global optimum, while GA is observed to not reliably discover the best solutions in problems with larger number of design variables.

Application of Optimization Methodology on Vehicular Crash Reconstruction

2009 ◽  
Author(s):  
Fengjiao Guan ◽  
Aditya Belwadi ◽  
Xu Han ◽  
King H. Yang
Keyword(s):  
Quadratic Programming ◽  
Sequential Quadratic Programming ◽  
Fe Simulation ◽  
Principal Direction ◽  
Accurate Determination ◽  
Kriging Model ◽  
Optimization Approach ◽  
Fe Model ◽  
Assessment Index ◽  
Optimization Methodology

In vehicular crash reconstruction, software packages such as PC-Crash, SMAC (Simulation Model of Automobile Collisions), WinSmash and HVE (Human Vehicle Environment) use physical evidences such as tire marks along with measurements of the deformed vehicles and photographs of the accident scene to determine the crash energy, impact velocity, and Principal Direction Of Force (PDOF). However, accurate determination of these parameters requires more sophisticated numerical methods, such as Finite Element (FE) modeling. At present, multiple runs of FE models need to be performed on a trial-and-error basis before the model predicted results are consistent with the actual ones. An optimization method to quickly and accurately determine key sensitive parameters in vehicular accident reconstruction is desired. We propose the use of Kriging model and sequential quadratic programming in conjunction with Latin Hypercube Sampling (LHS) to minimize the time needed for reconstruction and minimize the disparity between the actual and FE model predicted vehicular deformations. A selected number of modeling parameters, namely the velocity of impact, PDOF and initial impact position, are varied using this optimization approach until the deformation of six points measured on the impacted vehicle closely matches those measured in real world case. The optimization is performed in two stages. In the first stage, an approximated model was created by simplifying detailed FE models of the vehicles involved to reduce the simulation time without sacrificing accuracy. In the second stage, an assessment index ‘E’, the objective function, is maximized. To improve computational efficiency, the Kriging model is employed. The sampling points are distributed uniformly over the entire design space using the LHS. For evaluating the approximated model’s performance, the regression parameter is used as the error indicator. The objective functions based on approximated models are optimized using a sequential quadratic programming which has a higher efficiency and better convergence. Results show that through the application of this method, the deformations of the key points are in accord to the measured deformation within a small window of variability. The average difference between the deformation measured from the actual crash and that calculated from FE simulation using the optimum parameters as inputs is around 31 mm. The difference in the assessment index calculated from FE simulation with optimal assessment parameters and that from the Kriging model is only 1%. The proposed optimization methodology is a good tool to promptly reveal key parameters in a crash while simultaneously providing scientific basis for crash reconstruction.

Rotor blade aerodynamic shape optimization based on high-efficient optimization method

2019 ◽  
Vol 234 (2) ◽  
pp. 375-387 ◽  
Author(s):  
Qing Wang ◽  
Qijun Zhao
Keyword(s):  
Quadratic Programming ◽  
Sequential Quadratic Programming ◽  
Rotor Blade ◽  
High Performance ◽  
Kriging Model ◽  
Optimization Method ◽  
Tip Vortex ◽  
Design Variables ◽  
High Efficient ◽  
Blade Shape

In order to design a high-performance rotor, a high-efficient optimization method is established by coupling Kriging model and sequential quadratic programming with high-accuracy computational fluid dynamics method. In order to obtain the global optimal design point, the initial blade shape is optimized by using the Kriging model coupled with genetic algorithm based on the baseline rotor blade (Helishape 7A rotor). After that, the modified sequential quadratic programming method is employed to search the final blade shape based on the initial blade shape deeply. In the optimal process, the regions of design variables are restricted considering rotor dynamic characteristics. As a result, a new shape of rotor blade with characters of nonlinear twist, variational chord length, complex swept, and anhedral distributions is obtained. Compared with the baseline rotor, blade-tip vortex of the final optimized rotor is significantly weakened, the figure of merit of the final optimized rotor increases about 3.42%, and the peak of sound pressure decreases about 16.9%. At the same time, it is demonstrated that the final optimized rotor has better forward flight characteristics.

Design of evolutionary cubic spline intelligent solver for nonlinear Painlevé-I transcendent

2021 ◽  
Author(s):  
Sharafat Ali ◽  
Iftikhar Ahmad ◽  
Muhammad Asif Zahoor Raja ◽  
Siraj ul Islam Ahmad ◽  
Muhammad Shoaib
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Genetic Algorithms ◽  
Statistical Analysis ◽  
Quadratic Programming ◽  
Sequential Quadratic Programming ◽  
Process Variation ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Derivative ◽  
Search Technique

In this research paper, an innovative bio-inspired algorithm based on evolutionary cubic splines method (CSM) has been utilized to estimate the numerical results of nonlinear ordinary differential equation Painlevé-I. The computational mechanism is used to support the proposed technique CSM and optimize the obtained results with global search technique genetic algorithms (GAs) hybridized with sequential quadratic programming (SQP) for quick refinement. Painlevé-I is solved by the proposed technique CSM-GASQP. In this process, variation of splines is implemented for various scenarios. The CSM-GASQP produces an interpolated function that is continuous upto its second derivative. Also, splines proved to be stable than a single polynomial fitted to all points, and reduce wiggles between the tabulated points. This method provides a reliable and excellent procedure for adaptation of unknown coefficients of splines by searching globally exploiting the performance of GA-SQP algorithms. The convergence, exactness and accuracy of the proposed scheme are examined through the statistical analysis for the several independent runs.

Co-Optimization of Output and Reserve with Significant Penetration of Renewables

2013 ◽  
Vol 427-429 ◽  
pp. 341-345
Author(s):  
Xue Fei Chang ◽  
Zhe Yong Piao ◽  
Xiang Yu Lv ◽  
De Xin Li
Keyword(s):  
Quadratic Programming ◽  
Wind Power ◽  
Sequential Quadratic Programming ◽  
Renewable Generation ◽  
Renewable Sources ◽  
Reliability Level ◽  
Maximum Benefit ◽  
Proposed Model ◽  
Programming Techniques

Co-optimization of output and reserve is necessary in order to provide maximum benefit to both consumers and producers. Once renewable generation sources like wind or solar begin to make up a large proportion of the generation mix, this co-optimization becomes much more difficult since the output of renewable sources is not well-known in advance. In this paper, a uniform reliability level is used as a constraint in the process of output and reserve. The proposed model is tested on the modified 5-bus PJM system. The co-optimization is performed by sequential quadratic programming techniques. The results show that the co-optimization results are strongly related to the uncertainties of wind power, the reliability level of the system, and the reliability of generators when wind makes up a significant portion of the generation mix.

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