scholarly journals Constrained global optimization of multivariate polynomials using polynomial B-spline form and B-spline consistency prune approach

2021 ◽  
Author(s):  
Deepak Devidasrao Gawali ◽  
Bhagyesh V. Patil ◽  
Ahmed Zidna ◽  
P. S. V. Nataraj
Keyword(s):  
Global Optimization ◽  
Global Minimum ◽  
Performance Metrics ◽  
Test Problems ◽  
Constrained Global Optimization ◽  
Multivariate Polynomial ◽  
Basic Algorithm ◽  
B Spline ◽  
Optimal Value ◽  
Improved Algorithm

In this paper, we propose basic and improved algorithms based on polynomial B-spline form for constrained global optimization of multivariate polynomial functions. The proposed algorithms are based on a branch-and-bound framework. In improved algorithm we introduce several new ingredients, such as B-spline box consistency and B-spline hull consistency algorithm to prune the search regions and make the search more efficient. The performance of the basic and improved algorithm is tested and compared on set of test problems. The results of the tests show the superiority of the improved algorithm over the basic algorithm in terms of the chosen performance metrics. We compare optimal value of global minimum obtained using the proposed algorithms with CENSO, GloptiPoly and several state-of-the-art NLP solvers, on set of $11$ test problems. The results of the tests show the superiority of the proposed algorithm and CENSO solver (open source solver for global optimization of B-spline constrained problem) in that it always captures the global minimum to the user-specified accuracy.

scholarly journals A flexible generator of constrained global optimization test problems

2019 ◽  
Author(s):  
Victor Gergel ◽  
Konstantin Barkalov ◽  
Ilya Lebedev ◽  
Maria Rachinskaya ◽  
Alexander Sysoyev
Keyword(s):  
Global Optimization ◽  
Test Problems ◽  
Constrained Global Optimization

scholarly journals Global optimization of an encapsulated Si/SiO$$_2$$ L3 cavity with a 43 million quality factor

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
J. P. Vasco ◽  
V. Savona
Keyword(s):  
Global Optimization ◽  
Photonic Crystal ◽  
Quality Factor ◽  
State Of The Art ◽  
Optimization Strategy ◽  
High Quality ◽  
Optimal Value ◽  
Out Of Plane ◽  
Fabrication Tolerances ◽  
Structural Imperfections

AbstractWe optimize a silica-encapsulated silicon L3 photonic crystal cavity for ultra-high quality factor by means of a global optimization strategy, where the closest holes surrounding the cavity are varied to minimize out-of-plane losses. We find an optimal value of $$Q_c=4.33\times 10^7$$ Q c = 4.33 × 10 7 , which is predicted to be in the 2 million regime in presence of structural imperfections compatible with state-of-the-art silicon fabrication tolerances.

scholarly journals A Kriging-Assisted Multi-Objective Constrained Global Optimization Method for Expensive Black-Box Functions

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 149
Author(s):  
Yaohui Li ◽  
Jingfang Shen ◽  
Ziliang Cai ◽  
Yizhong Wu ◽  
Shuting Wang
Keyword(s):  
Global Optimization ◽  
Pareto Frontier ◽  
Kriging Model ◽  
Optimization Method ◽  
Function Evaluation ◽  
Sampling Point ◽  
Constrained Global Optimization ◽  
Multi Objective ◽  
Sample Data ◽  
Sampling Points

The kriging optimization method that can only obtain one sampling point per cycle has encountered a bottleneck in practical engineering applications. How to find a suitable optimization method to generate multiple sampling points at a time while improving the accuracy of convergence and reducing the number of expensive evaluations has been a wide concern. For this reason, a kriging-assisted multi-objective constrained global optimization (KMCGO) method has been proposed. The sample data obtained from the expensive function evaluation is first used to construct or update the kriging model in each cycle. Then, kriging-based estimated target, RMSE (root mean square error), and feasibility probability are used to form three objectives, which are optimized to generate the Pareto frontier set through multi-objective optimization. Finally, the sample data from the Pareto frontier set is further screened to obtain more promising and valuable sampling points. The test results of five benchmark functions, four design problems, and a fuel economy simulation optimization prove the effectiveness of the proposed algorithm.

Solution of non-linear time fractional telegraph equation with source term using B-spline and Caputo derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdul Majeed ◽  
Mohsin Kamran ◽  
Noreen Asghar
Keyword(s):  
Caputo Derivative ◽  
Numerical Solutions ◽  
Linear Time ◽  
Initial Boundary ◽  
Telegraph Equation ◽  
Test Problems ◽  
Von Neumann ◽  
B Spline ◽  
Nonlinear Part ◽  
Spline Basis

Abstract This article focusses on the implementation of cubic B-spline approach to investigate numerical solutions of inhomogeneous time fractional nonlinear telegraph equation using Caputo derivative. L1 formula is used to discretize the Caputo derivative, while B-spline basis functions are used to interpolate the spatial derivative. For nonlinear part, the existing linearization formula is applied after generalizing it for all positive integers. The algorithm for the simulation is also presented. The efficiency of the proposed scheme is examined on three test problems with different initial boundary conditions. The influence of parameter α on the solution profile for different values is demonstrated graphically and numerically. Moreover, the convergence of the proposed scheme is analyzed and the scheme is proved to be unconditionally stable by von Neumann Fourier formula. To quantify the accuracy of the proposed scheme, error norms are computed and was found to be effective and efficient for nonlinear fractional partial differential equations (FPDEs).

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