scholarly journals Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Chein-Shan Liu
Keyword(s):  
Optimal Algorithm ◽  
Optimal Parameter ◽  
Quadratic Function ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Iteration Step ◽  
Variable Metric ◽  
Nonlinear Minimization ◽  
Novel Algorithms ◽  
Variable Metric Method

To solve an unconstrained nonlinear minimization problem, we propose an optimal algorithm (OA) as well as a globally optimal algorithm (GOA), by deflecting the gradient direction to the best descent direction at each iteration step, and with an optimal parameter being derived explicitly. An invariant manifold defined for the model problem in terms of a locally quadratic function is used to derive a purely iterative algorithm and the convergence is proven. Then, the rank-two updating techniques of BFGS are employed, which result in several novel algorithms as being faster than the steepest descent method (SDM) and the variable metric method (DFP). Six numerical examples are examined and compared with exact solutions, revealing that the new algorithms of OA, GOA, and the updated ones have superior computational efficiency and accuracy.

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

scholarly journals Improved Immune Algorithm Combined with Steepest Descent Method for Optimal Design of IPMSM for FCEV Traction Motor

Energies ◽  
2021 ◽  
Vol 14 (13) ◽  
pp. 3904
Author(s):  
Ji-Chang Son ◽  
Myung-Ki Baek ◽  
Sang-Hun Park ◽  
Dong-Kuk Lim
Keyword(s):  
Optimal Design ◽  
Permanent Magnet Synchronous Motor ◽  
Torque Ripple ◽  
Descent Method ◽  
Immune Algorithm ◽  
Electric Machines ◽  
Steepest Descent Method ◽  
Superior Performance ◽  
Element Analysis ◽  
Traction Motor

In this paper, an improved immune algorithm (IIA) was proposed for the torque ripple reduction optimal design of an interior permanent magnet synchronous motor (IPMSM) for a fuel cell electric vehicle (FCEV) traction motor. When designing electric machines, both global and local solutions of optimal designs are required as design result should be compared in various aspects, including torque, torque ripple, and cogging torque. To lessen the computational burden of optimization using finite element analysis, the IIA proposes a method to efficiently adjust the generation of additional samples. The superior performance of the IIA was verified through the comparison of optimization results with conventional optimization methods in three mathematical test functions. The optimal design of an IPMSM using the IIA was conducted to verify the applicability in the design of practical electric machines.

Controlling Chaos by Hybrid System Based on FREN and Sliding Mode Control

2005 ◽  
Vol 128 (2) ◽  
pp. 352-358 ◽  
Author(s):  
C. Treesatayapun ◽  
S. Uatrongjit
Keyword(s):  
Sliding Mode ◽  
Adaptive Controller ◽  
Descent Method ◽  
Fuzzy Rules ◽  
System Stability ◽  
Steepest Descent Method ◽  
Upper And Lower Bounds ◽  
Adaptation Algorithm ◽  
Control Effort ◽  
Fine Tune

This paper presents a direct adaptive controller for chaotic systems. The proposed adaptive controller is constructed using the network called fuzzy rules emulated network (FREN). FREN’s structure is based on human knowledge in the form of fuzzy rules. Parameter adaptation algorithm based on the steepest descent method is presented to fine tune the controller’s performance. To improve the system stability, the modified sliding mode algorithm is applied to estimate the upper and lower bounds of the control effort. The suitable control effort is generated by FREN and kept within these bounds. Some computer simulations of using the controller to control the Hénon map have been performed to demonstrate the performance of the proposed controller.

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