A Family of Trust-Region-Based Algorithms for Unconstrained Minimization with Strong Global Convergence Properties

1985 ◽  
Vol 22 (1) ◽  
pp. 47-67 ◽  
Author(s):  
Gerald A. Shultz ◽  
Robert B. Schnabel ◽  
Richard H. Byrd
Keyword(s):  
Global Convergence ◽  
Trust Region ◽  
Unconstrained Minimization ◽  
Convergence Properties

scholarly journals A Stochastic Trust Region Method for Unconstrained Optimization Problems

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Ningning Li ◽  
Dan Xue ◽  
Wenyu Sun ◽  
Jing Wang
Keyword(s):  
Objective Function ◽  
Optimization Problems ◽  
Trust Region Method ◽  
Trust Region ◽  
Unconstrained Minimization ◽  
Quadratic Model ◽  
Convergence Properties ◽  
Region Method ◽  
Unconstrained Optimization Problems

In this paper, a stochastic trust region method is proposed to solve unconstrained minimization problems with stochastic objectives. Particularly, this method can be used to deal with nonconvex problems. At each iteration, we construct a quadratic model of the objective function. In the model, stochastic gradient is used to take the place of deterministic gradient for both the determination of descent directions and the approximation of the Hessians of the objective function. The behavior and the convergence properties of the proposed method are discussed under some reasonable conditions. Some preliminary numerical results show that our method is potentially efficient.

scholarly journals A new trust region method for nonsmooth equations

2003 ◽  
Vol 44 (4) ◽  
pp. 595-607 ◽  
Author(s):  
Y. F. Yang
Keyword(s):  
Global Convergence ◽  
Smooth Function ◽  
Trust Region Method ◽  
Trust Region ◽  
Nonsmooth Equations ◽  
Trust Region Methods ◽  
Convergence Properties ◽  
Trust Region Algorithm ◽  
Consistency Property ◽  
Quadratical Convergence

AbstractWe propose a new trust region algorithm for solving the system of nonsmooth equations F(x) = 0 by using a smooth function satisfying the Jacobian consistency property to approximate the nonsmooth function F(x). Compared with existing trust region methods for systems of nonsmooth equations, the proposed algorithm possesses some nice convergence properties. Global convergence is established and, in particular, locally superlinear or quadratical convergence is obtained if F is semismooth or strongly semismooth at the solution.

scholarly journals Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1551
Author(s):  
Bothina El-Sobky ◽  
Yousria Abo-Elnaga ◽  
Abd Allah A. Mousa ◽  
Mohamed A. El-Shorbagy
Keyword(s):  
Nonlinear Programming ◽  
Global Convergence ◽  
Programming Problem ◽  
Trust Region ◽  
Convergence Theory ◽  
Benchmark Problems ◽  
Nonlinear Programming Problem ◽  
Barrier Method ◽  
Starting Point ◽  
Constrained Nonlinear Programming

In this paper, a penalty method is used together with a barrier method to transform a constrained nonlinear programming problem into an unconstrained nonlinear programming problem. In the proposed approach, Newton’s method is applied to the barrier Karush–Kuhn–Tucker conditions. To ensure global convergence from any starting point, a trust-region globalization strategy is used. A global convergence theory of the penalty–barrier trust-region (PBTR) algorithm is studied under four standard assumptions. The PBTR has new features; it is simpler, has rapid convergerce, and is easy to implement. Numerical simulation was performed on some benchmark problems. The proposed algorithm was implemented to find the optimal design of a canal section for minimum water loss for a triangle cross-section application. The results are promising when compared with well-known algorithms.

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