scholarly journals A Simple Alternating Direction Method for the Conic Trust Region Subproblem

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Honglan Zhu ◽  
Qin Ni
Keyword(s):  
Trust Region ◽  
Alternating Direction Method ◽  
New Method ◽  
Quadratic Model ◽  
Descent Direction ◽  
Orthogonal Direction ◽  
Trust Region Subproblem ◽  
One Dimensional ◽  
Alternating Direction ◽  
Low Dimensional

A simple alternating direction method is used to solve the conic trust region subproblem of unconstrained optimization. By use of the new method, the subproblem is solved by two steps in a descent direction and its orthogonal direction, the original conic trust domain subproblem into a one-dimensional subproblem and a low-dimensional quadratic model subproblem, both of which are very easy to solve. Then the global convergence of the method under some reasonable conditions is established. Numerical experiment shows that the new method seems simple and effective.

scholarly journals SQP alternating direction method with a new optimal step size for solving variational inequality problems with separable structure

2018 ◽  
Vol 12 (1) ◽  
pp. 224-243 ◽  
Author(s):  
Abdellah Bnouhachem ◽  
Themistocles Rassias
Keyword(s):  
Variational Inequality ◽  
Alternating Direction Method ◽  
Convex Programming Problem ◽  
Descent Direction ◽  
Computational Results ◽  
Step Size ◽  
Sqp Method ◽  
Separable Structure ◽  
Alternating Direction ◽  
Optimal Step Size

In this paper, we suggest and analyze a new alternating direction scheme for the separable constrained convex programming problem. The theme of this paper is twofold. First, we consider the square-quadratic proximal (SQP) method. Next, by combining the alternating direction method with SQP method, we propose a descent SQP alternating direction method by using the same descent direction as in [6] with a new step size ?k. Under appropriate conditions, the global convergence of the proposed method is proved. We show the O(1/t) convergence rate for the SQP alternating direction method. Some preliminary computational results are given to illustrate the efficiency of the proposed method.

scholarly journals Exploring a new method based on generalized eigenvalues for the trust-region subproblem

2019 ◽  
Author(s):  
Jean Carlos Aparecido Medeiros ◽  
Sandra Augusta Santos
Keyword(s):  
Trust Region ◽  
Poor Performance ◽  
Original Method ◽  
New Method ◽  
Trust Region Subproblem ◽  
Generalized Eigenvalues ◽  
Geometric Characteristics

In 2017, a method for solving the trust-region subproblem using generalized eigenvalues was rediscovered and improved. In 1989, when the original method was proposed, it presented a poor performance, which was caused by the low quality of eigensolvers available at that time. In this work we explore some geometric characteristics of this method.

An adaptive nonmonotone trust region method based on a modified scalar approximation of the Hessian in the successive quadratic subproblems

2019 ◽  
Vol 53 (3) ◽  
pp. 829-839
Author(s):  
Saeed Rezaee ◽  
Saman Babaie-Kafaki
Keyword(s):  
Numerical Experiments ◽  
Trust Region Method ◽  
Trust Region ◽  
Quadratic Model ◽  
Test Problems ◽  
Trust Region Subproblem ◽  
Trust Region Algorithm ◽  
Scalar Approximation ◽  
Secant Equation ◽  
Modified Secant Equation

Based on a modified secant equation, we propose a scalar approximation of the Hessian to be used in the trust region subproblem. Then, we suggest an adaptive nonmonotone trust region algorithm with a simple quadratic model. Under proper conditions, it is briefly shown that the proposed algorithm is globally and locally superlinearly convergent. Numerical experiments are done on a set of unconstrained optimization test problems of the CUTEr collection, using the Dolan-Moré performance profile. They demonstrate efficiency of the proposed algorithm.

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