A New Technique for Estimating the Lower Bound of the Trust-Region Subproblem *

Trust-region methods are popular for nonlinear optimization problems. How to determine the predicted reduction of the trust-region subproblem is a key issue for trust-region methods. Powell gave an estimation of the lower bound of the trust-region subproblem by considering the negative gradient direction. In this article, we give an alternate way to estimate the same lower bound of the trust-region subproblem.


Introduction
There are many methods for solving an unconstrained optimization problem  , where f is a smooth function.Line search methods [1,2] and trust-region methods [3] are the two popular classes of methods for the problem (1).In this article, we consider the trust-region method for this problem.The trustregion method for the problem (1) need to solve the following trust-region subproblem at every step: g s s B s (2) subject to , where is the gradient of the objective fun- B is an n by n symmetric matrix which approximates the Hessian matrix of f , and It is a key issue to estimate the lower bound of the trust-region subproblem (2) and (3) for analyzing the global convergence of the trust-region methods for the problem (1) [1][2][3][4][5].Powell [5] obtained a lower bound of the subproblem ( 2) and ( 3) by considering the quadratic model   k s q along the negative gradient direction k g .
In the next section we give an alternate way to obtain the same estimation of its lower bound.Throughout the paper  denotes the Euclidean vector norm or its induced matrix norm.

A New Estimation Technique
In this section, we give an alternate way to estimate the lower bound and upper bound of q s , respectively, where k s is the solution of the subproblem (2) and (3).Firstly, we give the well-known properties of the trust-region problem [6,7]: and there exists where is a symmetric matrix.This lemma can be proved by the KKT (Karush-Kuhn-Tucker) conditions for the constraint optimization problem [4].
Powell [5] obtained an estimation of the lower bound of (2) and (3), then The estimation of the predicted reduction ( 7) is a key property to analyze the convergence of the trust-region method.Here, we are interested in the question whether we can obtain the analogous result of (7) if we directly consider the solution of ( 4) and (5).Our motivation is that we obtain the solution of the trust-region subproblem ( 2) and (3) by solving (4)-( 6), and we need to directly consider the search direction k s of ( 4) and its restricted step [2] or the parameter k  [8][9][10][11] for some trust region methods.Therefore, we give an alternate way to estimate the lower bound of q s by directly considering the solution ( 4)-( 6) of the trust-region subproblem ( 2) and (3).
We will prove this property by distinguishing two cases separately, namely min  is nonnegative or negative.In this case, combining 0 k   with ( 10)-( 13), we obtain The other case is < (10) and (11), we obtain Combining ( 14) with (15), we obtain which shows the constraint is inactive, from ( 4) and ( 5), and the definition of which shows that ( 8) is also true.□

Conclusions
In this article, we give an alternate way to estimate the lower bound of the trust-region method.This new technique can be applied to analyze the convergence of trajectory-following methods for unconstrained optimization problems [9][10][11].The interested future works is that this new technique is applied to estimate the lower bound of the trust-region subproblem of the constraint optimization.

Acknowledgments
We would like to acknowledge Jiawang Nie and Hongchao Zhang for their constructive suggestions.