Trust-region methods

Authors: Wenhe (Wayne) Ye (ChE 345 Spring 2014) Steward: Dajun Yue, Fengqi You Date Presented: Apr. 10, 2014

Introduction

Trust-region method (TRM) is one of the most important numerical optimization method in solving nonlinear programing (NLP) problems. It works in a way that first define a region around the current best solution, in which a certain model (usually a quadratic model) can to some extent approximate the original objective function. TRM then take a step forward according to the model depicts within the region. Unlike the line search method, TRM usually determines both the step size and direction at the same time. If a notable decrease (our following discussion will based on minimization problems) is gained after the step forward, then the model is believed to be a good representation of the original objective function. If the improvement is too subtle or even a negative improvement is gained, the region is not to be believed as a good representation of the original objective function. The convergence is thus ensured that the size of the “trust region” (usually the radius in Euclidean norm) in each iteration would depend on the improvement previously made.

Important Concepts

Trust-region

In most cases, the trust-region is defined as a spherical area of radius $\Delta_k$ in which the trust-region subproblem lies.

Trust-region subproblem

If we are using the quadratic model to approximate the original objective function, then our optimization problem is essentially reduced to solving a sequence of trust-region subporblems

$min~m_k(p)=f_k+{g_k}^Tp+\frac{1}{2}p^TB_kp$

$s.t.~||p||<=\Delta_k$

Where $\Delta_k$ is the trust region radius, $g_k$ is the gradient at current point and $B_k$ is the hessian (or a hessian approximation). It is easy to find the solution to the trust-region subproblem if $B_k$ is positive definite.

Actual reduction and predicted reduction

The most critical issue underlying the trust-region method is to update the size of the trust-region at every iteration. If the current iteration makes a satisfactory reduction, we may exploits our model more in the next iteration by setting a larger $\Delta_k$ . If we only achieved a limited improvement after the current iteration, the radius of the trust-region then should not have any increase, or in the worst cases, we may decrease the size of the trust-region by adjusting the radius to a smaller value to check the model’s validity.

$\rho_k=\frac{f(x_k)-f(x_k+p_k)}{m_k(0)-m_k(p_k)}$

Whether to take a more ambitious step or a more conservative one is depend on the ratio between the actual reduction gained by true reduction in the original objective function and the predicted reduction expected in the model function. Empirical threshold values of the ratio $\rho_k$ will guide us in determining the size of the trust-region.

The picture shows both the stepsize and the improving direction is a consequence of a pre-determined trust-region size

Trust Region Algorithm

Before implementing the trust-region algorithm, we should first determine several parameters. $\Delta_M$ is the upper bound for the size of the trust region. $\eta_1$ , $\eta_2$ and $\eta_3$ , $T_1$ , $T_2$ are the threshold values for evaluating the goodness of the quadratic model thus for determining the trust-region’s size in the next iteration.

Pseudo-code

Decide the starting point at $x_0$ , set the iteration number $k=1$

For $k=1,2...$

Get the improving step by solving trust-region subproblem ()

Evaluate $\rho_k$ from equation()

If $\rho_k<\eta_2$

$\Delta_k+1=T_1*\Delta_k$

Else

If $\rho_k<\eta_3$ and $p_k=||\Delta_k||$ (full step and model is a good approximation)

$\Delta_k+1=min(T_2\Delta_k,\Delta_M)$

Else

$\Delta_k+1=\Delta_k$

If $\rho_k>\eta_1$

$x_k+1=x_k+p_k$

Else

$x_k+1=x_k$ (the model is not a good approximation and need to solve another trust-region subproblem within a smaller trust-region)

end >

Methods of Solving the Trust-region Subproblem

Cauchy point calculation

In line search methods, we may find an improving direction from the gradient information, that is, by taking the steepest descent direction with regard to the maximum range we could make. We can solve the trust-region subproblem in an inexpensive way. This method is also denoted as the Cauchy point calculation. We can also express the improving step explicitly by the following closed-form equations

${p_k}^C=-\tau_k\dfrac{\Delta_k}{||g_k||}g_k$

if ${g_k}^TB_kg_k<=0$ $\tau_k=1$

otherwise $\tau_k=min~ ({||g_k||}^3/(\Delta_k{g_k}^TB_kg_k),1)$

Limitations and Further Improvements

Though Cauchy point is cheap to implement, like the steepest descent method, it performs poorly in some cases. Varies kinds of improvements are based on including the curvature information from $B_k$ .

Dogleg Method

If $B_k$ is positive definite (we can use quasi-Newton hessian approximation to guarantee), then a V-shaped trajectory can be determined by

if $0<=\tau<=1,~~p(\tau)=\tau p^U$

if $1<=\tau<=2,~~p(\tau)=\tau p^U+(\tau-1)(p^B-p^U)$

where $p^U=-\frac{g^Tg}{g^TBg}g$ is the steepest descent direction

Note that hessian or approximate hessian will be evaluated

Conjugated Gradient Steihaug’s Method

The most widely used method for solving a trust-region subproblem is by using the idea of conjugated gradient (CG) method for minimizing a quadratic function since CG guarantees a convergence within finite steps for a quadratic programing. Also, CG Steihaug’s method has the merit of Cauchy point and Dogleg method that both in terms of superlinear convergence rate and inexpensiveness to compute.

Pseudo-code for CG Steihaug method in solving trust region subproblem

Given tolerance $\epsilon_k > 0$ ;