Trust-region methods

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

Contents

Introduction

A pictorial view of trust-region method optimization trajectory

Trust-region method (TRM) is one of the most important numerical optimization methods in solving nonlinear programming (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 methods, TRM usually determines the step size before the improving direction (or 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, then the model is not to be believed as a good representation of the original objective function within that region. The convergence can be ensured that the size of the “trust region” (usually defined by the radius in Euclidean norm) in each iteration would depend on the improvement previously made.

Important Concepts

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

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.


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. A typical set for these values are 0=<\eta_1<=\eta_2, \eta_2=0.25 and \eta_3=0.75,t_1=0.25,t_2=2.0.


Pseudo-code

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

for k=1,2...

Get the improving step by solving trust-region sub-problem ()

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

Illustration of 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

Illustration of dogleg method trajectory.

If B_k is positive definite (we can use quasi-Newton Hessian approximation &updating 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 and p^B is the optimal solution of the quadratic model m_k(p). Therefore, a further improvement could be achieved compared to using only Cauchy point calculation method in one iteration. (Note that hessian or approximate hessian will be evaluated in dogleg method)


Conjugated Gradient Steihaug’s Method

The most widely used method for solving a trust-region sub-problem is by using the idea of conjugated gradient (CG) method for minimizing a quadratic function since CG guarantees convergence within a finite number of iterations for a quadratic programming. Also, CG Steihaug’s method has the merit of Cauchy point calculation and dogleg method that both in terms of super-linear convergence rate and inexpensiveness to compute.(No expensive Hessian evaluation)


Pseudo-code for CG Steihaug method in solving trust region sub-problem

Given tolerance \epsilon_k > 0;

Set z_0=0, r_0=\nabla f_k, d_0=-r_0=-\nabla f_k

if ||r_0|| <\epsilon_k

return p_k = z_0 = 0;

for j = 0, 1, 2, . . .

if {d_j}^TB_k d_j <= 0

Find \tau such that p_k = z_j + \tau d_j minimizes m_k(p_k)

and satisfies ||p_k|| =  \Delta_k ;

return p_k ;

Set \alpha_j = {r_j}^Tr_j /{d_j}^TB_kd_j;

Set z_{j+1} = z_j + \alpha_jd_j ;

if ||z_{j+1}|| >= \Delta_k

Find \tau >= 0 such that p_k = z_j + \tau d_j satisfies ||p_k|| =  \Delta_k ;

return p_k ;

Set r_{j+1} = r_j + \alpha_j B_kd_j ;

if ||r_{j+1}|| <\epsilon_k

return p_k = z_{j+1};

Set \beta_{j+1} = \frac{{r_{j+1}}^T{r_{j+1}}}{{r_j}^Tr_j} ;

Set d_{j+1}=-r_{j+1}+\beta_{j+1}d_j

end


Example

Contour of a 'Branin' function.

Here we use the trust-region method to solve an unconstrained problem as an example. The trust-region subproblems are solved by calculate the Cauchy point.

min~f(x_1,x_2)=(x_2-0.129{x_1}^2+1.6x_1-6)^2+6.07cos(x_1)+10 (This is the Branin function which is widely used as a test function. It has 3 global optima.)

Starting point x_1=6.00, x_2=14.00 The iteration stops when the stopping criteria ||g_k||<=0.01 is met.

\Delta_0=2.0, \Delta_M=5.0, t_1=0.25, t_2=2.0, \eta_1=0.2,\eta_2=0.25,\eta_3=0.75


Improving Process
Graphical illustration of iteration 1 (The quadratic model's contours are marked as red lines).
Graphical illustration of iteration2.
Graphical illustration of iteration3.
Graphical illustration of iteration4.
Graphical illustration of iteration5.
Graphical illustration of iteration6.
Optimization trajectory of the example function(unconstrained).

Iteration 1: The algorithm start from the initial point(marked as a green dot) x_1=6.00, x_2=14.00. The trust-region is defined as the area inside the circle centered at the starting point. The contour of the quadratic model can be visualized. After calculating the Cauchy point, \rho_k is evaluated and a full step was taken since the model gives a good prediction. Set x_{k+1}=x_k+p_k(x_1=5.767,x_2=12.014) and \Delta_k=min(2\Delta_k,\Delta_M)~(\rho_k>\eta_3 and a full step was taken.) (The current best solution is denoted as the red dot.)


Iteration 2: Start with x_1=5.767,x_2=12.014 and an enlarged trust-region. The new iteration gives a more ambitious full step to the new point(x_1=4.800,x_2=8.132). With \rho_k=0.980, the model is "trusted" again to increase its size in the next iteration.


Iteration 3: Start with x_1=4.800,x_2=8.132 and an enlarged trust-region. The new iteration gives a satisfactory but not good enough to the new point(x_1=1.668,x_2=4.235). With \rho_k=0.578, which is not high enough to trigger a new increment for the trust-region's size. So the radius is maintained in the next iteration.


Iteration 4: Start with x_1=1.668,x_2=4.235 and the maximum-sized trust-region. The new iteration gives a poor prediction. With \rho_k=-0.160, which incurs the decrease in the trust-region's size to improve the model's validity. Current best solution is unchanged and the radius for the trust-region is diminished to 1/4 of the current iteration.


Iteration 5: Start with x_1=1.668,x_2=4.235 and a shrinked trust-region. The new iteration gives a satisfactory but not good enough to the new point(x_1=2.887,x_2=3.956). With \rho_k=0.729, which is not high enough to trigger a new increment for the trust-region's size. So the radius is maintained in the next iteration.


Iteration 6: Start with x_1=2.887,x_2=3.956. The new iteration gives a satisfactory but not a full step to the new point(x_1=2.594,x_2=3.109). With \rho_k=0.989, which is high enough to trigger a new increment for the trust-region's size, however not a full step is taken thereby the radius is maintained in the next iteration.


....


The following table is a summary for the improving process. Notably the Cauchy point calculation does not give an efficient convergence rate especially at the end stage of the computation(not a full step was taken so it is essentially equivalent to a steepest decent line search algorithm). Dogleg and CG Steihaug's method will give faster convergence as explained previously.

TRM-Table.png

Another graphical illustration is available at Kranf site: [1]

Conclusion

Trust-Region vs. Line Search

Line search methods:

- Pick improving direction

- Pick step-size to minimize

- Update the incumbent solution

- Convergence rate is not guaranteed


Trust-region methods:

- Pick the step-size (the trust-region sub-problem is constrained)

- Solving the sub-problem using the approximated model

- If the improvement is acceptable, update the incumbent solution and the size of the trust-region

- Can have super-linear convergence rate when conjugated gradient method or dogleg method is used

References

[1] W. Sun and Y.-x. Yuan, Optimization theory and methods : nonlinear programming. New York: Springer, 2006.

[2] J. Nocedal, S. J. Wright, and SpringerLink (Online service). (2006). Numerical optimization (2nd ed.). Available: [2]

[3] L. Hei, "Practical techniques for nonlinear optimization," Ph D, Northwestern University, 2007.

[4] Wikipedia page for Trust region. Available: [3]