文档介绍：Models for trust-region methods
Models for trust-region methods
The trust-region Newton method has proved to be ints.
How to successively select N measurement points so that we can determine the smallest possible region of uncertainty in which the minimum must lie.
Fibonacci search
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Fibonacci search
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Golden section search
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If the number N of allowed measurement points in a Fibonacci search is made to approach infinity, we obtain the golden section method.
The interval of uncertainty at any point in the process has width:
Step length
Inexact line search
Two Stages :
A bracketing phase — finds an interval
A bisection or interpolation phase — computes a good step length
Termination condition
The Wolfe conditions
Armijo condition
Sufficient decrease condition
The Wolfe conditions
The Wolfe conditions
Curvature condition
The Wolfe conditions
The Wolfe conditions
The strong Wolfe conditions
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The existence of the step length
The existence of the step length
(1)
(2)
Combining (1) and (2) gives
The strong Wolfe conditions hold in the same interval.
The Goldstein conditions
Sufficient decrease condition
Introduced to control the
step length from below
The Goldstein conditions
Convergence of line search methods
Convergence of line search methods
Zoutendijk Condition
Convergence of line search methods
while the Lipschitz condition implies that
Substituting this inequality into the first Wolfe condition, we obtain
Convergence of line search methods
summing all
Convergence of line search methods
This limit can be used in turn to derive global convergence results for line search algorithms.
globally convergent
Convergence of line search methods
Steepest descent methods
Now we consider the ideal case, in which the objective function is quadratic and the line searches are exact.
symmetric and positive defini