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LineSearches

Description

This package provides an interface to line search algorithms implemented in Julia. The code was originally written as part of Optim, but has now been separated out to its own package.

Available line search algorithms

In Example 1 we show how to choose between the line search algorithms in `Optim`.

• `HagerZhang` (Taken from the Conjugate Gradient implementation by Hager and Zhang, 2006)
• `MoreThuente` (From the algorithm in More and Thuente, 1994)
• `BackTracking` (Described in Nocedal and Wright, 2006)
• `StrongWolfe` (Nocedal and Wright)
• `Static` (Takes the proposed initial step length.)

Available initial step length procedures

The package provides some procedures to calculate the initial step length that is passed to the line search algorithm. See Example 2 for its usage in `Optim`.

• `InitialPrevious` (Use the step length from the previous optimization iteration)
• `InitialStatic` (Use the same initial step length each time)
• `InitialHagerZhang` (Taken from Hager and Zhang, 2006)
• `InitialQuadratic` (Propose initial step length based on a quadratic interpolation)
• `InitialConstantChange` (Propose initial step length assuming constant change in step length)

Example 1

This example shows how to use `LineSearches` with `Optim`. We solve the Rosenbrock problem with two different line search algorithms.

First, run `Newton` with the default line search algorithm:

``````using Optim, LineSearches
prob = Optim.UnconstrainedProblems.examples["Rosenbrock"]

algo_hz = Newton(linesearch = HagerZhang())
res_hz = Optim.optimize(prob.f, prob.g!, prob.h!, prob.initial_x, method=algo_hz)
``````

This gives the result

``````Results of Optimization Algorithm
* Algorithm: Newton's Method
* Starting Point: [0.0,0.0]
* Minimizer: [0.9999999999999994,0.9999999999999989]
* Minimum: 3.081488e-31
* Iterations: 14
* Convergence: true
* |x - x'| ≤ 1.0e-32: false
|x - x'| = 3.06e-09
* |f(x) - f(x')| ≤ 1.0e-32 |f(x)|: false
|f(x) - f(x')| = 3.03e+13 |f(x)|
* |g(x)| ≤ 1.0e-08: true
|g(x)| = 1.11e-15
* Stopped by an increasing objective: false
* Reached Maximum Number of Iterations: false
* Objective Calls: 44
* Gradient Calls: 44
* Hessian Calls: 14
``````

Now we can try `Newton` with the cubic backtracking line search:

``````algo_bt3 = Newton(linesearch = BackTracking(order=3))
res_bt3 = Optim.optimize(prob.f, prob.g!, prob.h!, prob.initial_x, method=algo_bt3)
``````

This gives the following result, reducing the number of function and gradient calls:

``````Results of Optimization Algorithm
* Algorithm: Newton's Method
* Starting Point: [0.0,0.0]
* Minimizer: [0.9999999959215587,0.9999999918223065]
* Minimum: 1.667699e-17
* Iterations: 14
* Convergence: true
* |x - x'| ≤ 1.0e-32: false
|x - x'| = 1.36e-05
* |f(x) - f(x')| ≤ 1.0e-32 |f(x)|: false
|f(x) - f(x')| = 1.21e+08 |f(x)|
* |g(x)| ≤ 1.0e-08: true
|g(x)| = 4.16e-09
* Stopped by an increasing objective: false
* Reached Maximum Number of Iterations: false
* Objective Calls: 19
* Gradient Calls: 15
* Hessian Calls: 14
``````

Example 2

This example shows how to use the initial step length procedures with `Optim`. We solve the Rosenbrock problem with two different procedures.

First, run `Newton` with the (default) initial guess and line search procedures.

``````using Optim, LineSearches
prob = Optim.UnconstrainedProblems.examples["Rosenbrock"]

algo_st = Newton(alphaguess = InitialStatic(), linesearch = HagerZhang())
res_st = Optim.optimize(prob.f, prob.g!, prob.h!, prob.initial_x, method=algo_st)
``````

This gives the result

``````Results of Optimization Algorithm
* Algorithm: Newton's Method
* Starting Point: [0.0,0.0]
* Minimizer: [0.9999999999999994,0.9999999999999989]
* Minimum: 3.081488e-31
* Iterations: 14
* Convergence: true
* |x - x'| ≤ 1.0e-32: false
|x - x'| = 3.06e-09
* |f(x) - f(x')| ≤ 1.0e-32 |f(x)|: false
|f(x) - f(x')| = 3.03e+13 |f(x)|
* |g(x)| ≤ 1.0e-08: true
|g(x)| = 1.11e-15
* Stopped by an increasing objective: false
* Reached Maximum Number of Iterations: false
* Objective Calls: 44
* Gradient Calls: 44
* Hessian Calls: 14
``````

We can now try with the initial step length guess from Hager and Zhang.

``````algo_prev = Newton(alphaguess = InitialHagerZhang(α0=1.0), linesearch = HagerZhang())
res_prev = Optim.optimize(prob.f, prob.g!, prob.h!, prob.initial_x, method=algo_prev)
``````

This gives the following result, reducing the number of function and gradient calls, but increasing the number of iterations.

``````Results of Optimization Algorithm
* Algorithm: Newton's Method
* Starting Point: [0.0,0.0]
* Minimizer: [0.9999999974436653,0.9999999948855858]
* Minimum: 6.535152e-18
* Iterations: 15
* Convergence: true
* |x - x'| ≤ 1.0e-32: false
|x - x'| = 1.09e-05
* |f(x) - f(x')| ≤ 1.0e-32 |f(x)|: false
|f(x) - f(x')| = 8.61e+08 |f(x)|
* |g(x)| ≤ 1.0e-08: true
|g(x)| = 4.41e-09
* Stopped by an increasing objective: false
* Reached Maximum Number of Iterations: false
* Objective Calls: 36
* Gradient Calls: 21
* Hessian Calls: 15
``````

References

• W. W. Hager and H. Zhang (2006) "Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent." ACM Transactions on Mathematical Software 32: 113-137.
• Moré, Jorge J., and David J. Thuente. "Line search algorithms with guaranteed sufficient decrease." ACM Transactions on Mathematical Software (TOMS) 20.3 (1994): 286-307.
• Nocedal, Jorge, and Stephen Wright. "Numerical optimization." Springer Science & Business Media, 2006.

09/07/2016

2 days ago

180 commits