Univariate and multivariate optimization in Julia.
Optim.jl is part of the JuliaNLSolvers family.
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Optim.jl is a package for univariate and multivariate optimization of functions. A typical example of the usage of Optim.jl is
using Optim
rosenbrock(x) = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2
result = optimize(rosenbrock, zeros(2), BFGS())This minimizes the Rosenbrock function
with a = 1, b = 100 and the initial values x=0, y=0. The minimum is at (a,a^2).
The above code gives the output
* Status: success
* Candidate solution
Minimizer: [1.00e+00, 1.00e+00]
Minimum: 5.471433e-17
* Found with
Algorithm: BFGS
Initial Point: [0.00e+00, 0.00e+00]
* Convergence measures
|x - x'| = 3.47e-07 ≰ 0.0e+00
|x - x'|/|x'| = 3.47e-07 ≰ 0.0e+00
|f(x) - f(x')| = 6.59e-14 ≰ 0.0e+00
|f(x) - f(x')|/|f(x')| = 1.20e+03 ≰ 0.0e+00
|g(x)| = 2.33e-09 ≤ 1.0e-08
* Work counters
Seconds run: 0 (vs limit Inf)
Iterations: 16
f(x) calls: 53
∇f(x) calls: 53
To get information on the keywords used to construct method instances, use the Julia REPL help prompt (?)
help?> LBFGS
search: LBFGS
LBFGS
≡≡≡≡≡≡≡
Constructor
=============
LBFGS(; m::Integer = 10,
alphaguess = LineSearches.InitialStatic(),
linesearch = LineSearches.HagerZhang(),
P=nothing,
precondprep = (P, x) -> nothing,
manifold = Flat(),
scaleinvH0::Bool = true && (typeof(P) <: Nothing))
LBFGS has two special keywords; the memory length m, and
the scaleinvH0 flag. The memory length determines how many
previous Hessian approximations to store. When scaleinvH0
== true, then the initial guess in the two-loop recursion
to approximate the inverse Hessian is the scaled identity,
as can be found in Nocedal and Wright (2nd edition) (sec.
7.2).
In addition, LBFGS supports preconditioning via the P and
precondprep keywords.
Description
=============
The LBFGS method implements the limited-memory BFGS
algorithm as described in Nocedal and Wright (sec. 7.2,
2006) and original paper by Liu & Nocedal (1989). It is a
quasi-Newton method that updates an approximation to the
Hessian using past approximations as well as the gradient.
References
============
• Wright, S. J. and J. Nocedal (2006), Numerical
optimization, 2nd edition. Springer
• Liu, D. C. and Nocedal, J. (1989). "On the
Limited Memory Method for Large Scale
Optimization". Mathematical Programming B. 45
(3): 503–528
For more details and options, see the documentation
- STABLE — most recently tagged version of the documentation.
- LATEST — in-development version of the documentation.
The package is a registered package, and can be installed with Pkg.add.
julia> using Pkg; Pkg.add("Optim")or through the pkg REPL mode by typing
] add Optim
If you use Optim.jl in your work, please cite the following.
@article{mogensen2018optim,
author = {Mogensen, Patrick Kofod and Riseth, Asbj{\o}rn Nilsen},
title = {Optim: A mathematical optimization package for {Julia}},
journal = {Journal of Open Source Software},
year = {2018},
volume = {3},
number = {24},
pages = {615},
doi = {10.21105/joss.00615}
}