scipy.optimize.dual_annealing#
- scipy.optimize.dual_annealing(func, bounds, args=(), maxiter=1000, minimizer_kwargs=None, initial_temp=5230.0, restart_temp_ratio=2e-05, visit=2.62, accept=-5.0, maxfun=10000000.0, seed=None, no_local_search=False, callback=None, x0=None)[source]#
Find the global minimum of a function using Dual Annealing.
- Parameters:
- funccallable
The objective function to be minimized. Must be in the form
f(x, *args)
, wherex
is the argument in the form of a 1-D array andargs
is a tuple of any additional fixed parameters needed to completely specify the function.- boundssequence or
Bounds
Bounds for variables. There are two ways to specify the bounds:
Instance of
Bounds
class.Sequence of
(min, max)
pairs for each element in x.
- argstuple, optional
Any additional fixed parameters needed to completely specify the objective function.
- maxiterint, optional
The maximum number of global search iterations. Default value is 1000.
- minimizer_kwargsdict, optional
Extra keyword arguments to be passed to the local minimizer (
minimize
). Some important options could be:method
for the minimizer method to use andargs
for objective function additional arguments.- initial_tempfloat, optional
The initial temperature, use higher values to facilitates a wider search of the energy landscape, allowing dual_annealing to escape local minima that it is trapped in. Default value is 5230. Range is (0.01, 5.e4].
- restart_temp_ratiofloat, optional
During the annealing process, temperature is decreasing, when it reaches
initial_temp * restart_temp_ratio
, the reannealing process is triggered. Default value of the ratio is 2e-5. Range is (0, 1).- visitfloat, optional
Parameter for visiting distribution. Default value is 2.62. Higher values give the visiting distribution a heavier tail, this makes the algorithm jump to a more distant region. The value range is (1, 3].
- acceptfloat, optional
Parameter for acceptance distribution. It is used to control the probability of acceptance. The lower the acceptance parameter, the smaller the probability of acceptance. Default value is -5.0 with a range (-1e4, -5].
- maxfunint, optional
Soft limit for the number of objective function calls. If the algorithm is in the middle of a local search, this number will be exceeded, the algorithm will stop just after the local search is done. Default value is 1e7.
- seed{None, int,
numpy.random.Generator
,numpy.random.RandomState
}, optional If seed is None (or np.random), the
numpy.random.RandomState
singleton is used. If seed is an int, a newRandomState
instance is used, seeded with seed. If seed is already aGenerator
orRandomState
instance then that instance is used. Specify seed for repeatable minimizations. The random numbers generated with this seed only affect the visiting distribution function and new coordinates generation.- no_local_searchbool, optional
If no_local_search is set to True, a traditional Generalized Simulated Annealing will be performed with no local search strategy applied.
- callbackcallable, optional
A callback function with signature
callback(x, f, context)
, which will be called for all minima found.x
andf
are the coordinates and function value of the latest minimum found, andcontext
has value in [0, 1, 2], with the following meaning:0: minimum detected in the annealing process.
1: detection occurred in the local search process.
2: detection done in the dual annealing process.
If the callback implementation returns True, the algorithm will stop.
- x0ndarray, shape(n,), optional
Coordinates of a single N-D starting point.
- Returns:
- resOptimizeResult
The optimization result represented as a
OptimizeResult
object. Important attributes are:x
the solution array,fun
the value of the function at the solution, andmessage
which describes the cause of the termination. SeeOptimizeResult
for a description of other attributes.
Notes
This function implements the Dual Annealing optimization. This stochastic approach derived from [3] combines the generalization of CSA (Classical Simulated Annealing) and FSA (Fast Simulated Annealing) [1] [2] coupled to a strategy for applying a local search on accepted locations [4]. An alternative implementation of this same algorithm is described in [5] and benchmarks are presented in [6]. This approach introduces an advanced method to refine the solution found by the generalized annealing process. This algorithm uses a distorted Cauchy-Lorentz visiting distribution, with its shape controlled by the parameter \(q_{v}\)
\[g_{q_{v}}(\Delta x(t)) \propto \frac{ \ \left[T_{q_{v}}(t) \right]^{-\frac{D}{3-q_{v}}}}{ \ \left[{1+(q_{v}-1)\frac{(\Delta x(t))^{2}} { \ \left[T_{q_{v}}(t)\right]^{\frac{2}{3-q_{v}}}}}\right]^{ \ \frac{1}{q_{v}-1}+\frac{D-1}{2}}}\]Where \(t\) is the artificial time. This visiting distribution is used to generate a trial jump distance \(\Delta x(t)\) of variable \(x(t)\) under artificial temperature \(T_{q_{v}}(t)\).
From the starting point, after calling the visiting distribution function, the acceptance probability is computed as follows:
\[p_{q_{a}} = \min{\{1,\left[1-(1-q_{a}) \beta \Delta E \right]^{ \ \frac{1}{1-q_{a}}}\}}\]Where \(q_{a}\) is a acceptance parameter. For \(q_{a}<1\), zero acceptance probability is assigned to the cases where
\[[1-(1-q_{a}) \beta \Delta E] < 0\]The artificial temperature \(T_{q_{v}}(t)\) is decreased according to
\[T_{q_{v}}(t) = T_{q_{v}}(1) \frac{2^{q_{v}-1}-1}{\left( \ 1 + t\right)^{q_{v}-1}-1}\]Where \(q_{v}\) is the visiting parameter.
New in version 1.2.0.
References
[1]Tsallis C. Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 52, 479-487 (1998).
[2]Tsallis C, Stariolo DA. Generalized Simulated Annealing. Physica A, 233, 395-406 (1996).
[3]Xiang Y, Sun DY, Fan W, Gong XG. Generalized Simulated Annealing Algorithm and Its Application to the Thomson Model. Physics Letters A, 233, 216-220 (1997).
[4]Xiang Y, Gong XG. Efficiency of Generalized Simulated Annealing. Physical Review E, 62, 4473 (2000).
[5]Xiang Y, Gubian S, Suomela B, Hoeng J. Generalized Simulated Annealing for Efficient Global Optimization: the GenSA Package for R. The R Journal, Volume 5/1 (2013).
[6]Mullen, K. Continuous Global Optimization in R. Journal of Statistical Software, 60(6), 1 - 45, (2014). DOI:10.18637/jss.v060.i06
Examples
The following example is a 10-D problem, with many local minima. The function involved is called Rastrigin (https://en.wikipedia.org/wiki/Rastrigin_function)
>>> import numpy as np >>> from scipy.optimize import dual_annealing >>> func = lambda x: np.sum(x*x - 10*np.cos(2*np.pi*x)) + 10*np.size(x) >>> lw = [-5.12] * 10 >>> up = [5.12] * 10 >>> ret = dual_annealing(func, bounds=list(zip(lw, up))) >>> ret.x array([-4.26437714e-09, -3.91699361e-09, -1.86149218e-09, -3.97165720e-09, -6.29151648e-09, -6.53145322e-09, -3.93616815e-09, -6.55623025e-09, -6.05775280e-09, -5.00668935e-09]) # random >>> ret.fun 0.000000