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Minimization of scalar function of one or more variables.
| Parameters: | fun : callable x0 : ndarray
args : tuple, optional
method : str or callable, optional
Photoshop software for mac. jac : bool or callable, optional
hess, hessp : callable, optional
bounds : sequence, optional
constraints : dict or sequence of dict, optional
tol : float, optional
options : dict, optional
callback : callable, optional
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|---|---|
| Returns: | res : OptimizeResult
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Notes
This section describes the available solvers that can be selected by the‘method’ parameter. The default method is BFGS.
Unconstrained minimization
Method Nelder-Mead uses the Simplex algorithm [R123], [R124]. Thisalgorithm has been successful in many applications but other algorithmsusing the first and/or second derivatives information might be preferredfor their better performances and robustness in general.
Method Powell is a modification of Powell’s method [R125], [R126] whichis a conjugate direction method. It performs sequential one-dimensionalminimizations along each vector of the directions set (direc field inoptions and info), which is updated at each iteration of the mainminimization loop. The function need not be differentiable, and noderivatives are taken.
Method CG uses a nonlinear conjugate gradient algorithm by Polak andRibiere, a variant of the Fletcher-Reeves method described in [R127] pp.120-122. Only the first derivatives are used.
Method BFGS uses the quasi-Newton method of Broyden, Fletcher,Goldfarb, and Shanno (BFGS) [R127] pp. 136. It uses the first derivativesonly. BFGS has proven good performance even for non-smoothoptimizations. This method also returns an approximation of the Hessianinverse, stored as hess_inv in the OptimizeResult object.
Method Newton-CG uses a Newton-CG algorithm [R127] pp. 168 (also knownas the truncated Newton method). It uses a CG method to the compute thesearch direction. See also TNC method for a box-constrainedminimization with a similar algorithm.
Method Anneal uses simulated annealing, which is a probabilisticmetaheuristic algorithm for global optimization. It uses no derivativeinformation from the function being optimized.
Method dogleg uses the dog-leg trust-region algorithm [R127]for unconstrained minimization. This algorithm requires the gradientand Hessian; furthermore the Hessian is required to be positive definite.
Method trust-ncg uses the Newton conjugate gradient trust-regionalgorithm [R127] for unconstrained minimization. This algorithm requiresthe gradient and either the Hessian or a function that computes theproduct of the Hessian with a given vector.
Constrained minimization
Method L-BFGS-B uses the L-BFGS-B algorithm [R128], [R129] for boundconstrained minimization.
Method TNC uses a truncated Newton algorithm [R127], [R130] to minimize afunction with variables subject to bounds. This algorithm usesgradient information; it is also called Newton Conjugate-Gradient. Itdiffers from the Newton-CG method described above as it wraps a Cimplementation and allows each variable to be given upper and lowerbounds.
Method COBYLA uses the Constrained Optimization BY LinearApproximation (COBYLA) method [R131], [10], [11]. The algorithm isbased on linear approximations to the objective function and eachconstraint. The method wraps a FORTRAN implementation of the algorithm.
Method SLSQP uses Sequential Least SQuares Programming to minimize afunction of several variables with any combination of bounds, equalityand inequality constraints. The method wraps the SLSQP Optimizationsubroutine originally implemented by Dieter Kraft [12]. Note that thewrapper handles infinite values in bounds by converting them into largefloating values.
Custom minimizers
It may be useful to pass a custom minimization method, for examplewhen using a frontend to this method such as scipy.optimize.basinhoppingor a different library. You can simply pass a callable as the methodparameter.
The callable is called as method(fun,x0,args,**kwargs,**options)where kwargs corresponds to any other parameters passed to minimize(such as callback, hess, etc.), except the options dict, which hasits contents also passed as method parameters pair by pair. Also, ifjac has been passed as a bool type, jac and fun are mangled so thatfun returns just the function values and jac is converted to a functionreturning the Jacobian. The method shall return an OptimizeResultobject.
The provided method callable must be able to accept (and possibly ignore)arbitrary parameters; the set of parameters accepted by minimize mayexpand in future versions and then these parameters will be passed tothe method. You can find an example in the scipy.optimize tutorial.
References
| [R123] | (1, 2) Nelder, J A, and R Mead. 1965. A Simplex Method for FunctionMinimization. The Computer Journal 7: 308-13. |
| [R124] | (1, 2) Wright M H. 1996. Direct search methods: Once scorned, nowrespectable, in Numerical Analysis 1995: Proceedings of the 1995Dundee Biennial Conference in Numerical Analysis (Eds. D FGriffiths and G A Watson). Addison Wesley Longman, Harlow, UK.191-208. |
| [R125] | (1, 2) Powell, M J D. 1964. An efficient method for finding the minimum ofa function of several variables without calculating derivatives. TheComputer Journal 7: 155-162. |
| [R126] | (1, 2) Press W, S A Teukolsky, W T Vetterling and B P Flannery.Numerical Recipes (any edition), Cambridge University Press. |
| [R127] | (1, 2, 3, 4, 5, 6, 7, 8) Nocedal, J, and S J Wright. 2006. Numerical Optimization.Springer New York. |
| [R128] | (1, 2) Byrd, R H and P Lu and J. Nocedal. 1995. A Limited MemoryAlgorithm for Bound Constrained Optimization. SIAM Journal onScientific and Statistical Computing 16 (5): 1190-1208. |
| [R129] | (1, 2) Zhu, C and R H Byrd and J Nocedal. 1997. L-BFGS-B: Algorithm778: L-BFGS-B, FORTRAN routines for large scale bound constrainedoptimization. ACM Transactions on Mathematical Software 23 (4):550-560. |
| [R130] | (1, 2) Nash, S G. Newton-Type Minimization Via the Lanczos Method.1984. SIAM Journal of Numerical Analysis 21: 770-778. |
| [R131] | (1, 2) Powell, M J D. A direct search optimization method that modelsthe objective and constraint functions by linear interpolation.1994. Advances in Optimization and Numerical Analysis, eds. S. Gomezand J-P Hennart, Kluwer Academic (Dordrecht), 51-67. |
| [10] | (1, 2) Powell M J D. Direct search algorithms for optimizationcalculations. 1998. Acta Numerica 7: 287-336. |
| [11] | (1, 2) Powell M J D. A view of algorithms for optimization withoutderivatives. 2007.Cambridge University Technical Report DAMTP2007/NA03 |
| [12] | (1, 2) Kraft, D. A software package for sequential quadraticprogramming. 1988. Tech. Rep. DFVLR-FB 88-28, DLR German AerospaceCenter – Institute for Flight Mechanics, Koln, Germany. |
Examples
Let us consider the problem of minimizing the Rosenbrock function. Thisfunction (and its respective derivatives) is implemented in rosen(resp. rosen_der, rosen_hess) in the scipy.optimize.
A simple application of the Nelder-Mead method is:
Now using the BFGS algorithm, using the first derivative and a fewoptions:
Next, consider a minimization problem with several constraints (namelyExample 16.4 from [R127]). The objective function is:
There are three constraints defined as:
And variables must be positive, hence the following bounds:
The optimization problem is solved using the SLSQP method as:
It should converge to the theoretical solution (1.4 ,1.7).
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