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Minimization of scalar function of one or more variables.

Using graphical method, Minimize Z=10x1 + 4x2 Subject to 3x + 2x2 60 7x + 2x2 84 3x +6X22 72 X120,X920. Get more help from Chegg. Minimize a function func using the L-BFGS-B algorithm. Fmintnc (func, x0, fprime, args, ) Minimize a function with variables subject to bounds, using gradient information in a truncated Newton algorithm. Fmincobyla (func, x0, cons, args, ) Minimize a function using the Constrained Optimization BY Linear Approximation (COBYLA) method. Define minimizing. Minimizing synonyms, minimizing pronunciation, minimizing translation, English dictionary definition of minimizing. Minimized, minimizing, minimizes 1. To reduce to the smallest possible amount, extent, size, or degree.

In general, the optimization problems are of the form:

minimize f(x)

subject to:

Where x is a vector of one or more variables.g_i(x) are the inequality constraints.h_j(x) are the equality constrains.

Optionally, the lower and upper bounds for each element in x can also be specified using the bounds argument.

See also

minimize_scalar

Interface to minimization algorithms for scalar univariate functions

show_options

Additional options accepted by the solvers

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 theSimplex algorithm [R142], [R143]. This algorithm has been successfulin many applications but other algorithms using the first and/orsecond derivatives information might be preferred for their betterperformances and robustness in general.

Method Powell is a modificationof Powell’s method [R144], [R145] which is a conjugate directionmethod. It performs sequential one-dimensional minimizations alongeach vector of the directions set (direc field in options andinfo), 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 conjugategradient algorithm by Polak and Ribiere, a variant of theFletcher-Reeves method described in [R146] pp. 120-122. Only thefirst derivatives are used.

Method BFGS uses the quasi-Newtonmethod of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) [R146]pp. 136. It uses the first derivatives only. BFGS has proven goodperformance even for non-smooth optimizations. This method alsoreturns an approximation of the Hessian inverse, stored ashess_inv in the OptimizeResult object.

Method Newton-CG Downcast 2 9 36. uses aNewton-CG algorithm [R146] pp. 168 (also known as the truncatedNewton method). It uses a CG method to the compute the searchdirection. See also TNC method for a box-constrainedminimization with a similar algorithm.

Method dogleg uses the dog-legtrust-region algorithm [R146] for unconstrained minimization. Thisalgorithm requires the gradient and Hessian; furthermore theHessian is required to be positive definite.

Method trust-ncg uses theNewton conjugate gradient trust-region algorithm [R146] forunconstrained minimization. This algorithm requires the gradientand either the Hessian or a function that computes the product ofthe Hessian with a given vector.

Constrained minimization

Method L-BFGS-B uses the L-BFGS-Balgorithm [R147], [R148] for bound constrained minimization.

Method TNC uses a truncated Newtonalgorithm [R146], [R149] to minimize a function with variables subjectto bounds. This algorithm uses gradient information; it is alsocalled Newton Conjugate-Gradient. It differs from the Newton-CGmethod described above as it wraps a C implementation and allowseach variable to be given upper and lower bounds.

Method COBYLA uses theConstrained Optimization BY Linear Approximation (COBYLA) method[R150], [10], [11]. The algorithm is based on linearapproximations to the objective function and each constraint. Themethod wraps a FORTRAN implementation of the algorithm. Theconstraints functions ‘fun’ may return either a single numberor an array or list of numbers.

Method SLSQP uses SequentialLeast SQuares Programming to minimize a function of severalvariables with any combination of bounds, equality and inequalityconstraints. The method wraps the SLSQP Optimization subroutineoriginally implemented by Dieter Kraft [12]. Note that thewrapper handles infinite values in bounds by converting them intolarge floating 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.

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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 Printworks 2 0 8 0.

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: Wheel of fortune comcom.

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Now using the BFGS algorithm, using the first derivative and a fewoptions:

Next, consider a minimization problem with several constraints (namelyExample 16.4 from [R146]). 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:

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It should converge to the theoretical solution (1.4 ,1.7).