Optimization and root finding (scipy.optimize)


Local Optimization

minimize(fun, x0[, args, method, jac, hess, …]) Minimization of scalar function of one or more variables.
minimize_scalar(fun[, bracket, bounds, …]) Minimization of scalar function of one variable.
OptimizeResult Represents the optimization result.

The minimize function supports the following methods:

Constraints are passed to minimize function as a single object or as a list of objects from the following classes:

NonlinearConstraint(fun, lb, ub[, jac, …]) Nonlinear constraint on the variables.
LinearConstraint(A, lb, ub[, keep_feasible]) Linear constraint on the variables.

Simple bound constraints are handled separately and there is a special class for them:

Bounds(lb, ub[, keep_feasible]) Bounds constraint on the variables.

Quasi-Newton strategies implementing HessianUpdateStrategy interface can be used to approximate the Hessian in minimize function (available only for the ‘trust-constr’ method). Available quasi-Newton methods implementing this interface are:

BFGS([exception_strategy, min_curvature, …]) Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian update strategy.
SR1([min_denominator, init_scale]) Symmetric-rank-1 Hessian update strategy.

The minimize_scalar function supports the following methods:

The specific optimization method interfaces below in this subsection are not recommended for use in new scripts; all of these methods are accessible via a newer, more consistent interface provided by the functions above.

General-purpose multivariate methods:

fmin(func, x0[, args, xtol, ftol, maxiter, …]) Minimize a function using the downhill simplex algorithm.
fmin_powell(func, x0[, args, xtol, ftol, …]) Minimize a function using modified Powell’s method.
fmin_cg(f, x0[, fprime, args, gtol, norm, …]) Minimize a function using a nonlinear conjugate gradient algorithm.
fmin_bfgs(f, x0[, fprime, args, gtol, norm, …]) Minimize a function using the BFGS algorithm.
fmin_ncg(f, x0, fprime[, fhess_p, fhess, …]) Unconstrained minimization of a function using the Newton-CG method.

Constrained multivariate methods:

fmin_l_bfgs_b(func, x0[, fprime, args, …]) Minimize a function func using the L-BFGS-B algorithm.
fmin_tnc(func, x0[, fprime, args, …]) Minimize a function with variables subject to bounds, using gradient information in a truncated Newton algorithm.
fmin_cobyla(func, x0, cons[, args, …]) Minimize a function using the Constrained Optimization BY Linear Approximation (COBYLA) method.
fmin_slsqp(func, x0[, eqcons, f_eqcons, …]) Minimize a function using Sequential Least SQuares Programming
differential_evolution(func, bounds[, args, …]) Finds the global minimum of a multivariate function.

Univariate (scalar) minimization methods:

fminbound(func, x1, x2[, args, xtol, …]) Bounded minimization for scalar functions.
brent(func[, args, brack, tol, full_output, …]) Given a function of one-variable and a possible bracket, return the local minimum of the function isolated to a fractional precision of tol.
golden(func[, args, brack, tol, …]) Return the minimum of a function of one variable using golden section method.

Equation (Local) Minimizers

leastsq(func, x0[, args, Dfun, full_output, …]) Minimize the sum of squares of a set of equations.
least_squares(fun, x0[, jac, bounds, …]) Solve a nonlinear least-squares problem with bounds on the variables.
nnls(A, b[, maxiter]) Solve argmin_x || Ax - b ||_2 for x>=0.
lsq_linear(A, b[, bounds, method, tol, …]) Solve a linear least-squares problem with bounds on the variables.

Global Optimization

basinhopping(func, x0[, niter, T, stepsize, …]) Find the global minimum of a function using the basin-hopping algorithm
brute(func, ranges[, args, Ns, full_output, …]) Minimize a function over a given range by brute force.
differential_evolution(func, bounds[, args, …]) Finds the global minimum of a multivariate function.

Rosenbrock function

rosen(x) The Rosenbrock function.
rosen_der(x) The derivative (i.e.
rosen_hess(x) The Hessian matrix of the Rosenbrock function.
rosen_hess_prod(x, p) Product of the Hessian matrix of the Rosenbrock function with a vector.


curve_fit(f, xdata, ydata[, p0, sigma, …]) Use non-linear least squares to fit a function, f, to data.

Root finding

Scalar functions

brentq(f, a, b[, args, xtol, rtol, maxiter, …]) Find a root of a function in a bracketing interval using Brent’s method.
brenth(f, a, b[, args, xtol, rtol, maxiter, …]) Find root of f in [a,b].
ridder(f, a, b[, args, xtol, rtol, maxiter, …]) Find a root of a function in an interval.
bisect(f, a, b[, args, xtol, rtol, maxiter, …]) Find root of a function within an interval.
newton(func, x0[, fprime, args, tol, …]) Find a zero using the Newton-Raphson or secant method.

Fixed point finding:

fixed_point(func, x0[, args, xtol, maxiter, …]) Find a fixed point of the function.


General nonlinear solvers:

root(fun, x0[, args, method, jac, tol, …]) Find a root of a vector function.
fsolve(func, x0[, args, fprime, …]) Find the roots of a function.
broyden1(F, xin[, iter, alpha, …]) Find a root of a function, using Broyden’s first Jacobian approximation.
broyden2(F, xin[, iter, alpha, …]) Find a root of a function, using Broyden’s second Jacobian approximation.

The root function supports the following methods:

Large-scale nonlinear solvers:

newton_krylov(F, xin[, iter, rdiff, method, …]) Find a root of a function, using Krylov approximation for inverse Jacobian.
anderson(F, xin[, iter, alpha, w0, M, …]) Find a root of a function, using (extended) Anderson mixing.

Simple iterations:

excitingmixing(F, xin[, iter, alpha, …]) Find a root of a function, using a tuned diagonal Jacobian approximation.
linearmixing(F, xin[, iter, alpha, verbose, …]) Find a root of a function, using a scalar Jacobian approximation.
diagbroyden(F, xin[, iter, alpha, verbose, …]) Find a root of a function, using diagonal Broyden Jacobian approximation.

Additional information on the nonlinear solvers

Linear Programming

General linear programming solver:

linprog(c[, A_ub, b_ub, A_eq, b_eq, bounds, …]) Minimize a linear objective function subject to linear equality and inequality constraints.

The linprog function supports the following methods:

The simplex method supports callback functions, such as:

linprog_verbose_callback(xk, **kwargs) A sample callback function demonstrating the linprog callback interface.

Assignment problems:

linear_sum_assignment(cost_matrix) Solve the linear sum assignment problem.


approx_fprime(xk, f, epsilon, *args) Finite-difference approximation of the gradient of a scalar function.
bracket(func[, xa, xb, args, grow_limit, …]) Bracket the minimum of the function.
check_grad(func, grad, x0, *args, **kwargs) Check the correctness of a gradient function by comparing it against a (forward) finite-difference approximation of the gradient.
line_search(f, myfprime, xk, pk[, gfk, …]) Find alpha that satisfies strong Wolfe conditions.
show_options([solver, method, disp]) Show documentation for additional options of optimization solvers.
LbfgsInvHessProduct(sk, yk) Linear operator for the L-BFGS approximate inverse Hessian.
HessianUpdateStrategy Interface for implementing Hessian update strategies.