class scipy.optimize.NonlinearConstraint(fun, lb, ub, jac='2-point', hess=<scipy.optimize._hessian_update_strategy.BFGS object>, keep_feasible=False, finite_diff_rel_step=None, finite_diff_jac_sparsity=None)[source]

Nonlinear constraint on the variables.

The constraint has the general inequality form:

lb <= fun(x) <= ub

Here the vector of independent variables x is passed as ndarray of shape (n,) and fun returns a vector with m components.

It is possible to use equal bounds to represent an equality constraint or infinite bounds to represent a one-sided constraint.

fun : callable

The function defining the constraint. The signature is fun(x) -> array_like, shape (m,).

lb, ub : array_like

Lower and upper bounds on the constraint. Each array must have the shape (m,) or be a scalar, in the latter case a bound will be the same for all components of the constraint. Use np.inf with an appropriate sign to specify a one-sided constraint. Set components of lb and ub equal to represent an equality constraint. Note that you can mix constraints of different types: interval, one-sided or equality, by setting different components of lb and ub as necessary.

jac : {callable, ‘2-point’, ‘3-point’, ‘cs’}, optional

Method of computing the Jacobian matrix (an m-by-n matrix, where element (i, j) is the partial derivative of f[i] with respect to x[j]). The keywords {‘2-point’, ‘3-point’, ‘cs’} select a finite difference scheme for the numerical estimation. A callable must have the following signature: jac(x) -> {ndarray, sparse matrix}, shape (m, n). Default is ‘2-point’.

hess : {callable, ‘2-point’, ‘3-point’, ‘cs’, HessianUpdateStrategy, None}, optional

Method for computing the Hessian matrix. The keywords {‘2-point’, ‘3-point’, ‘cs’} select a finite difference scheme for numerical estimation. Alternatively, objects implementing HessianUpdateStrategy interface can be used to approximate the Hessian. Currently available implementations are:

A callable must return the Hessian matrix of dot(fun, v) and must have the following signature: hess(x, v) -> {LinearOperator, sparse matrix, array_like}, shape (n, n). Here v is ndarray with shape (m,) containing Lagrange multipliers.

keep_feasible : array_like of bool, optional

Whether to keep the constraint components feasible throughout iterations. A single value set this property for all components. Default is False. Has no effect for equality constraints.

finite_diff_rel_step: None or array_like, optional

Relative step size for the finite difference approximation. Default is None, which will select a reasonable value automatically depending on a finite difference scheme.

finite_diff_jac_sparsity: {None, array_like, sparse matrix}, optional

Defines the sparsity structure of the Jacobian matrix for finite difference estimation, its shape must be (m, n). If the Jacobian has only few non-zero elements in each row, providing the sparsity structure will greatly speed up the computations. A zero entry means that a corresponding element in the Jacobian is identically zero. If provided, forces the use of ‘lsmr’ trust-region solver. If None (default) then dense differencing will be used.


Finite difference schemes {‘2-point’, ‘3-point’, ‘cs’} may be used for approximating either the Jacobian or the Hessian. We, however, do not allow its use for approximating both simultaneously. Hence whenever the Jacobian is estimated via finite-differences, we require the Hessian to be estimated using one of the quasi-Newton strategies.

The scheme ‘cs’ is potentially the most accurate, but requires the function to correctly handles complex inputs and be analytically continuable to the complex plane. The scheme ‘3-point’ is more accurate than ‘2-point’ but requires twice as many operations.