BFGS

The Broyden-Fletcher-Goldfarb-Shanno algorithm, also known as the BFGS algorithm, is a gradient-based optimization algorithm. This solver uses the ‘BFGS’ algorithm from the Scipy library.

Note

BFGS is a quasi-Newton optimization algorithm for unconstrained problems. Therefore, it does not support bounds or constraints. Please use general nonlinear programming algorithms such as PySLSQP or IPOPT, if your problem has bounds or constraints.

To use the BFGS solver, start by importing it as shown in the following code:

from modopt import BFGS

Options could be set by just passing them within the solver_options dictionary when instantiating the BFGS optimizer object. For example, we can set the maximum number of iterations maxiter and the tolerance on the gradient norm gtol as shown below.

optimizer = BFGS(prob, solver_options={'maxiter':1000, 'gtol':1e-6})

The options available for the BFGS solver in modOpt are given in the following table. For more information on the Scipy ‘BFGS’ algorithm, visit Scipy documentation.

BFGS solver options
Option	Type (default value)	Description
`maxiter`	int (`200`)	Maximum number of iterations.
`gtol`	float(`1e-5`)	Terminate successfully if: `norm[gradient] <= gtol`.
`xrtol`	float (`0.0`)	Terminate successfully if `norm[alphapk] <= (norm[xk] + xrtol) xrtol`, where `alpha*pk` is the step computed and `xk` is the current optimization variable vector.
`norm`	float (`np.inf`)	Order of the norm to be used with the two termination criteria above. `np.inf` represents maximum and `-np.inf` represents minimum.
`c1`	float (`1e-4`)	Armijo condition parameter.
`c2`	float (`0.9`)	Curvature condition parameter. Must satisfy `0 < c1 < c2 < 1`.
`hess_inv0`	np.ndarray (`np.identity(n)`)	Initial estimate of the objective Hessian inverse. Default is the identity matrix.
`return_all`	bool (`False`)	Set to `True` to return a list containing the best solution from each major iteration in the final results dict.
`disp`	bool (`False`)	Set to `True` to print convergence messages. If `False`, no console outputs will be generated.
`callback`	callable (`None`)	Function to be called after each major iteration. The function is called as`callback(xk, fk)`, where `xk` is the optimization variable vector from the current major iteration, and `fk` is the corresponding objective value.