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In numerical analysis, a quasi-Newton method is an iterative numerical method used either to find zeroes or to find local maxima and minima of functions via an iterative recurrence formula much like the one for Newton's method, except using approximations of the derivatives of the functions in place of exact derivatives.
In SciPy, the scipy.optimize.fmin_bfgs function implements BFGS. [14] It is also possible to run BFGS using any of the L-BFGS algorithms by setting the parameter L to a very large number. It is also one of the default methods used when running scipy.optimize.minimize with no constraints. [15]
Since BFGS (and hence L-BFGS) is designed to minimize smooth functions without constraints, the L-BFGS algorithm must be modified to handle functions that include non-differentiable components or constraints. A popular class of modifications are called active-set methods, based on the concept of the active set. The idea is that when restricted ...
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
A simple work-around is to negate the function, substituting -D T (D X) for D T (D X) and thus reversing the order of the eigenvalues, since LOBPCG does not care if the matrix of the eigenvalue problem is positive definite or not. [9] LOBPCG for PCA and SVD is implemented in SciPy since revision 1.4.0 [13]
The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then the gradient of the function (at that point) can be expressed as a linear combination of the gradients of the constraints (at that point), with the ...
A practical way to enforce this is by requiring that the next search direction be built out of the current residual and all previous search directions. The conjugation constraint is an orthonormal-type constraint and hence the algorithm can be viewed as an example of Gram-Schmidt orthonormalization. This gives the following expression:
Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization which may be considered a quasi-Newton method.SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable, but not necessarily convex.