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Let : be a convex function with domain . A classical subgradient method iterates (+) = () where () denotes any subgradient of at (), and () is the iterate of . If is differentiable, then its only subgradient is the gradient vector itself.
The Barzilai-Borwein method [1] is an iterative gradient descent method for unconstrained optimization using either of two step sizes derived from the linear trend of the most recent two iterates.
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.
At each iteration, there is a set of "working points" in which we know the value of f (and possibly also its derivative). Based on these points, we can compute a polynomial that fits the known values, and find its minimum analytically. The minimum point becomes a new working point, and we proceed to the next iteration: [1]: sec.5
So in each iteration is updated as + = However beyond such problems projection operators are not appropriate and more general operators are required to tackle them. Among the various generalizations of the notion of a convex projection operator that exist, proximal operators are best suited for other purposes.
This can be seen in the following tables, the left of which shows Newton's method applied to the above f(x) = x + x 4/3 and the right of which shows Newton's method applied to f(x) = x + x 2. The quadratic convergence in iteration shown on the right is illustrated by the orders of magnitude in the distance from the iterate to the true root (0,1 ...
The fixed point iteration x n+1 = cos x n with initial value x 1 = −1.. An attracting fixed point of a function f is a fixed point x fix of f with a neighborhood U of "close enough" points around x fix such that for any value of x in U, the fixed-point iteration sequence , (), (()), ((())), … is contained in U and converges to x fix.
In order to be able to use the Bregman method, one must frame the problem of interest as finding () + (), where is a regularizing function such as . [3]The Bregman distance is defined as (,):= (() + , ) where belongs to the subdifferential of at (which we denoted ()).