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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. This method, and modifications, are globally convergent under mild conditions, [ 2 ] [ 3 ] and perform competitively with conjugate gradient methods ...
The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a trajectory that maximizes that function; the procedure is then known as gradient ascent .
2 Equivalence to gradient descent. 3 See also. ... Download as PDF; Printable version; ... A step of gradient descent is ...
Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics , the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations , namely those whose matrix is positive-semidefinite .
In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems. [1] Like the related Davidon–Fletcher–Powell method, BFGS determines the descent direction by preconditioning the gradient with curvature information.
Proximal gradient method — use splitting of objective function in sum of possible non-differentiable pieces; Subgradient method — extension of steepest descent for problems with a non-differentiable objective function; Biconvex optimization — generalization where objective function and constraint set can be biconvex
In optimization, a gradient method is an algorithm to solve problems of the form min x ∈ R n f ( x ) {\displaystyle \min _{x\in \mathbb {R} ^{n}}\;f(x)} with the search directions defined by the gradient of the function at the current point.
When the damping factor is large relative to ‖ ‖, inverting + is not necessary, as the update is well-approximated by the small gradient step [()]. To make the solution scale invariant Marquardt's algorithm solved a modified problem with each component of the gradient scaled according to the curvature.