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The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. Despite differences in their approaches, these derivations share a common topic—proving the orthogonality of the ...
The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method [1] for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. The intent of this article is to document the important steps in these derivations.
For high-dimensional problems, the exact computation of the Hessian is usually prohibitively expensive, and even its storage can be problematic, requiring () memory (but see the limited-memory L-BFGS quasi-Newton method). The conjugate gradient method can also be derived using optimal control theory. [6]
In optimization, a gradient method is an algorithm to solve problems of the form with the search directions defined by the gradient of the function at the current point. Examples of gradient methods are the gradient descent and the conjugate gradient.
Typical examples involve using non-linear iterative methods, e.g., the conjugate gradient method, as a part of the preconditioner construction. Such preconditioners may be practically very efficient, however, their behavior is hard to predict theoretically.
Powell's method — based on conjugate gradient descent; Rosenbrock methods — derivative-free method, similar to Nelder–Mead but with guaranteed convergence; Augmented Lagrangian method — replaces constrained problems by unconstrained problems with a term added to the objective function; Ternary search; Tabu search
The prototypical method in this class is the conjugate gradient method (CG) which assumes that the system matrix is symmetric positive-definite. For symmetric (and possibly indefinite) A {\displaystyle A} one works with the minimal residual method (MINRES).
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 ...