Journal
SIAM JOURNAL ON OPTIMIZATION
Volume 25, Issue 1, Pages 234-260Publisher
SIAM PUBLICATIONS
DOI: 10.1137/140955501
Keywords
linear programming; quasi-Newton methods; conjugate gradient; Gaussian inference
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This paper proposes a probabilistic framework for algorithms that iteratively solve unconstrained linear problems Bx = b with positive definite B for x. The goal is to replace the point estimates returned by existing methods with a Gaussian posterior belief over the elements of the inverse of B, which can be used to estimate errors. Recent probabilistic interpretations of the secant family of quasi-Newton optimization algorithms are extended. Combined with properties of the conjugate gradient algorithm, this leads to uncertainty-calibrated methods with very limited cost overhead over conjugate gradients, a self-contained novel interpretation of the quasi-Newton and conjugate gradient algorithms, and a foundation for new nonlinear optimization methods.
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