Titlebook: Numerical Optimization; Theoretical and Prac J. Frédéric Bonnans,J. Charles Gilbert,Claudia A. Textbook 20031st edition Springer-Verlag Be

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J. Frédéric Bonnans,J. Charles Gilbert,Claude Lemaréchal,Claudia A. Sagastizábal by the authorsThe emergence of topological quantum ?eld theory has been one of the most important breakthroughs which have occurred in the context of ma- ematical physics in the last century, a century characterizedbyindependent developments of the main ideas in both disciplines, physics and mathem

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General Introductiont will be the usual dot-product: .); | · | or ∥ · ∥ will denote the associated norm. The . (vector of partial derivatives) of a function .: ℝ. → ℝ will be denoted by ∇ . or .′; the . (matrix of second derivatives) by ∇.. or .″. We will also use continually the notation .(.) = .′(.).

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Line-Searches be studied in the next chapters), we focus on the computation of the stepsize. Here appear the most serious practical difficulties, while directions are generally easy to compute, once the theory is well-mastered. A firm experience is required to write a good computer code for line-searches, which

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Conjugate Gradientith symmetric positive definite matrix (or, equivalently, to minimize in . iterations a quadratic strongly convex function on ℝ.), without storing an additional matrix, without even storing the matrix of the system. In fact, to solve . + . = 0 (. symmetric positive definite), the conjugate gradient

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Special Methodsmely important, and might supersede line-searches, sooner or later. The other methods deal with the direction; they are either classical (Gauss-Newton) or recent (limited-memory quasi-Newton, truncated Newton) and apply only in some well-defined subclasses of problems.

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