By Dimitri P. Bertsekas
ISBN-10: 0120934809
ISBN-13: 9780120934805
This broadly referenced textbook, first released in 1982 via educational Press, is the authoritative and entire therapy of a few of the main familiar restricted optimization equipment, together with the augmented Lagrangian/multiplier and sequential quadratic programming equipment. between its particular positive aspects, the ebook: 1) treats widely augmented Lagrangian tools, together with an exhaustive research of the linked convergence and cost of convergence homes 2) develops comprehensively sequential quadratic programming and different Lagrangian tools three) presents a close research of differentiable and nondifferentiable specific penalty tools four) offers nondifferentiable and minimax optimization tools in accordance with smoothing five) includes a lot intensive examine now not present in the other textbook
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Extra info for Constrained Optimization and Lagrange Multiplier Methods
Sample text
A dk such that Vf{xk)'dk < 0 and d'kV2f(xk)dk < 0. This can be done in a numerically stable and efficient manner via a form of triangular factorization of V2/(xfc). For a detailed presentation we refer to Fletcher and Freeman (1977), Mon? and Sorensen (1979), and Goldfarb (1980). Periodic Réévaluation of the Hessian Finally, we mention that a Newton-type method, which in many cases is considerably more efficient computationally than those described above, is obtained if the Hessian matrix V2/is recomputed every p iterations (p > 2) rather than at every iteration.
INTRODUCTION for the method to start generating nonsensical and inefficient directions of search after a few iterations. For this reason it is important to operate the method in cycles of conjugate direction steps given by (80), with the first step in the cycle being a steepest descent step. Some possible restarting policies are: (a) Restart with a steepest descent step n iterations after the preceding restart. (b) Restart with a steepest descent step k iterations after the preceding restart with k < n.
An important practical issue relates to the line search accuracy that is necessary for efficient computation. An elementary calculation shows that if line search is carried out to the extent that mxk)'dk-,<\mxk-i)\\ then dki generated by (80) and (81), satisfies Vf(xk)'dk < 0 and is a direction of descent. On the other hand, a much more accurate line search may be necessary in order to keep loss of direction conjugacy and deterioration of rate of convergence within a reasonable level. At the same time, insisting on a very accurate line search can be computationally expensive.