By A.G. Buckley, J-.L. Goffin

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**Extra resources for Algorithms for Constrained Minimization of Smooth Nonlinear Functions (Mathematical programming study)**

**Sample text**

42) 39 D. 39), showing the superlinear convergence of the modified method. 23). e. 43) is equivalent to the quasi-Newton method along geodesics of [1] where only one step of the restoration phase is performed (the stepsize being taken as 1). 17). 5. 35) is satisfied). , ~ [ci(x)[. 29). Instead of requiring tk to achieve an approximate minimization of the form proposed in [10], we follow the spirit of Powell [19] and select tk = 2 -t for the first index I of the sequence {0, 1, 2 .... 5) 40 D. Gabay/ Reduced quasi-Newton methods with a ~ (0, ~) and ~(x, d, r) defined by ~(x, d, r) = r ~ Ic,(x)l-

D. agrangian functions", Mathematical Programming 14 (1978) 224-248. D. A. , Numerical analysis (Springer-Verlag, Heidelberg 1978) pp. 144-157. [20] A. E. V. Levy, "Use of the augmented penalty function in mathematical programming problems, Part. II", Journal of, Optimization Theory and Applications 8 (1971) 336-349. E. J. Mort, "Quasi-Newton methods, motivation and theory", SIAM Review 19 (1977) 46--89. E. Byrd, "Local convergence of the diagonalized method of multipliers", Journal off Optimization Theory and Applications 26 (1978) 485-500.

E. lim Ilxk§ x*ll 0. 22) k~§ IIx k-'- x*tl-- Proof. 24) 3It is sufficient to assume that f and c are q~2differentiable and have Lipschitz continuous secondorder derivatives. 23) and (424) (for index k - 1) yields Ilx " + ' - ~*11 < r ~ [~ + K3]/,.. ~K7 K2 IId~-'ll. 21) holds. 2) depends only upon how the restriction of M~ to the subspace Tk approximates the similar restriction of the Hessian of the Lagrangian L ( x k, AE*~). Actually it only requires that this approximation be adequate along the direction pk; this result, mentioned by Powell for a particular method where the orthogona~ restriction was considered thus generalizes a similar condition given by Dennis-Mor~ 121] for quasi-Newton methods for unconstrained minimization.