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Algorithms for Constrained Minimization of Smooth Nonlinear - download pdf or read online

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

ISBN-10: 0444863907

ISBN-13: 9780444863904

ISBN-10: 3642008127

ISBN-13: 9783642008122

ISBN-10: 3642008135

ISBN-13: 9783642008139

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

Example text

5) after a finite number of iterations. 2. 5) [or k ~- N. such that the stepsize tk = 1 Proof. 36), f ( x k + d k + e k) = f ( x k) + ( V f ( x k ) , d k) + ( V f ( x k ) , e k) + ~ § ~(lld~ll3). 29) of e k and h k'~. Using the third-order Taylor expansions of c~, we obtain ~ ( X k + d k + e k, r) = qb(x k, r ) - llz(xk, d k, r) + '(d k, L(x ~, x~") d ~) + ~(lld~ll3). 26) with p k = Hkgk. 28) D. e. after a finite number of iterations since {xk}~ x* satisfying the first-order optimality conditions).

25) Either l(x) or E(x) may be empty; indeed for almost all x in ~ " there will exist only one integer in l ( x ) tO E(x) which is therefore a (small) subset of the active constraints. The following assumptions suffice to establish global convergence (in the sense of Theorem 1): (HI) The functions f, g, h, are continuously differentiable. (H2) For all x the vectors {Vgi(x), j E l(x); Vhi(x), j E E(x)} are linearly independent. Note that, because of the almost everywhere low cardinality of I ( x ) U E(x), (H2) is relatively mild given that F is non empty; it is certainly much less mild than linear independence of all equality and active inequality constraints, and cannot be much further relaxed without destroying the ability of the algorithm to determine feasible points.

Lim I I x ~ " - x * l l _ 0. 39) IIx ~ - x*ll - Proof. lldkll 2-< K,011d~ll. Ilek-,ll} K2 Ilxk"--xkll + Ka + E (K, + Kt)lldkll. 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 ....

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Algorithms for Constrained Minimization of Smooth Nonlinear Functions (Mathematical programming study) by A.G. Buckley, J-.L. Goffin

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