Download e-book for kindle: Convex Analysis and Minimization Algorithms II: Advanced by Jean-Baptiste Hiriart-Urruty, Claude Lemarechal

By Jean-Baptiste Hiriart-Urruty, Claude Lemarechal

ISBN-10: 3642081622

ISBN-13: 9783642081620

ISBN-10: 366206409X

ISBN-13: 9783662064092

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Additional info for Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods

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1. " "" ...... IIt ... -----------------____ orthogonal black box support-black bOx ----------------------------___ _ -- .... 10 50 100 k 150 Fig. 2. 1 shows the two corresponding speeds of convergence of {dk} to o. In this test, 0 is actually an interior point of S C ]R48. This explains the very fast 34 IX. Inner Construction of the Subdifferential decrease of IIdk II at the end of the first variant (finite convergence, cf. 8). 2, 0 is a boundary point of S c lR lo • The results are plotted in Fig.

1) holds for example if 1 E Conv]Rn. 1) is the function j* defined by ]Rn 3 s ~ I*(s) := sup {(s, x) - I(x) : x E dom f}. 2) For simplicity, we may also let x run over the whole space instead of dom I. The mapping 1 ~ 1* will often be called the conjugacy operation, or the Legendre-Fenchel transform. 0 A very first observation is that a conjugate function is associated with a scalar product on ]Rn. 3) that I*(s) = - inf {f(x) - (s, x) : x E dom f}. 2), we have for all (x, s) I*(s) E dom 1 x ]Rn + I(x) ~ (s, x) .

XIV will globalize the approach and enlarge this 8 so as to definitely escape from the steepest-descent concept. Here we give some comments on (i) and (ii), and we start with an important remark. 4, each cycle of the bundling mechanism generates a subgradient Sk+1 lying in the face of CJf(x) exposed by the direction dk. This sk+ I is interesting for dk itself: not only is dk uphill (because (sk+ I, dk) ~ 0), but we can say more. In terms of the descent property of dk, the subgradient Sk+1 is the worst possible; or, reverting the argument, sk+1 is the best possible in terms of the useful information concerning dk.

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Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods by Jean-Baptiste Hiriart-Urruty, Claude Lemarechal

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