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

From the experiences: "The account is sort of special and is written in a fashion that would entice analysts and numerical practitioners alike...they comprise every thing from rigorous proofs to tables of numerical calculations.... one of many robust gains of those books...that they're designed no longer for the specialist, yet when you whish to profit the subject material ranging from very little background...there are quite a few examples, and counter-examples, to again up the theory...To my wisdom, no different authors have given this kind of transparent geometric account of convex analysis." "This leading edge textual content is easily written, copiously illustrated, and obtainable to a large audience"

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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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