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

By Jean-Baptiste Hiriart-Urruty, Claude Lemarechal

ISBN-10: 3642081622

ISBN-13: 9783642081620

ISBN-10: 366206409X

ISBN-13: 9783662064092

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

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2, represents the problem to be solved: it is in charge of providing the rest with information concerning the function f to minimize, or the set of (x) to separate. 6 is the "decision maker", which decides whether to stop, or what other direction to try, etc. It might even decide what norm to use for the projection. ). Given d, it computes s E S yielding as(d), or detects that as(d) < O. In other words, our present mechanism is adapted to situations in which S is not known explicitly, but in which the only available information is pointwise and concerns the support function of S.

And" ~ 81 . 1) tends to 0, with an asymptotic speed of 1/ k. This is very slow indeed; it is fair to say that an algorithm behaving so slowly must be considered as not convergent at all. We now proceed to relate the bundling algorithm to the above analysis. Worst Possible Behaviour. 5 1 A II skll ~ PROOF. 11) ~ (1 JLr /LM. 3) = 1. 12) to obtain with A. := M 2/(I-m')2. 1 applies (with8k = IIskll2 and observing that A. 3) follows by taking square roots. 0 Thus a bundling algorithm must reach its stopping criterion after not more than [M / (1- m')8]2 iterations - and we already mentioned that this is a prohibitive number.

1 that of (x) may contain some points with negative ordinates, without changing the sequence {Sk}. 2 that, if x = i actually minimizes f, the property 0 E ri of (i) is natural. We illustrate this by the following numerical experiments. e. in their black box. 1) at each iteration. - In the second form, Step 2 merely computes an sk+1 such that (Sk+l, dk) = 0 (note: since 0 E S, oS(d) ~ 0 for all d). 6. 10). 3. 2. o -1 -2 support-black box -3 100 200 400 600 k Fig. 1. " "" ...... IIt ... -----------------____ orthogonal black box support-black bOx ----------------------------___ _ -- ....

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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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