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

By Jean-Baptiste Hiriart-Urruty, Claude Lemarechal

From the experiences: "The account is kind of distinctive and is written in a way that would entice analysts and numerical practitioners alike...they include every thing from rigorous proofs to tables of numerical calculations.... one of many powerful beneficial properties of those books...that they're designed no longer for the professional, yet should you whish to benefit 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 one of these transparent geometric account of convex analysis." "This leading edge textual content is definitely written, copiously illustrated, and available to a large audience"

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

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Remember that -dk E S. As always, (dk. Sj) ~ - IIdkll2 for all k and i ~ k. 5dk/lldkll E B(O. 5). 5) E aft'S. Then. 5 is small enough. 5 > 0 for all k and i ~ k • which implies that the compact sequence {Sk} is actually finite. 6: observe in Fig. 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.

E. I*(so) ~ - ro < +00. 1). Thus, dom j* - the set where j* is finite - is the set of slopes of all the possible affine functions minorizing lover ]Rn. Likewise, any s E dom 1 is the slope of an affine function smaller than j*. 38 X. 2 For f satisfYing (1. 1), the conjugate f* isaclosedconvexfunction: f* E Conv lRn. PROOF. 3. 4) . 2), and the resulting f* is f*(s) = 4(s - b, Q-I (s - b») for all s E lRn. In particular, the function 1/211 • 112 is its own conjugate. Needless to say, the Legendre transformation is present here, in a particularly simple setting: V f is the affine mapping x 1-+ Qx +b, its inverse is the affine mapping s 1-+ Q-I(S - b), the gradient of f*.

5 is at least as "good" as 1/ k, even if the bundle SA: is kept minimal, just containing two subgradients. On the other hand, IISkll2 can be as bad as 1/ k2, at least if is kept minimal, or also when the dimension of the space is really large. 8) with a rate q < 1 independent of the particular function f, cannot hold for our algorithm at least when the space is infinite-dimensional. 8) must tend to 1 when the dimension n of the space tends to infinity; in fact q must be "at best" of the form q ~ 1 - 1/ n.

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