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

By Jean-Baptiste Hiriart-Urruty, Claude Lemarechal

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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.