Convex Analysis and Minimization Algorithms I: Fundamentals by Jean-Baptiste Hiriart-Urruty, Claude Lemarechal

By Jean-Baptiste Hiriart-Urruty, Claude Lemarechal

Convex research will be regarded as a refinement of ordinary calculus, with equalities and approximations changed by means of inequalities. As such, it may possibly simply be built-in right into a graduate research curriculum. Minimization algorithms, extra in particular these tailored to non-differentiable capabilities, offer a right away program of convex research to numerous fields with regards to optimization and operations examine. those subject matters making up the name of the booklet, mirror the 2 origins of the authors, who belong respectively to the tutorial international and to that of purposes. half i will be able to be used as an introductory textbook (as a foundation for classes, or for self-study); half II maintains this at a better technical point and is addressed extra to experts, gathering effects that to date haven't seemed in books.

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F(a) We take + feb)• - f(a) (x - a). + (1 - a)b. 5) 36 I. Convex Functions of One Real Variable g(x) := f(x) - f(a) - feb) - f(a) (x - a) , b-a and we prove g ~ 0 on la, b[. 12f. 12g(x) = ~f(x) > 0 for all x E la, b[. 6). 5) is proved. k(x) := f(x) + l/kx 2. k is convex. 1). 3) represents one more "curvature" estimate. 4) and force Pc to coincide with f atx, x - t, x + t: we again obtain c = i12f(x, x - t, x + t). 0 6 First Steps into the Theory of Conjugate Functions On several occasions, we have encountered the conjugate jUnction of f, defined by R 3 S H- f*(s) := sup {sx - f(x) : x E dom f} .

3) at the same time. 1. 2) written at appropriate points to obtain D+/~)~ ~ I(x) - I(a) ~D_/~)~D+/~)~ x-a I(x') - I(x) , x -x ~ , D_/(x ) ~ etc. note that the relevant inequalities hold as well if x L = max{-D+/(a), D_/(b)}. ~ = D_/(b); a. 1) with 0 A sort of differentiability being thus established on the interior of dom I, what can be said about its endpoints? Let again a be its left endpoint, as in Fig. 1. First of all, the whole concept is meaningless if a f/. dom I (case 3), and the very definition shows that D _/(a) = -00.

8) is met. ~kEN u oscillates between the functions x and x I (1 + x), see Fig. 2). 4 of increasing slopes), has O-derivative at 0, but is differentiable on no segment ]0. 2.... 0 (qJ x .. 2. 1 For I E Cony 1R, the follOWing properties hold: al(x) behaves 4 First-Order Differentiation 25 (i) The multifunction al is increasing on its domain, in the sense that Sl ~ Sl whenever Sl al(xd, E S2 E al(X2) and XI < Xl. 1) (ii) The set ofpoints where I fails to be differentiable is at most countable. 2) likewise, al(x) converges decreasingly to D+/(xo) when X ,j.

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