By Jean-Baptiste Hiriart-Urruty, Claude Lemarechal

Convex research might be regarded as a refinement of ordinary calculus, with equalities and approximations changed via inequalities. As such, it will possibly simply be built-in right into a graduate examine curriculum. Minimization algorithms, extra particularly these tailored to non-differentiable capabilities, supply a right away program of convex research to varied fields relating to optimization and operations examine. those subject matters making up the name of the publication, mirror the 2 origins of the authors, who belong respectively to the educational global and to that of functions. 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 the next technical point and is addressed extra to experts, amassing effects that thus far haven't seemed in books.

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**Example text**

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} .

1) t~o t 5 Second-Order Differentiation Vs(t) E [D_f(x + td), D+f(x + td)], . o + td) - . o t f(x) - tf'(x, d) . 2) are just equivalent: if one of the limits exists, the other two exist as well and are the same; this is the so-called point of view of Dini. 3) exists and is denoted by f"(x, d) (';3 0), then the other limits exist as well and are equal to f" (x, d). PROOF. 2, without bothering with the sign ofh. 9 and modify cp by settingcp(u) = 0 for u ~ O. Then the new I has the two "half-second derivatives" 1"(0, -1) = 0 and 1"(0,1) = 1.

O The dual version of this result is that, if II and 12 are two closed convex functions finite at some common point, then (/1 + 12)* = It t g . 1, and their conjugates are II and fz respectively; hence (/t t Iz*)* = II + fz . 2). 2). 5), we have inf[fl(x) + hex)] XER = -(/1 + 12)*(0) = SER inf[ft(s) + g(-s)] , which is known as (the univariate version of) Fenchel's duality theorem - but once again, beware that it does not extend readily to several variables. 2) show that the addition offunctions and their infimal convolution are operations dual to each other.