By V. M. Tikhomirov (auth.), R. V. Gamkrelidze (eds.)

Intended for quite a lot of readers, this ebook covers the most principles of convex research and approximation concept. the writer discusses the resources of those developments in mathematical research, develops the most techniques and effects, and mentions a few appealing theorems. the connection of convex research to optimization difficulties, to the calculus of adaptations, to optimum keep watch over and to geometry is taken into account, and the evolution of the guidelines underlying approximation concept, from its origins to the current day, is mentioned. The publication is addressed either to scholars who are looking to acquaint themselves with those developments and to teachers in mathematical research, optimization and numerical tools, in addition to to researchers in those fields who wish to take on the subject as an entire and search suggestion for its additional development.

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**Example text**

Xn } c) = op = I Ilxll =oN = In a)-c) X = p(x) = N(x) = 5) Support functions. normed linear space X* its dual. a) A = B; = t I}; {Y ERn: Yi = 1==1 B*, B* = {X*: IIx*l! = W, in d) ~ Y = sA(y) = lip' n- 1 = IIYli p ' + lip = B:::'(a) = sA(y) =: l(p = X ~ I}. (tI 6) SubdifJerentials of functions at a point. In a)-b) X = Y = R, in c) X = Yis a Hilbert space. -1, x < 0, { a) f(x) = Ixl = af(x) = • [ - ] , 1J, x = 0, x> 0; 1, b) f(x) = max {eX, 1 - x} = cf(x) = {~~ x> 0, 1, 1J, -1, x = 0, x < 0; 37 Chapter 1.

Let fl, .. ~,fn be convex functions, and moreover at some point x E n7=1 domf, let all the fimctions, except possibly one, be continuous. Then for any x EX aCt /;) it (x) = aJ;(x). Theorem 7 (the subdifferential of the maximum). Let f1 , ... , f" be convex functions continuous at a point x. Then aU1 V"'V fn)(x) = {. I. 'E T(xj AiX;: ;'i E R+,. where T(x) = {i: f,(x) = (f1 I. 'E T(Xj ;'i = 1, x' E o/;(X)}, v.. ·V fnHx)}. The theorems stated here are fundamental to the subdifferential calculus. They are all proved by similar means and, as a matter of fact, aU are equivalent to the separation theorem.

A) cl co A = A <=> A b) cl co A = cl(co A). 3. Topological Properties of Convex Functions. In the following theorem X is a linear topological (not necessarily locally convex) space. Theorem 1. Let f be a proper convex function on X. Then the following are equivalent: a) f is bounded above in the neighbourhood of some point X; b) f is continuous at some point X; c) intepif -# 0; d) int(domf) -# 0 and f is continuous on int(domf). Here int(epif) = {(a, x): x E int(domf), x > f(x)}. Let us prove this important theorem.