Download e-book for iPad: Fundamentals of Convex Analysis by Jean-Baptiste Hiriart-Urruty

Download e-book for iPad: Fundamentals of Convex Analysis by Jean-Baptiste Hiriart-Urruty

By Jean-Baptiste Hiriart-Urruty

This ebook is an abridged model of the 2 volumes "Convex research and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306). It provides an creation to the elemental innovations in convex research and a research of convex minimization difficulties (with an emphasis on numerical algorithms). The "backbone" of bot volumes used to be extracted, a few fabric deleted which used to be deemed too complicated for an advent, or too heavily connected to numerical algorithms. a few workouts have been integrated and at last the index has been significantly enriched, making it a great selection for the aim of studying and educating.

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3 A set C C IRn is convex combination of its elements. if and only if it contains every convex 28 A. Convex Sets Proof. The condition is sufficient: convex combinations of two elements just make up the segment joining them. To prove necessity, take Xl, . . ,Xk in C and Q (Q1 , . ,Qk ) E L\k. One at least of the Q ;'S is positive, say Q1 > 0. 1 itself. Therefore , Y3 := + Q2 + Q 2 + Q 3 Y2 + Q1 Q1 Q3 Ql + Q 2 + Q 3 X3 is in C for the same reason ; and so on until Yk := Ql + . . + Qk 1 l Yk-l + Qk T Xk o The working argument of the above proof is longer to write than to understand.

Then, aXk + (1 - a)xA, is in C and converges to x", which is therefore in cl C. D ° x 1-a a Fig. 3. 1. Observe the nonsymmetric character of x and x' in Fig. 3. 1. 26 A. 2 took convex sets and made new convex sets with them. The present section is devoted to another operation, which takes a nonconvex set and makes a convex set with it. First, let us recall the following basic facts from linear algebra. (i) A linear combination of elements Xl , .. , Xk of IRn is an element 2::=1 O:iX i, where the coefficients O:i are arbitrary real numbers.

1) with C 1 C C 2 ) . We restrict our next statements to the case of the relative interior. 8 help in carrying them over to the closure operation. 11 For i = 1, . . , k, let C, C ffi,n i be convex sets. Then ri(Cl x . . x Ck) = (riCl ) x .. x (riCk) ' Proof. 12 Let A : ffi,n -+ ffi,m be an affinemapping and C a convex set of ffi,n . Then ri(A( C)] = A(ri C) . 3) "# 0, then ri[A(D)] = A(riD). 4) Proof. First, note that the continuity of A implies A(clS) c cl[A(S)] for any S C ffi,n. 8) , and use the monotonicity of the closure operation: A(C) c A(clC) = A[cl(riC)] c cl[A(riC)] c cl[A(C)] ; the closed set cl [A(ri C)] is therefore cl [A(C)].

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