Download e-book for kindle: Convex Analysis and Nonlinear Optimization: Theoryand by Jonathan Borwein, Adrian S. Lewis

Download e-book for kindle: Convex Analysis and Nonlinear Optimization: Theoryand by Jonathan Borwein, Adrian S. Lewis

By Jonathan Borwein, Adrian S. Lewis

A cornerstone of recent optimization and research, convexity pervades functions ranging via engineering and computation to finance.

This concise creation to convex research and its extensions goals firstly yr graduate scholars, and contains many guided routines. The corrected moment variation provides a bankruptcy emphasizing concrete types. New issues contain monotone operator concept, Rademacher's theorem, proximal basic geometry, Chebyshev units, and amenability. the ultimate fabric on "partial smoothness" gained a 2005 SIAM notable Paper Prize.

Jonathan M. Borwein, FRSC is Canada learn Chair in Collaborative expertise at Dalhousie collage. A Fellow of the AAAS and a international member of the Bulgarian Academy of technology, he got his Doctorate from Oxford in 1974 as a Rhodes pupil and has labored at Waterloo, Carnegie Mellon and Simon Fraser Universities. attractiveness for his large courses in optimization, research and computational arithmetic comprises the 1993 Chauvenet prize.

Adrian S. Lewis is a Professor within the tuition of Operations study and business Engineering at Cornell. Following his 1987 Doctorate from Cambridge, he has labored at Waterloo and Simon Fraser Universities. He obtained the 1995 Aisenstadt Prize, from the collage of Montreal, and the 2003 Lagrange Prize for non-stop Optimization, from SIAM and the Mathematical Programming Society.

About the 1st Edition:

"...a very profitable ebook, and that i hugely suggest it... "

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- L. Qi, within the Australian Mathematical Society Gazette

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- J.-P. Penot, in Canadian Mathematical Society Notes

"There is an interesting interweaving of conception and applications..."

- J.R. Giles, in Mathematical Reviews

" excellent introductory educating text..."

- S. Cobzas, in Studia Universitatis Babes-Bolyai Mathematica

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Extra info for Convex Analysis and Nonlinear Optimization: Theoryand Examples

Example text

The proof of t he calculus rule in t he second part of the t heore m is a simple consequence of t he first part (Exercise 9). D T he case of the Fen chel theorem above, when t he fun ction 9 is simply the indicator fun ction of a point , gives the following part icularly elegant and useful corollary. 11 (Fenchel duality for linear constraints) Given any fu nction f : E --+ (00, +00]' any lin ear map A : E --+ Y , and an y element b of Y , th e weak duality inequality inf {f( x) I A x xE E = b} 2:: sup {(b,¢» - j* (A*¢>)} ¢E Y hold s.

9) has a nonzero optimal so lution. (d) Use the Karush-Kuhn-Tucker conditions to find the unique optimal solution. 32 2. Inequality Const raints (e) Deduce that any vect ors x and y in R " satisfy (y, x) ::; II Yll qll x ll p . ) 7. * Cons ider a matrix A in S++ and a real b » O. (a) Assuming t he pr oblem inf{- log det X I t r A X < b, X E S ~ +} has a solut ion, find it . (b) Repeat usin g the obj ective func tion tr X -I. (c) Prove the problems in parts (a) and (b) have optimal solutions. ) 8.

On 9. * (Coercivity) Suppose that the fun ction f : E ---+ R is differentiable and satisfies the growt h condition limllxll-+oo f( x) /ll xll = +00. Prove that the gradient map V'f has range E . ) 10. + ---+ R defined by f(X) = tr X - I is differentiable on S+'+ . + ---+ R by f(X) = logdetX. Prove V' f(I) = I . + . 11. * * (Kirchhoff's law [9, Chapter 1]) Consider a finit e, undirected, connec te d graph with vertex set V and edge set E. Suppose t hat a a nd (3 in V are distinct vertices and that each edge ij in E has an assoc iate d "res istance" r ij > 0 in R .

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