Convex Analysis and Nonlinear Optimization: Theory and by Jonathan M. Borwein, Adrian S. Lewis

By Jonathan M. Borwein, Adrian S. Lewis

Optimization is a wealthy and thriving mathematical self-discipline, and the underlying conception of present computational optimization recommendations grows ever extra refined. This e-book goals to supply a concise, obtainable account of convex research and its purposes and extensions, for a large viewers. each one part concludes with a regularly broad set of non-compulsory routines. This re-creation provides fabric on semismooth optimization, in addition to numerous new proofs.

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Additional resources for Convex Analysis and Nonlinear Optimization: Theory and Examples - Second edition (CMS Books in Mathematics)

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We denote the set of points where a fun ction 9 : Y ---+ [-00, + 00] is finite and conti nuous by contg . 5 (Fenchel duality and convex calculus) For given function s f : E ---+ (00, +00] and 9 : Y ---+ (00, +00] and a lin ear map A : E ---+ Y , let p, dE [- 00, +00] be primal and dual valu es defin ed, respectively, by the Fenchel problems p = d = inf {f (x) xE E + g(Axn sup {- j*(A* ¢) - g*( - ¢n . >EY These values satisfy the weak duality in equality p 2:: d. 7} is atta in ed if fin it e. 9} holds.

3) Geome trically, Go rd an 's t heo rem says that the origin do es not lie in the convex hull of t he set {aO , aI, .. , arT! } if and only if ther e is an open halfsp ace {y I (y, x ) < O} containing {aO , a I , .. , am} (and hence it s convex hull) . This is another illustration of t he idea of separa tio n (in t his case we separate the origin a nd t he convex hull) . Theorems of t he alternative like Gordan 's theorem m ay be proved in a vari ety of ways , including separation and algorit h mic a pproaches .

Deduce that if 9 is twi ce d iffer entiable then 9 is convex if a nd only if o" is nonnegative on I , and 9 is strictly convex if s" is st r ict ly positive on I . (c) Deduce that if f is twice cont inuo usly differentiable on S t hen f is convex if and only if it s Hess ian matrix is po sitive semide finite everyw here on S, and f is st rict ly convex if it s Hessian matrix is positive definite every where on S . 1 Subgr adien ts a nd Convex Functi ons (d) Find a strict ly convex fun ct ion 1 : (- 1,1 ) ---.

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