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 advent to convex research and its extensions goals at the beginning yr graduate scholars, and contains many guided routines. The corrected moment version provides a bankruptcy emphasizing concrete versions. New subject matters comprise monotone operator thought, Rademacher's theorem, proximal general geometry, Chebyshev units, and amenability. the ultimate fabric on "partial smoothness" received a 2005 SIAM remarkable Paper Prize.

Jonathan M. Borwein, FRSC is Canada study Chair in Collaborative expertise at Dalhousie college. A Fellow of the AAAS and a overseas member of the Bulgarian Academy of technological know-how, he obtained his Doctorate from Oxford in 1974 as a Rhodes student and has labored at Waterloo, Carnegie Mellon and Simon Fraser Universities. popularity for his huge guides in optimization, research and computational arithmetic comprises the 1993 Chauvenet prize.

Adrian S. Lewis is a Professor within the tuition of Operations learn 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.

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**Example text**

4) is convex. We say h is proper if dom h is nonempty and h never takes t he valu e - 00 : if we wish t o demonstra te the existence of subgradients for v using t he results in t he previous section t hen we need to excl ude -00 . 6 If th e f un ction h : E -+ [-00, +00] is convex and some point fj in core (dom h) satisfies h (fj) > - 00, th en h never tak es th e value -00 . Proof. Suppose som e point y in E satisfies h(y) = - 00. Since fj lies in core (dom h) , there is a real t > 0 with fj + t(fj - y) in dom (h) , and hence a real r with (fj + t( fj - y ), r) in epi (h) .

Prove the function if x E R+. otherwise is convex. 19. (Domain of subdifferential) If the fun ction f: R 2 ~ (00, +00] is defin ed by f( Xl , X 2 ) -- {max{1 +00 prove that JXl,IX 21} if Xl :::: 0 otherwise, f is convex but that dom f)f is not convex. 20. * (Monotonicity of gradients) Suppose that the set 5 c R " is op en a nd convex and that the fun ction f : 5 ~ R is differ entiabl e. Prove f is convex if and only if (\1 f( x) - \1 fey) , X - y ) :::: 0 for all x , y E 5, a nd f is st rict ly convex if and only if the ab ove inequ ality holds st rict ly whenever X #- y.

2, each Pk is every where finite and sublinear. 7 we know linpk => linpk - I + span {ek} for k = 1,2, . . ,n, so Pn is linear. Thus ther e is an eleme nt 4> of E sa ti sfyi ng (4), ,) = PnO. 7 implies Pn :::; Pn-I :::; . . 6, any point x in E sa t isfies Pn( x - x ) :::; Po(x - x ) = f'( x ; x - x ) :::; f( x) - f( x) . Thus 4> is a subgradient. 7 we see Pn(d) :::; Po(d) = po( ed = -p~ (e l ; - el) = f'( x ;0) . = -PI (-ed = - PI (-d) :::; -Pn( -d) = Pn(d) , wh enc e Pn(d) = Po(d) = f'( x ;d).