By Heinrich Dorrie

Difficulties that beset Archimedes, Newton, Euler, Cauchy, Gauss, Monge and different greats, able to problem today's would-be challenge solvers. between them: How is a sundial developed? how will you calculate the logarithm of a given quantity with no using logarithm desk? No complicated math is needed. contains a hundred issues of proofs.

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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). , «(X), + 00] is convex and th e point x lies in core (dom f). Th en f is Gateau x differentiabl e at x ex ac tly wh en f has a uniqu e subgradient at x (in which case th is subgradient is th e deri va tive) . 1 Subgradients and Convex Functions 37 We say the convex function f is essentially smooth if it is Gateaux differentiable on dom of. 1, Exercise 21) that a function is essentially smooth if and only if its subdifferential is always singleton or empty.

Subgradients and normal cones) If a point x lies in a set C c E , prove oi5c(x) = Nc(x). 5. Prove the following functions x E R t-+ of: f( x) are convex and calc ulate (a) Ixl (b) i5 R + - ,fi if x 2: 0 { +00 otherwise (c) 3. Fenchel Duality 38 o (d) 1 { +00 if x

15) is solvable. 16) is un solvabl e. Hint : Com p lete t he following steps . 1. (b) Prove (ii) implies (iii). 17) i =O where K is the subs pace {( (ai ,x)) ~o I x E E} c Rm+l . 16) is solvable. Gen er aliz e by considering t he problem inf{f(x ) I X j 2:: 0 (j 9. *. (Schur-convexity) The dual cone of t he cone R~ E J)} . is defin ed by (R~ ) + = {y E R " I (x , y) 2:: 0 for all x in R~ }. (a) Prove a vect or Y lies in ( R~)+ if and only if j L Yi 2:: 0 for j = 1,2, . , n - 1, 1 (b) By wr iting I:{[x li = m ax k(ak, x ) for some suit a ble set of vectors a k , prove t hat t he function x f--t I:{[Xli is convex.

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