By Jonathan Borwein, Adrian S. Lewis

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

This concise advent to convex research and its extensions goals in the beginning 12 months graduate scholars, and comprises many guided routines. The corrected moment version provides a bankruptcy emphasizing concrete versions. New subject matters comprise monotone operator idea, Rademacher's theorem, proximal common geometry, Chebyshev units, and amenability. the ultimate fabric on "partial smoothness" gained a 2005 SIAM remarkable Paper Prize.

Jonathan M. Borwein, FRSC is Canada examine Chair in Collaborative expertise at Dalhousie collage. A Fellow of the AAAS and a overseas member of the Bulgarian Academy of technology, he got his Doctorate from Oxford in 1974 as a Rhodes student 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 university of Operations study and commercial Engineering at Cornell. Following his 1987 Doctorate from Cambridge, he has labored at Waterloo and Simon Fraser Universities. He got the 1995 Aisenstadt Prize, from the college of Montreal, and the 2003 Lagrange Prize for non-stop Optimization, from SIAM and the Mathematical Programming Society.

About the 1st Edition:

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**Extra info for Convex Analysis and Nonlinear Optimization: Theoryand Examples**

**Example text**

8) are a converse to the above resul t wh en the functions f , g1, g2, ... , 9TTl ar e convex a nd differentiable. We next follow a very different, and sur prising, route t o this result , circumvent ing differentiability. 1), and analyze the resulting (optimal) value fun ction v : R TTl -+ [-00 , + 00], defined by the equat ion v(b) = inf{j(x ) I g( x) :s; b}. 4) We show that Lagran ge mu ltiplier vectors 5. corr esp ond to subgradient s of v (Exerc ise 9). Our old definition of convex ity for func tions does not naturally extend to fun ctions h : E -+ [-00, +ooJ (due t o t he p ossible occ urre nce of 00 - (0).

0 in R and nonzero x in R ". Prove f has a minimizer. (b) Given a matrix A in S" , define a function g(x) = x TA x /llxl 12 for nonzero x in R " . Prove 9 has a minimizer. (c) Calculate "V9 (x) for nonzero x. (d) Deduce t hat minimizers of 9 must be eigenvectors, and calculate the minimum value. (e) Find an alternative proof of part (d) by using a spectral decomposition of A . ) 7. Suppose a convex function 9 : [0,1] ----? R satisfies g(O) = O. Prove the function t E (0,1] t----+ g(t) /t is nondecreasing.

At each ste p we guarantee on e ext ra dir ection of linearity. The basic st ep is summarized in the following exercise. 7 Suppo se that th e function p : E ----; (00, + ooJ is sublin ear and that the point x lies in core (dom p). ) satisfies the conditions 36 3. x) = Ap(X) fo r all real ).. , (ii) q :::; p , and (iii) lin q => lin p + span {x}. With t his tool we are now ready for t he main result, whi ch gives conditions guaranteeing the existe nce of a subgradi ent . 6 showed how t o identify subgradients from directional derivatives; t his next result shows how t o move in the reverse direction .