By Jonathan M. Borwein

Optimization is a wealthy and thriving mathematical self-discipline, and the underlying conception of present computational optimization concepts grows ever extra refined. This e-book goals to supply a concise, obtainable account of convex research and its functions and extensions, for a vast viewers. every one part concludes with a frequently vast set of not obligatory workouts. This new version provides fabric on semismooth optimization, in addition to numerous new proofs.

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**Extra resources for Convex analysis and nonlinear optimization: Theory and examples**

**Example text**

With compact level sets. f' (Xi x - x ) = - 00 for any point x on the boundary of 42 3. ' is linear with Gx for any vector c in R n that the problem inf { f (x ) + (c, x) I Gx = b, has a unique optimal solut ion X, lying in = b. Prove x E R"} R~ + . 3 (First order condit ions for linear const raint s) to prove that some vector A in R '" sa t isfies V'f( x) = G*A - c, and deduce Xi = exp(G* A - ck 28. ** (DAD problems [36]) Consider the following example of Exercis e 27 (Maximum entropy) . Suppose the k x k matrix A has each ent ry aij nonnegative.

5) is convex on R " by calculat ing its Hessian . 23. * If the fun ction f : E ---7 (00, +00] is essentially st rict ly convex , prove all distinct point s X and y in E satisfy f)f( x) n f)f(y) = 0. Deduce that f has at most on e minimizer . 24. (Minimizers of essentially smooth functions) Prove that any minimizer of an essent ially smooth fun ction f must lie in core (dom J) . 25. ** (Convex matrix functions) Consid er a matrix C in S+.. 1 Subgradients and Convex Functions 41 (a) For m atrices X in S+.

C) Prove similarly the function X E S" t---+ tr (CX 2 ) and the function XE S+' t---+ -tr (CX 1 / 2 ) are convex. 26. ** (Lo g-convexit y) Given a convex set C c E , we say that a function f : C ----t R ++ is log-conv ex if log f(-) is convex. 1 , Exercise 9 (Composing convex fun ctions) . (b) If a polynomial p : R ----t R has all real roots, prove l ip is logconvex on any interval on which p is strictly positive. (c) One version of Holder's in equality states, for real p ,q > 1 satisfying »: ' + «: ' = 1 and functions u , v : R + ----t R , J ~ (Jlul (J Ivl uv q P ) l /p ) l /q when the right hand side is well-defined.