# Convex analysis and nonlinear optimization: Theory and by Jonathan M. Borwein 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.

Similar mathematics books

Mathematik für Physiker 2: Basiswissen für das Grundstudium der Experimentalphysik

Die für Studienanfanger geschriebene „Mathematik für Physiker'' wird in Zukunftvom Springer-Verlag betreut. Erhalten bleibt dabei die Verbindung einesakademischen Lehrbuches mit einer detaillierten Studienunterstützung. DieseKombination hat bereits vielen Studienanfangern geholfen, sich die Inhalte desLehrbuches selbständig zu erarbeiten.

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 ) 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.