# Convex Analysis and Non Linear Optimization by Jonathan Borwein, Adrian S. Lewis

By Jonathan Borwein, Adrian S. Lewis

Optimization is a wealthy and thriving mathematical self-discipline, and the underlying thought of present computational optimization thoughts grows ever extra subtle. This publication goals to supply a concise, available account of convex research and its functions and extensions, for a wide viewers. every one part concludes with a frequently wide set of non-compulsory workouts. This re-creation provides fabric on semismooth optimization, in addition to numerous new proofs.

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26. ∗∗ (Log-convexity) Given a convex set C ⊂ E, we say that a function f : C → R++ is log-convex if log f (·) is convex. 1, Exercise 9 (Composing convex functions). (b) If a polynomial p : R → R has all real roots, prove 1/p is logconvex on any interval on which p is strictly positive. (c) One version of H¨ older’s inequality states, for real p, q > 1 satisfying −1 −1 p + q = 1 and functions u, v : R+ → R, uv ≤ 1/p |u|p |v|q 1/q when the right-hand-side is well-deﬁned. Use this to prove the Gamma function Γ : R → R given by ∞ Γ(x) = tx−1 e−t dt 0 is log-convex.

16) ai , x ≤ 0 for i = 0, 1, . . , m, not all 0, x ∈ E. 16) is unsolvable. Hint: complete the following steps. 1. (b) Prove (ii) implies (iii). 17) inf exp yi | y ∈ K , i=0 m+1 . 16) is solvable. 2 Theorems of the alternative 35 Generalize by considering the problem inf{f (x) | xj ≥ 0 (j ∈ J)}. 9. ∗∗ (Schur-convexity) The dual cone of the cone Rn≥ is deﬁned by (Rn≥ )+ = {y ∈ Rn | x, y ≥ 0, for all x in Rn≥ }. (a) Prove (Rn≥ )+ = {y | j 1 yi ≥ 0 (j = 1, 2, . . , n − 1), n 1 yi = 0}. (b) By writing j1 [x]i = maxk ak , x for some suitable set of vectors ak , prove that the function x → j1 [x]i is convex.

K, x ∈ RZ . (a) Suppose A has doubly stochastic pattern. Prove there is a point xˆ in the interior of RZ+ which is feasible for the problem above. Deduce that the problem has a unique optimal solution x¯ satisfying, for some vectors λ and µ in Rk , x¯ij = aij exp(λi + µj ), for (i, j) ∈ Z. (b) Deduce that A has doubly stochastic pattern if and only if there are diagonal matrices D1 and D2 with strictly positive diagonal entries and D1 AD2 doubly stochastic. 1 Subgradients and convex functions 29.