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.
Read Online or Download Convex Analysis and Non Linear Optimization PDF
Similar linear programming books
This publication provides a landscape of operator thought. It treats a number of sessions of linear operators which illustrate the richness of the idea, either in its theoretical advancements and its functions. for every of the sessions quite a few differential and crucial operators encourage or illustrate the most effects.
Basic Classes of Linear Operators
A comprehensive graduate textbook that introduces functional research with an emphasis at the idea of linear operators and its program to differential equations, vital equations, countless structures of linear equations, approximation conception, and numerical research. As a textbook designed for senior undergraduate and graduate scholars, it starts off with the geometry of Hilbert areas and proceeds to the idea of linear operators on those areas together with Banach areas.
Creation making plans difficulties containing distinct features from strategy industries are addressed during this e-book. the most topic is the advance of mathematical programming versions that permit to version construction plans which aren't disrupted by means of discretization of time. even if, discrete time versions are used as a foundation and are hence better to incorporate features of time continuity.
- Metaheuristics:: Progress as Real Problem Solvers
- Transport Equations and Multi-D Hyperbolic Conservation Laws
- Linear Programming, Volume 1: Introduction (Springer Series in Operations Research and Financial Engineering)
- A mathematical view of interior-point methods in convex optimization
- Handbook on Data Envelopment Analysis
Additional resources for Convex Analysis and Non Linear Optimization
Example text
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.