Solution Manual For Introduction to Linear Optimization byDimitris英文清晰版.pdf

Solution Manual For: Introduction to Linear Optimization by Dimitris Bertsimas & John N. Tsitsiklis John L. Weatherwax∗ November 22, 2007 Introduction Acknowledgements Special thanks to Dave Monet for helping find and correct various typos in these solutions. Chapter 1 (Introduction) Exercise 1.1 Since f (·) is convex we have that f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) . (1) Since f (·) is concave we also have that f (λx + (1 − λ)y) ≥ λf (x) + (1 − λ)f (y) . (2) Combining these two expressions we have that f must satisfy each with equality or f (λx + (1 − λ)y) = λf (x) + (1 − λ)f (y) . (3) This implies that f must be linear and the expression given in the book holds. ∗wax@alum.mit.edu 1 Exercise 1.2 Part (a): We are told that fi is convex so we have that f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) , (4) i i i for every i. For our function f (·) we have that m f (λx + (1 − λ)y) = f (λx + (1 − λ)y) (5) i i=1 m ≤ λf (x) + (1 − λ)f (y) (6) i i