1. Linear Programming Computational Geometry, WS 2006/07 Lecture 5, Part I Prof. Dr. Thomas Ottmann Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg

2. Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann2 Overview Problem formulation and example. Incremental, deterministic algorithm. Randomized algorithm. Unbounded linear programs. Linear programming in higher dimensions.

3. Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann3 Problem Description Maximize c 1 x 1 + c 2 x 2 +... + c d x d Subject to the conditions: a 1,1 x 1 +... a 1,d x d b 1 a 2,1 x 1 +... a 2,d x d b 2 ::: a n,1 x 1 +... a n,d x d b n Linear program of dimension d: c = (c 1,c 2,...,c d ) h i = {(x 1,...,x d ) ; a i,1 x 1 +... + a i,d x d b i } l i = hyperplane that bounds h i ( straight lines, if d=2 ) H = {h 1,..., h n }

4. Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann4 Example Production of two goods A and B using four raw materials Value of A: 6 CU, value of B: 3 CU Rm1Rm2Rm3Rm4 Prod A Prod B Reserve 245245 212212 6246242 223223 Maximize profit: f c (x) = 6x A + 3x B under the conditions: x A = 0, x B x A, x B = 0 2x A + 4x B 5 2x A + 1x B 2 6x A + 2x B 4 2x A + 2x B 3 x A, x B 0

5. Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann5 Chart IiIi (x A, x B ) 1(0, 5/4)(5/2, 0) 2(0, 2)(1, 0) 3(0, 2)(2/3, 0) 4(0, 3/2)(3/2, 0) 2 3/2 5/4 1 1/2 2/313/225/2 xBxB xAxA

6. Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann6 Chart

7. Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann7 Structure of Feasible Region 1. Bounded 2. Unbounded 3. Empty C C C

8. Computational Geometry, WS 2006/07 Prof. Dr. Thomas Ottmann8 Result Four possibilities for the solution of a linear program 1.A vertex of the feasible region is the only solution. 2.One edge of the feasible region contains all solutions. 3.There are no solutions. 4.The feasible region is unbounded toward the direction of optimization. In case 2: Choose the lexicographically minimum solution = > corner