1. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 1 Linear Programming Overview Formulation of the problem and example Incremental, deterministic algorithm Randomized algorithm Unbounded linear programs Linear programming in higher dimensions

2. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 2 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 }

3. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 3 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 624624 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

4. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 4 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

5. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 5

6. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 6 Structure of the feasible region 1. Bounded 2. Unbounded 3. Empty C C C

7. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 7 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

8. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 8 Structure of the feasible region 1. Bounded 2. Unbounded 3. Empty C C C