Linear Programming

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

2. Linear program of dimension d: c = (c1,c2,...,cd) hi = {(x1,...,xd) ; ai,1x1 + ... + ai,dxd bi} Problem description Maximize c1x1 + c2x2 + ... + cdxd Subject to the conditions: a1,1x1 + ... a1,dxd b1 a2,1x1 + ... a2,dxd b2 : : : an,1x1 + ... an,dxd bn li = hyperplane that bounds hi (straight lines, ifd=2) H = {h1, ... , hn} 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 ofB: 3 CU Maximize profit: fc (x) = 6xA+ 3xB under the conditions: 2xA + 4xB 52xA + 1xB 26xA + 2xB 42xA + 2xB 3 xA, xB 0 Computational Geometry Prof. Dr. Th. Ottmann

4. xA Chart 2 3/2 5/4 1 1/2 xB 1/2 2/3 1 3/2 2 5/2 Computational Geometry Prof. Dr. Th. Ottmann

5. Computational Geometry Prof. Dr. Th. Ottmann

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

7. Result • Four possibilities for the solution of a linear program • A vertex of the feasible region is the only solution. • One edge of the feasible region contains all solutions. • There are no solutions. • The feasible region is unbounded toward the direction of optimization. • In case 2: Choose the lexicographically minimum solution = > corner Computational Geometry Prof. Dr. Th. Ottmann

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