1. Approximation of Non-linear Functions in Mixed Integer Programming Alexander Martin TU Darmstadt Workshop on Integer Programming and Continuous Optimization Chemnitz, November 7-9, 2004 Joint work with Markus Möller and Susanne Moritz

2. A. Martin 2 1.Non-linear Functions in MIPs - design of sheet metal - gas optimization - traffic flows 2.Modelling Non-linear Functions - with binary variables - with SOS constraints 3.Polyhedral Analysis 4.Computational Results Outline

3. A. Martin 3 1.Non-linear Functions in MIPs - design of sheet metal - gas optimization - traffic flows 2.Modelling Non-linear Functions - with binary variables - with SOS constraints 3.Polyhedral Analysis 4.Computational Results Outline

4. A. Martin 4 Design of Transport Channels -Bounds on the perimeters -Bounds on the area(s) -Bounds on the centre of gravity Goal Subject To Maximize stiffness Variables -topology -material

5. A. Martin 5 - contracts - physical constraints Goal Subject To Minimize fuel gas consumption Optimization of Gas Networks

6. A. Martin 6 Gas Network in Detail

7. A. Martin 7 Gas Networks: Nature of the Problem Non-linear - fuel gas consumption of compressors - pipe hydraulics - blending, contracts Discrete - valves - status of compressors - contracts

8. A. Martin 8 Pressure Loss in Gas Networks stationary case horizontal pipes p out p in q

9. A. Martin 9 1.Non-linear Functions in MIPs - design of sheet metal - gas optimization - traffic flows 2.Modelling Non-linear Functions - with binary variables - with SOS constraints 3.Polyhedral Analysis 4.Computational Results Outline

10. A. Martin 10 Approximation of Pressure Loss: Binary Approach p in p out q

11. A. Martin 11 Approximation of Pressure Loss: SOS Approach p out p in q

12. A. Martin 12 Branching on SOS Constraints

13. A. Martin 13 1.Non-linear Functions in MIPs - design of sheet metal - gas optimization - traffic flows 2.Modelling Non-linear Functions - with binary variables - with SOS constraints 3.Polyhedral Analysis 4.Computational Results Outline

14. A. Martin 14 The SOS Constraints: General Definition

15. A. Martin 15 The SOS Constraints: Special Cases SOS Type 2 constraints SOS Type 3 constraints

16. A. Martin 16 The Binary Polytope

17. A. Martin 17 The Binary Polytope: Inequalities

18. A. Martin 18 The SOS Polytope Pipe 1Pipe 2

19. A. Martin 19  |Y|Vertices Facets Max. Coeff. 812161825 1618494742 24 7390670 32 1421049250640 The SOS Polytope: Increasing Complexity

20. A. Martin 20 The SOS Polytope: Properties Theorem. There exist only polynomially many vertices The vertices can be determined algorithmically This yields a polynomial separation algorithm by solving for given and

21. A. Martin 21 The SOS Polytope: Generalizations Pipe to pipe with respect to pressure and flow Several pipes to several pipes Pipes to compressors (SOS constraints of Type 4) General Mixed Integer Programs: Consider Ax=b and a set I of SOS constraints of Type for such that each variable is contained in exactly one SOS constraint. If the rank of A (incl. I ) and are fixed then has only polynomial many vertices.

22. A. Martin 22 Binary versus SOS Approach Binary - more (binary) variables - more constraints - complex facets - LP solutions with fractional y variables and correct variables SOS + no binary variables + triangle condition can be incorporated within branch & bound + underlying polyhedra are tractable

23. A. Martin 23 1.Non-linear Functions in MIPs - design of sheet metal - gas optimization - traffic flows 2.Modelling Non-linear Functions - with binary variables - with SOS constraints 3.Polyhedral Analysis 4.Computational Results Outline

24. A. Martin 24 Computational Results Nr of Pipes Nr of Compressors Total length of pipes Time (  = 0.05) Time (  = 0.01) 11 3 920 1.2 sec 2.0 sec 20 31200 1.2 sec 9.9 sec 3115220011.5 sec104.4 sec