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Approximation of Non-linear Functions in Mixed Integer Programming

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

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Approximation of Non-linear Functions in Mixed Integer Programming

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

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

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

  4. Design of Transport Channels Goal Maximize stiffness Subject To - Bounds on theperimeters • Bounds on thearea(s) • Bounds on thecentre of gravity Variables - topology • material A. Martin

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

  6. Gas Network in Detail 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 A. Martin

  8. Pressure Loss in Gas Networks pout horizontal pipes stationary case pin q A. Martin

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

  10. Approximation of Pressure Loss: Binary Approach pout pin q A. Martin

  11. Approximation of Pressure Loss: SOS Approach pout pin q A. Martin

  12. Branching on SOS Constraints A. Martin

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

  14. The SOS Constraints: General Definition A. Martin

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

  16. The Binary Polytope A. Martin

  17. The Binary Polytope: Inequalities A. Martin

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

  19. The SOS Polytope: Increasing Complexity 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 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. A. Martin

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

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

  24. Computational Results A. Martin

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