1 / 24

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.

alair
Télécharger la présentation

Approximation of Non-linear Functions in Mixed Integer Programming

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related