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MIP Lifting Techniques for Mixed Integer Nonlinear Programs Jean-Philippe P. Richard* School of Industrial Engineering, Purdue University Mohit Tawarmalani.

Published byGarey Richardson Modified over 7 years ago

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Presentation on theme: "MIP Lifting Techniques for Mixed Integer Nonlinear Programs Jean-Philippe P. Richard* School of Industrial Engineering, Purdue University Mohit Tawarmalani."— Presentation transcript:

1 MIP Lifting Techniques for Mixed Integer Nonlinear Programs Jean-Philippe P. Richard* School of Industrial Engineering, Purdue University Mohit Tawarmalani Krannert School of Management, Purdue University *Supported by NSF DMI0348611

2 MIP 2006, Thursday June 8th 2006 2 Structure of the Talk

3 MIP 2006, Thursday June 8th 2006 3

4 4 A Motivation in Integer Programming

5 MIP 2006, Thursday June 8th 2006 5 A Motivation in Integer Programming

6 MIP 2006, Thursday June 8th 2006 6 What is MIP Lifting?

7 MIP 2006, Thursday June 8th 2006 7 Some Literature on MIP lifting

8 MIP 2006, Thursday June 8th 2006 8 What is Hard about MINLP Lifting?

9 MIP 2006, Thursday June 8th 2006 9 Overview & Goal of Our Work

10 MIP 2006, Thursday June 8th 2006 10 Some Nice Features of MINLP Lifting

11 MIP 2006, Thursday June 8th 2006 11

12 MIP 2006, Thursday June 8th 2006 12 Goal of Part II

13 MIP 2006, Thursday June 8th 2006 13

14 MIP 2006, Thursday June 8th 2006 14 Mixed Integer Nonlinear Knapsack

15 MIP 2006, Thursday June 8th 2006 15 Mixed Integer Nonlinear Knapsack

16 MIP 2006, Thursday June 8th 2006 16 Generating Valid Inequalities for PS

17 MIP 2006, Thursday June 8th 2006 17 A General Lifting Result for PS

18 MIP 2006, Thursday June 8th 2006 18 Advantages and Limitations of the Lifting Scheme

19 MIP 2006, Thursday June 8th 2006 19 A Superadditive Lifting Result for PS

20 MIP 2006, Thursday June 8th 2006 20 A Superadditive Lifting Result for PS

21 MIP 2006, Thursday June 8th 2006 21 A Superadditive Lifting Result for PS

22 MIP 2006, Thursday June 8th 2006 22

23 MIP 2006, Thursday June 8th 2006 23 Application: Bilinear Mixed Integer Knapsack Problem (BMIKP)

24 MIP 2006, Thursday June 8th 2006 24 BMIKP: Comments

25 MIP 2006, Thursday June 8th 2006 25 The Convex Hull of PT’ is a Polyhedron

26 MIP 2006, Thursday June 8th 2006 26 Obtaining Facets of PT using Superadditive Lifting

27 MIP 2006, Thursday June 8th 2006 27 Obtaining Facets of PT using Superadditive Lifting

28 MIP 2006, Thursday June 8th 2006 28 Example

29 MIP 2006, Thursday June 8th 2006 29 Obtaining Facets of PT using Superadditive Lifting

30 MIP 2006, Thursday June 8th 2006 30 Another Family of Strong Inequalities for PT

31 MIP 2006, Thursday June 8th 2006 31 Another Family of Strong Inequalities for PT

32 MIP 2006, Thursday June 8th 2006 32 Example

33 MIP 2006, Thursday June 8th 2006 33

34 MIP 2006, Thursday June 8th 2006 34

35 MIP 2006, Thursday June 8th 2006 35 An Equivalent Integer Programming Formulation for PT

36 MIP 2006, Thursday June 8th 2006 36 An Equivalent Integer Programming Formulation for PT

37 MIP 2006, Thursday June 8th 2006 37 Strong Rank-1 Inequalities

38 MIP 2006, Thursday June 8th 2006 38 High Rank Certificate

39 MIP 2006, Thursday June 8th 2006 39 Lifted Cover Cuts for BMIKP are not Strong Rank-1

40 MIP 2006, Thursday June 8th 2006 40 Towards the Next Step…

41 MIP 2006, Thursday June 8th 2006 41

42 MIP 2006, Thursday June 8th 2006 42 Goal of Part III

43 MIP 2006, Thursday June 8th 2006 43 A General Procedure

44 MIP 2006, Thursday June 8th 2006 44 Deriving Nonlinear Cuts for Mixed Integer Programs: Applications

45 MIP 2006, Thursday June 8th 2006 45 Obtaining the Convex Hull of a Simple Bilinear Knapsack Set

46 MIP 2006, Thursday June 8th 2006 46 Obtaining the Convex hull of a Simple Bilinear Knapsack Set

47 MIP 2006, Thursday June 8th 2006 47 Obtaining the Convex hull of a Simple Bilinear Knapsack Set

48 MIP 2006, Thursday June 8th 2006 48 Obtaining the Convex hull of a Simple Bilinear Knapsack Set

49 MIP 2006, Thursday June 8th 2006 49 Obtaining Convex Hulls of Disjunctive Sets: An Example

50 MIP 2006, Thursday June 8th 2006 50 Obtaining Convex Hulls of Disjunctive Sets: An Example

51 MIP 2006, Thursday June 8th 2006 51 Obtaining Convex Hulls of Disjunctive Sets: An Example

52 MIP 2006, Thursday June 8th 2006 52 Obtaining the Convex Hull of Disjunctive Sets: A Result

53 MIP 2006, Thursday June 8th 2006 53 Obtaining the Convex Hull of Disjunctive Sets: An Example

54 MIP 2006, Thursday June 8th 2006 54 Obtaining the Convex Hull of Disjunctive Sets: An Example

55 MIP 2006, Thursday June 8th 2006 55 Some Comments

56 MIP 2006, Thursday June 8th 2006 56

57 MIP 2006, Thursday June 8th 2006 57 Goal of Part IV

58 MIP 2006, Thursday June 8th 2006 58 Generalizing the Theory…

59 MIP 2006, Thursday June 8th 2006 59 Generalizing the Superadditive Lifting Theory…

60 MIP 2006, Thursday June 8th 2006 60 Application: Sequence Independent lifting for single-constraint problems

61 MIP 2006, Thursday June 8th 2006 61 Application: Sequence Independent lifting for single-constraint problems

62 MIP 2006, Thursday June 8th 2006 62

63 MIP 2006, Thursday June 8th 2006 63 Conclusion & Future Work

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