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Integer Programming, a Technology

Integer Programming, a Technology. Anureet Saxena ACO PhD Student, Tepper School of Business, Carnegie Mellon University. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A. Acknowledgements. Parents Dr. Egon Balas Thesis Committee

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Integer Programming, a Technology

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  1. Integer Programming, a Technology AnureetSaxena ACO PhD Student, Tepper School of Business, Carnegie Mellon University. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

  2. Acknowledgements • Parents • Dr. EgonBalas • Thesis Committee • Dr. EgonBalas • Dr. Samuel Burer • Dr. Gerard Cornuejols • Dr. Francois Margot • Faculty Members, staff and friends … Anureet Saxena, TSoB

  3. Disjunctive Cuts for Non-Convex Mixed Integer Quadratically Constrained Problems Anureet Saxena, TSoB

  4. A. Saxena, P. Bonami and J. Lee, Disjunctive Cuts for non-convex Mixed Integer Quadratically Constrained Problems (to appear in IPCO 2008). Disjunctive Cuts for Non-Convex MIQCP Anureet Saxena, TSoB

  5. MIQCP Integer Constrained Variables Symmetric Matrices NOT necessarily positive semidefinite Anureet Saxena, TSoB

  6. MIQCP Anureet Saxena, TSoB

  7. Research Question? Determine lower bounds on theoptimal value of MIQCP byconstructing strong convex relaxations of MIQCP. Disjunctive Programming Anureet Saxena, TSoB

  8. Disjunctive Programming Polyhedral Relaxation Disjunction Separation Problem Given x2P show that x2PD or find an inequality which is satisfied by all points in PD and is violated by x. Anureet Saxena, TSoB

  9. Disjunctive Programming CGLP Anureet Saxena, TSoB

  10. Disjunctive Programming Polyhedral Relaxation Disjunction Outer Approximationof MIQCP defined by the incumbent solution Anureet Saxena, TSoB

  11. Disjunctive Programming Polyhedral Relaxation Disjunction What are the sources of non-convexity in MIQCP? Anureet Saxena, TSoB

  12. Disjunctive Programming Polyhedral Relaxation Disjunction Integrality Constraints Y=xxT • xj2 Z j2NI • Elementary 0-1 disjunction • (xj· 0) OR (xj¸ 1) • Split Disjunctions • GUB Disjunctions ? Anureet Saxena, TSoB

  13. Y=xxT Y=xxT Eigenvectors of Y-xxT associated with non-zero eigenvalues can be used as sources of cuts All eigenvalues of Y-xxT are equal to zero. Anureet Saxena, TSoB

  14. Y=xxT Ohh!! I don’t like fractional components. I can use them to get good cuts MILP Anureet Saxena, TSoB

  15. Y=xxT Ohh!! I don’t like non-zero eigenvalues. I can use them to get good cuts MIQCP Anureet Saxena, TSoB

  16. Negative Eigenvalues of Y-xxT Anureet Saxena, TSoB

  17. Positive Eigenvalues of Y-xxT Univariate non-convex expression Anureet Saxena, TSoB

  18. Positive Eigenvalues of Y-xxT Anureet Saxena, TSoB

  19. Positive Eigenvalues of Y-xxT Secant Approximation Y.ccT·p(cTx) + q Anureet Saxena, TSoB

  20. Positive Eigenvalues of Y-xxT Anureet Saxena, TSoB

  21. Positive Eigenvalues of Y-xxT Anureet Saxena, TSoB

  22. Cutting Plane Algorithm Convex Quadratic Cut Derive Disjunction CGLP Derive Disjunctive Cut AnureetSaxena, TSoB

  23. Cutting Plane Algorithm Y.ccT· (cTx)2 Can we improve the disjunctive cuts by choosing c more carefully? Convex Quadratic Cut Derive Disjunction CGLP Derive Disjunctive Cut AnureetSaxena, TSoB

  24. A Lesson from MILP Lift and Project Cuts Intersection Cuts Elementary 0-1 Disjunctions Balas ’72 Balas and Peregaard ‘02 Anderson, Cornuejols & Li ‘04 Intersection cuts are canonicaldisjunctive cuts which are used to determine improving directions orreduced costs for improving L&P cuts. Anureet Saxena, TSoB

  25. A Lesson from MILP Lift and Project Cuts ? Y.ccT· (cx)2 Anureet Saxena, TSoB

  26. A Lesson from MILP Lift and Project Cuts Secant Approximation Y.ccT· (cx)2 Y.ccT·p(cTx) + q Anureet Saxena, TSoB

  27. A Lesson from MILP Lift and Project Cuts Secant Approximation Y.ccT· (cx)2 Error a b width Anureet Saxena, TSoB

  28. A Lesson from MILP This condition is always satisfied if c belongs to vector space spanned by eigenvectors of Y-xxT associated with positiveeigenvalues. This can be calculated by solving a linear program whose right hand side is a linear function of c Anureet Saxena, TSoB

  29. A Lesson from MILP This problem can be formulated as a mixed integer linear program!! Univariate Expression Generating Mixed Integer Program (UGMIP) This condition is always satisfied if c belongs to vector space spanned by eigenvectors of Y-xxT associated with positiveeigenvalues. This can be calculated by solving a linear program whose right hand side is a linear function of c Anureet Saxena, TSoB

  30. Cutting Plane Algorithm UGMIP Convex Quadratic Cut Derive Disjunction CGLP Derive Disjunctive Cut AnureetSaxena, TSoB

  31. Cutting Plane Algorithm Version 1 UGMIP Convex Quadratic Cut Derive Disjunction CGLP Derive Disjunctive Cut AnureetSaxena, TSoB

  32. Cutting Plane Algorithm Version 2 UGMIP Convex Quadratic Cut Derive Disjunction CGLP Derive Disjunctive Cut AnureetSaxena, TSoB

  33. Cutting Plane Algorithm Version 3 UGMIP Convex Quadratic Cut Derive Disjunction CGLP Derive Disjunctive Cut AnureetSaxena, TSoB

  34. Computational Results • Solvers • Convex Relaxations – IpOpt • Eigenvalue Computations – LAPACK • Linear Programs & Mixed Integer Programs– CPLEX 10.1 • COIN-OR / Bonminbased implementation • Test Bed • 160 GlobalLIB Instances • 4 Chemical Process Design instances from Lee & Grossman • 40 Box QP Instances AnureetSaxena, TSoB

  35. Computational Results • Experiment Setup • 1 Hour Time limit on each instance • Initial Relaxation strengthened by RLT inequalities AnureetSaxena, TSoB

  36. GlobalLIB Instances 160 = 129 + 24 + 7 • All MIQCP Instances with upto 50 variables • x1x2x3x4x5 • (x1+x2)/x3¸2x1 • x0.75 Numerical Problems Zero Duality Gap Non-Zero Duality Gap AnureetSaxena, TSoB

  37. GlobalLIB Instances AnureetSaxena, TSoB

  38. GlobalLIB Instances Anureet Saxena, TSoB

  39. Version (2,3) vs Version 1 Anureet Saxena, TSoB

  40. Version (2,3) vs Version 1 Observation • Either version 2 or version 3 closes >99% of the duality gap on 44 instances on which version 1 is unable to close any gap. • The relaxation obtained by adding disjunctive cuts can be substantially stronger than the SDP relaxation!! Anureet Saxena, TSoB

  41. Version 2 vs Version 3 Anureet Saxena, TSoB

  42. Version 2 vs Version 3 Observation • Version 3 closes 10% more duality gap than version 2 on 23 instances. • The effort spent on finding c vectors with small width pays off!! Anureet Saxena, TSoB

  43. Linear Complementarity Disjunctions • Some problems have linear complementarity constraints • xixj = 0 • These constraints can be used to derive the linear complementarity disjunctions • (xi=0) OR (xj=0) • which can be used with the medley of other disjunctions to derive disjunctive cuts Anureet Saxena, TSoB

  44. Linear Complementarity Disjunctions Observation Linear Complementarity conditions can be exploited effectively within a disjunctive programming framework to derive strong cuts AnureetSaxena, TSoB

  45. Recap Eigenvalue computation Disjunctive Programming Research Question What is the marginal value of Disjunctive Programming in this framework? AnureetSaxena, TSoB

  46. Marginal Value of Disjunctive Programming Secant Approximation Y.ccT·p(cTx) + q Anureet Saxena, TSoB

  47. Marginal Value of Disjunctive Programming UGMIP Convex Quadratic Cut Derive Disjunction Secant Approximation CGLP Derive Disjunctive Cut AnureetSaxena, TSoB

  48. Marginal Value of Disjunctive Programming AnureetSaxena, TSoB

  49. Marginal Value of Disjunctive Programming 32% 28% AnureetSaxena, TSoB

  50. Summary Disjunctive Programming Anureet Saxena, TSoB

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