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Convex Relaxations of Non-Convex Mixed Integer Quadratically Constrained Problems

Convex Relaxations of Non-Convex Mixed Integer Quadratically Constrained Problems. Dr. Anureet Saxena Associate, Research Axioma Inc. (Joint Work with Pierre Bonami and Jon Lee) Dedicated to Prof. Egon Balas. TexPoint fonts used in EMF.

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Convex Relaxations of Non-Convex Mixed Integer Quadratically Constrained Problems

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  1. Convex Relaxations of Non-Convex Mixed Integer Quadratically Constrained Problems Dr. Anureet Saxena Associate, Research Axioma Inc. (Joint Work with Pierre Bonami and Jon Lee) Dedicated to Prof. EgonBalas TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAA

  2. MIQCP Integer Constrained Variables Symmetric Matrices NOT necessarily positive semidefinite Anureet Saxena, Axioma Inc.

  3. MIQCP Anureet Saxena, Axioma Inc.

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

  5. 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, Axioma Inc.

  6. Disjunctive Programming CGLP Anureet Saxena, Axioma Inc.

  7. Disjunctive Programming Polyhedral Relaxation Disjunction Outer Approximationof MIQCP defined by the incumbent solution Anureet Saxena, Axioma Inc.

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

  9. 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, Axioma Inc.

  10. 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, Axioma Inc.

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

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

  13. Negative Eigenvalues of Y-xxT Anureet Saxena, Axioma Inc.

  14. Positive Eigenvalues of Y-xxT Univariate non-convex expression Anureet Saxena, Axioma Inc.

  15. Positive Eigenvalues of Y-xxT Anureet Saxena, Axioma Inc.

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

  17. Positive Eigenvalues of Y-xxT Anureet Saxena, Axioma Inc.

  18. Positive Eigenvalues of Y-xxT Anureet Saxena, Axioma Inc.

  19. Cutting Plane Algorithm Convex Quadratic Cut Derive Disjunction CGLP Derive Disjunctive Cut Anureet Saxena, Axioma Inc.

  20. Sequential Convexification Y.ccT· (cTx)2 Can we improve the disjunctive cuts by choosing c more carefully?

  21. Improving Disjunctions? 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, Axioma Inc.

  22. Improving Disjunctions? 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, Axioma Inc.

  23. Cutting Plane Algorithm UGMIP Convex Quadratic Cut Derive Disjunction CGLP Derive Disjunctive Cut Anureet Saxena, Axioma Inc.

  24. MIQCP Reformulations MIQCP (x,Y) RLT + SDP Disjunctive Cuts Strengthening MIQCP (x,Y) Heavy Relaxation Projection Lifting B & B Light Relaxation Strengthening ? MIQCP (x) Projected Ineq MIQCP (x) Anureet Saxena, Axioma Inc.

  25. MIQCP Reformulations MIQCP (x,Y) RLT + SDP Disjunctive Cuts Strengthening MIQCP (x,Y) Convex Relaxations of Non-Convex Mixed Integer Quadratically Constrained Problems: Projected Formulations A. Saxena, P. Bonami and J. Lee Heavy Relaxation Projection Lifting B & B Light Relaxation Strengthening ? MIQCP (x) Projected Ineq MIQCP (x) Anureet Saxena, Axioma Inc.

  26. Projecting the RLT Formulation RLT Inequalities Anureet Saxena, Axioma Inc.

  27. Projecting the RLT Formulation Separation Problem Given x show that x2Qxor find an inequality which is satisfied by all points in Qxand is violated by x. Anureet Saxena, Axioma Inc.

  28. Projecting the RLT Formulation ProjLP Anureet Saxena, Axioma Inc

  29. Projecting the RLT Formulation Dual Solution (u, B, C) Projected Inequality Anureet Saxena, Axioma Inc

  30. Projecting the RLT Formulation • A linear programming separation algorithm • Handles large number O(n2) of RLT inequalities as bound constraints • No of constraints = No of quadratic constraints in the original problem Anureet Saxena, Axioma Inc

  31. Surrogate Constraints Surrogate Constraint Surrogate Constraint Can we extract the convex part of the surrogate constraint A = B – C B, C ¸ 0

  32. Surrogate Constraints Surrogate Constraint What happens if we add all such convex quadratic cuts? A = B + C – D B ¸SDP 0 C, D ¸ 0

  33. Projecting the SDP Formulation Dual Solution (u, B, C, D) ProjSDP Separation Problem is a SDP Anureet Saxena, Axioma Inc

  34. Projecting the SDP Formulation Unconstrained Convex Optimization Problem over the Cartessian product of a simplex and cone of PSD matrices Anureet Saxena, Axioma Inc

  35. Projecting the SDP Formulation • Projected Sub Gradient Heuristic • Initialize B = Projection of A to the cone of PSD matrices • Compute a sub gradient of F(u,B) at B • Perform line search along the sub gradient direction • Update B and goto 2 Anureet Saxena, Axioma Inc.

  36. Limitations of Projection Theorems Surrogate Constraint Once the surrogate constraint has been produced very little global information is used in the convexification process Anureet Saxena, Axioma Inc.

  37. Limitations of Projection Theorems • st_e23 instances from GlobalLib • OPT = -1.08 • RLT = -3 • SDP + RLT = -1.5 • x = ( 0.811, 0.689, -1.500) The non-convex quadratic constraint and the bound constraints cannot cut off x Anureet Saxena, Axioma Inc.

  38. Limitations of Projection Theorems • st_e23 instances from GlobalLib • OPT = -1.08 • RLT = -3 • SDP + RLT = -1.5 • x = ( 0.811, 0.689, -1.500) Global Information We need a technique for engaging additional constraints in the problem during the convexification process Anureet Saxena, Axioma Inc.

  39. Limitations of Projection Theorems Univariate non-convex expression Anureet Saxena, Axioma Inc.

  40. Limitations of Projection Theorems Anureet Saxena, Axioma Inc.

  41. Limitations of Projection Theorems Secant Approximation Anureet Saxena, Axioma Inc.

  42. Limitations of Projection Theorems Cuts off the incumbent solution Anureet Saxena, Axioma Inc.

  43. Eigen Reformulation Anureet Saxena, Axioma Inc.

  44. Eigen Reformulation Directions of maximal non-convexity Anureet Saxena, Axioma Inc.

  45. Eigen Reformulation Geometric correlations along directions of maximal non-convexity Anureet Saxena, Axioma Inc.

  46. Eigen Reformulation • st_glmp_kky instances from GlobalLib • OPT = -2.5 • RLT = RLT+SDP = -3.0 Can we exploit these correlations in deriving strong cutting planes? Projection along y1 and y2 Anureet Saxena, Axioma Inc.

  47. Polarity Cuts Projection Determine Extreme Points Lifting Convexification Anureet Saxena, Axioma Inc.

  48. Polarity Cuts Projection Determine Extreme Points Lifting Convexification Projection Anureet Saxena, Axioma Inc.

  49. Polarity Cuts Projection Determine Extreme Points Lifting Convexification Anureet Saxena, Axioma Inc.

  50. Polarity Cuts Projection Determine Extreme Points Lifting Convexification Anureet Saxena, Axioma Inc.

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