1. Frank Edward Curtis Northwestern University Joint work with Richard Byrd and Jorge Nocedal February 12, 2007 Inexact Methods for PDE-Constrained Optimization Emory University

2. Nonlinear Optimization “One” problem

3. Circuit Tuning Building blocks:  Transistors (switches) and Gates (logic units) Improve aspects of the circuit – speed, area, power – by choosing transistor widths AT1 AT3 AT2 d1 d2 w1w2 (A. Wächter, C. Visweswariah, and A. R. Conn, 2005)

4. Circuit Tuning Building blocks:  Transistors (switches) and Gates (logic units) Improve aspects of the circuit – speed, area, power – by choosing transistor widths Formulate an optimization problem AT1 AT3 AT2 d1 d2 w1w2 (A. Wächter, C. Visweswariah, and A. R. Conn, 2005)

5. Strategic Bidding in Electricity Markets Independent operator collects bids and sets production schedule and “spot price” to minimize cost to consumers (Pereira, Granville, Dix, and Barroso, 2004)

6. Strategic Bidding in Electricity Markets Independent operator collects bids and sets production schedule and “spot price” to minimize cost to consumers (Pereira, Granville, Dix, and Barroso, 2004) Electricity production companies “bid” on how much they will charge for one unit of electricity

7. Strategic Bidding in Electricity Markets Independent operator collects bids and sets production schedule and “spot price” to minimize cost to consumers Bilevel problem Equivalent to MPCC Hard geometry! (Pereira, Granville, Dix, and Barroso, 2004) Electricity production companies “bid” on how much they will charge for one unit of electricity

8. Challenges for NLP algorithms Very large problems Numerical noise Availability of derivatives Degeneracies Difficult geometries Expensive function evaluations Real-time solutions needed Integer variables Negative curvature

9. Outline Problem Formulation  Equality constrained optimization  Sequential Quadratic Programming Inexact Framework  Unconstrained optimization and nonlinear equations  Stopping conditions for linear solver Global Behavior  Merit function and sufficient decrease  Satisfying first order conditions Numerical Results  Model inverse problem  Accuracy tradeoffs Final Remarks  Future work  Negative curvature

10. Outline Problem Formulation  Equality constrained optimization  Sequential Quadratic Programming Inexact Framework  Unconstrained optimization and nonlinear equations  Stopping conditions for linear solver Global Behavior  Merit function and sufficient decrease  Satisfying first order conditions Numerical Results  Model inverse problem  Accuracy tradeoffs Final Remarks  Future work  Negative curvature

11. Equality constrained optimization e.g., minimize the difference between observed and expected behavior, subject to atmospheric flow equations (Navier-Stokes) Goal: solve the problem

12. Equality constrained optimization Define: the Lagrangian Define: the derivatives Goal: solve KKT conditions

13. Sequential Quadratic Programming (SQP) Algorithm: Newton’s methodAlgorithm: the SQP subproblem Two “equivalent” step computation techniques

14. Sequential Quadratic Programming (SQP) Algorithm: Newton’s methodAlgorithm: the SQP subproblem Two “equivalent” step computation techniques KKT matrix Cannot be formed Cannot be factored

15. Sequential Quadratic Programming (SQP) Algorithm: Newton’s methodAlgorithm: the SQP subproblem Two “equivalent” step computation techniques KKT matrix Cannot be formed Cannot be factored Linear system solve Iterative method Inexactness

16. Outline Problem Formulation  Equality constrained optimization  Sequential Quadratic Programming Inexact Framework  Unconstrained optimization and nonlinear equations  Stopping conditions for linear solver Global Behavior  Merit function and sufficient decrease  Satisfying first order conditions Numerical Results  Model inverse problem  Accuracy tradeoffs Final Remarks  Future work  Negative curvature

17. Unconstrained optimization Goal: minimize a nonlinear objective Algorithm: Newton’s method (CG)

18. Unconstrained optimization Goal: minimize a nonlinear objective Algorithm: Newton’s method (CG) Note: choosing any intermediate step ensures global convergence to a local solution of NLP (Steihaug, 1983)

19. Nonlinear equations Goal: solve a nonlinear system Algorithm: Newton’s method

20. any step with and ensures descent Nonlinear equations Goal: solve a nonlinear system Algorithm: Newton’s method (Eisenstat and Walker, 1994) (Dembo, Eisenstat, and Steihaug, 1982)

21. Line Search SQP Framework Define “exact” penalty function

22. Line Search SQP Framework Define “exact” penalty function

23. for k = 0, 1, 2, …  Compute step by…  Set penalty parameter to ensure descent on…  Perform backtracking line search to satisfy…  Update iterate Algorithm Outline (exact steps)

24. Exact Case

25. Exact step minimizes the objective on the linearized constraints

26. Exact Case Exact step minimizes the objective on the linearized constraints … which may lead to an increase in the model objective

27. Quadratic/linear model of merit function Create model Quantify reduction obtained from step

28. Quadratic/linear model of merit function Create model Quantify reduction obtained from step

29. Exact Case Exact step minimizes the objective on the linearized constraints … which may lead to an increase in the model objective

30. Exact Case Exact step minimizes the objective on the linearized constraints … which may lead to an increase in the model objective … but this is ok since we can account for this conflict by increasing the penalty parameter

31. Exact Case Exact step minimizes the objective on the linearized constraints … which may lead to an increase in the model objective … but this is ok since we can account for this conflict by increasing the penalty parameter

32. for k = 0, 1, 2, …  Compute step by…  Set penalty parameter to ensure descent on…  Perform backtracking line search to satisfy…  Update iterate Algorithm Outline (exact steps)

33. First attempt Proposition: sufficiently small residual 1e-81e-71e-61e-51e-41e-31e-21e-1 Success100% 97% 90%85%72%38% Failure0% 3% 10%15%28%62% Test: 61 problems from CUTEr test set

34. First attempt… not robust Proposition: sufficiently small residual … not enough for complete robustness  We have multiple goals (feasibility and optimality)  Lagrange multipliers may be completely off … may not have descent!

35. Recall the line search condition Second attempt Step computation: inexact SQP step We can show

36. Recall the line search condition Second attempt Step computation: inexact SQP step We can show... but how negative should this be?

37. for k = 0, 1, 2, …  Compute step  Set penalty parameter to ensure descent  Perform backtracking line search  Update iterate Algorithm Outline (exact steps)

38. for k = 0, 1, 2, …  Compute step and set penalty parameter to ensure descent and a stable algorithm  Perform backtracking line search  Update iterate Algorithm Outline (inexact steps)

39. Inexact Case

41. Step is acceptable if for

42. Inexact Case Step is acceptable if for

43. Inexact Case Step is acceptable if for

44. for k = 0, 1, 2, …  Iteratively solve  Until  Update penalty parameter  Perform backtracking line search  Update iterate Algorithm Outline or

45. Observe KKT conditions Termination Test

46. Outline Problem Formulation  Equality constrained optimization  Sequential Quadratic Programming Inexact Framework  Unconstrained optimization and nonlinear equations  Stopping conditions for linear solver Global Behavior  Merit function and sufficient decrease  Satisfying first order conditions Numerical Results  Model inverse problem  Accuracy tradeoffs Final Remarks  Future work  Negative curvature

47. The sequence of iterates is contained in a convex set and the following conditions hold:  the objective and constraint functions and their first and second derivatives are bounded  the multiplier estimates are bounded  the constraint Jacobians have full row rank and their smallest singular values are bounded below by a positive constant  the Hessian of the Lagrangian is positive definite with smallest eigenvalue bounded below by a positive constant Assumptions

48. Sufficient Reduction to Sufficient Decrease Taylor expansion of merit function yields Accepted step satisfies

49. Intermediate Results is bounded below by a positive constant is bounded above

50. Sufficient Decrease in Merit Function

51. Step in Dual Space (for sufficiently small and ) Therefore, We converge to an optimal primal solution, and

52. Outline Problem Formulation  Equality constrained optimization  Sequential Quadratic Programming Inexact Framework  Unconstrained optimization and nonlinear equations  Stopping conditions for linear solver Global Behavior  Merit function and sufficient decrease  Satisfying first order conditions Numerical Results  Model inverse problem  Accuracy tradeoffs Final Remarks  Future work  Negative curvature

53. Problem Formulation Tikhonov-style regularized inverse problem  Want to solve for a reasonably large mesh size  Want to solve for small regularization parameter SymQMR for linear system solves Input parameters: orRecall: (Curtis and Haber, 2007)

54. Numerical Results Iters.TimeTotal LS Iters. Avg. LS Iters. Avg. Rel. Res. 0.52929.5s145250.13.12e-1 0.11211.37s65454.56.90e-2 0.01911.60s68175.76.27e-3 n1024 m512 1e-6 (Curtis and Haber, 2007)

55. Numerical Results Iters.TimeTotal LS Iters. Avg. LS Iters. Avg. Rel. Res. 0.52929.5s145250.13.12e-1 0.11211.37s65454.56.90e-2 0.01911.60s68175.76.27e-3 n1024 m512 1e-6 (Curtis and Haber, 2007)

56. Numerical Results Iters.TimeTotal LS Iters. Avg. LS Iters. Avg. Rel. Res. 1e-61211.40s65454.56.90e-2 1e-71114.52s84076.46.99e-2 1e-8810.57s63979.96.15e-2 1e-91118.52s11391048.65e-2 1e-101944.41s27081438.90e-2 n1024 m512 1e-1 (Curtis and Haber, 2007)

57. Numerical Results Iters.TimeTotal LS Iters. Avg. LS Iters. Avg. Rel. Res. 1e-615264.47s19921338.13e-2 1e-711236.51s17761616.89e-2 1e-89204.51s15671746.77e-2 1e-911347.66s26812448.29e-2 1e-1016805.14s62493918.93e-2 n8192 m4096 1e-1 (Curtis and Haber, 2007)

58. Numerical Results Iters.TimeTotal LS Iters. Avg. LS Iters. Avg. Rel. Res. 1e-6155055.9s43652918.46e-2 1e-7104202.6s36303638.87e-2 1e-8125686.2s48254027.96e-2 1e-9126678.7s56334698.77e-2 1e-101414783s125258958.63e-2 n65536 m32768 1e-1 (Curtis and Haber, 2007)

59. Outline Problem Formulation  Equality constrained optimization  Sequential Quadratic Programming Inexact Framework  Unconstrained optimization and nonlinear equations  Stopping conditions for linear solver Global Behavior  Merit function and sufficient decrease  Satisfying first order conditions Numerical Results  Model inverse problem  Accuracy tradeoffs Final Remarks  Future work  Negative curvature

60. Review and Future Challenges Review  Defined a globally convergent inexact SQP algorithm  Require only inexact solutions of primal-dual system  Require only matrix-vector products involving objective and constraint function derivatives  Results also apply when only reduced Hessian of Lagrangian is assumed to be positive definite  Numerical experience on model problem is promising Future challenges  (Nearly) Singular constraint Jacobians  Inexact derivative information  Negative curvature  etc., etc., etc….

61. Negative Curvature Big question  What is the best way to handle negative curvature (i.e., when the reduced Hessian may be indefinite)? Small question  What is the best way to handle negative curvature in the context of our inexact SQP algorithm?  We have no inertia information! Smaller question  When can we handle negative curvature in the context of our inexact SQP algorithm with NO algorithmic modifications?  When do we know that a given step is OK?  Our analysis of the inexact case leads to a few observations…

62. Why Quadratic Models?

63. Provides a good… direction? Yes step length? Yes Provides a good… direction? Maybe step length? Maybe

64. Why Quadratic Models? One can use our stopping criteria as a mechanism for determining which are good directions All that needs to be determined is whether the step lengths are acceptable

65. Unconstrained Optimization Direct method is the angle test Indirect method is to check the conditions or

66. Unconstrained Optimization Direct method is the angle test Indirect method is to check the conditions or step qualitystep length

67. Constrained Optimization Step quality determined by Step length determined by or

68. Thanks!

69. Actual Stopping Criteria Stopping conditions: Model reduction condition or

70. Constraint Feasible Case If feasible, conditions reduce to

71. Constraint Feasible Case If feasible, conditions reduce to

72. Constraint Feasible Case If feasible, conditions reduce to Some region around the exact solution

73. Constraint Feasible Case If feasible, conditions reduce to Ellipse distorted toward the linearized constraints

74. Constraint Feasible Case If feasible, conditions reduce to