1. Inexact SQP Methods for Equality Constrained Optimization Frank Edward Curtis Department of IE/MS, Northwestern University with Richard Byrd and Jorge Nocedal November 6, 2006 INFORMS Annual Meeting 2006

2. Outline Introduction  Problem formulation  Motivation for inexactness  Unconstrained optimization and nonlinear equations Algorithm Development  Step computation  Step acceptance Global Analysis  Merit function and sufficient decrease  Satisfying first-order conditions Conclusions/Final remarks

3. Outline Introduction  Problem formulation  Motivation for inexactness  Unconstrained optimization and nonlinear equations Algorithm Development  Step computation  Step acceptance Global Analysis  Merit function and sufficient decrease  Satisfying first-order conditions Conclusions/Final remarks

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

5. Equality constrained optimization Algorithm: Newton’s methodAlgorithm: the SQP subproblem Two “equivalent” step computation techniques

6. Equality constrained optimization Algorithm: Newton’s methodAlgorithm: the SQP subproblem Two “equivalent” step computation techniques KKT matrix Cannot be formed Cannot be factored

7. Equality constrained optimization 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

8. 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)

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

10. Outline Introduction/Motivation  Unconstrained optimization  Nonlinear equations  Constrained optimization Algorithm Development  Step computation  Step acceptance Global Analysis  Merit function and sufficient decrease  Satisfying first-order conditions Conclusions/Final remarks

11. Equality constrained optimization Algorithm: Newton’s methodAlgorithm: the SQP subproblem Two “equivalent” step computation techniques Question: can we ensure convergence to a local solution by choosing any step into the ball?

12. Globalization strategy: exact merit function … with Armijo line search condition Globalization strategy Step computation: inexact SQP step

13. 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

14. 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

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

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

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

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

19. Exact case

20. Exact step minimizes the objective on the linearized constraints

21. Exact case Exact step minimizes the objective on the linearized constraints … which may lead to an increase in the objective (but that’s ok)

22. Inexact case

23. Option #1: current penalty parameter

24. Step is acceptable if for

25. Option #2: new penalty parameter

26. Step is acceptable if for

27. Option #2: new penalty parameter Step is acceptable if for

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

29. Observe KKT conditions Termination test

30. Outline Introduction/Motivation  Unconstrained optimization  Nonlinear equations  Constrained optimization Algorithm Development  Step computation  Step acceptance Global Analysis  Merit function and sufficient decrease  Satisfying first-order conditions Conclusions/Final remarks

31. The sequence of iterates is contained in a convex set over which the following hold:  the objective function is bounded below  the objective and constraint functions and their first and second derivatives are uniformly bounded in norm  the constraint Jacobian has full row rank and its smallest singular value is bounded below by a positive constant  the Hessian of the Lagrangian is positive definite with smallest eigenvalue bounded below by a positive constant Assumptions

32. Sufficient reduction to sufficient decrease Taylor expansion of merit function yields Accepted step satisfies

33. Intermediate results is bounded below by a positive constant is bounded above

34. Sufficient decrease in merit function

35. Step in dual space (for sufficiently small and ) Therefore, We converge to an optimal primal solution, and

36. Outline Introduction/Motivation  Unconstrained optimization  Nonlinear equations  Constrained optimization Algorithm Development  Step computation  Step acceptance Global Analysis  Merit function and sufficient decrease  Satisfying first-order conditions Conclusions/Final remarks

37. Conclusion/Final remarks Review  Defined a globally convergent inexact SQP algorithm  Require only inexact solutions of KKT 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 Future challenges  Implementation and appropriate parameter values  Nearly-singular constraint Jacobian  Inexact derivative information  Negative curvature  etc., etc., etc….