New Optimality Conditions and Methods for State-Constrained Elliptic Optimal Control Problems

1. New Optimality Conditions and Methods for State-Constrained Elliptic Optimal Control Problems Building Bridges between ODE and PDE Optimal Control Michael Frey, Simon Bechmann, Hans Josef Pesch, Armin Rund Chair of Mathematics in Engineering Sciences University of Bayreuth, Germany Brose, Coburg, Germany University of Graz, Austria hans-josef.pesch@uni-bayreuth.de Torrey Pines State Park July 7, 2013

2. Outline • Introduction • Split method • Bryson-Denham-Dreyfus approach (BDD) • Shape calculus and optimization on vector bundles • Numerics

3. Outline • Introduction • Split method • Bryson-Denham-Dreyfus approach (BDD) • Shape calculus and optimization on vector bundles • Numerics

4. subject to with Elliptic optimal control problem with state constraints Minimize

5. Well-known first-order necessary conditions Theorem (Casas 1986): Slater condition such that low regularity causes problems in numerical treatment

6. Goals and ideas • Goals • new necessary conditions with higher regularity of Lagrange multipliers • formulate efficient numerical algorithms, which • don’t require any regularization technique • and exploit the structure of the multiplier • Ideas • geometric split set optimal control problem • BDD approach higher regularity • shape calculus necessary conditions • optimization on vector bundles design of algorithms

7. Outline • Introduction • Split method • Bryson-Denham-Dreyfus approach (BDD) • Shape calculus and optimization on vector bundles • Numerics

8. Definition of active setand assumptions Bergounioux, Kunisch, 2003 Definition: active / inactive set / interface Assumption on admissible active sets No degeneracy. No active set of zero measure. No common points with boundary.

9. Reformulation of the model problem (Analog to the multipoint-boundary-value-problem formulation) Minimize subject to

10. Reformulation as set optimal control problem Minimize subject to Theorem: The original problem and the set optimal control problem possess the same unique solution

11. outer Set optimal control problem as bilevel optimization problem Minimize inner subject to constraint of outer optimization Theorem:The inner optimization problem possesses a unique solution for any

12. subject to Set optimal control problem (shape-/topology-optimization) Consequences: existence of a geometry to solution operator Reduced functional: is well-defined on .

13. Theorem: For any there exist Lagrange multipliers associated with the equality constraints of the inner optimization problem Necessary conditions are sufficient Inner optimization problem is strictly convex Replace inner optimization problem by its necessary conditions Analysis of the set optimal control problem

14. Set optimal control problem subject to and no measures involved unusual boundary conditions everything else can be computed a posteriori

15. Theorem: For each admissible the objective is shape differentiable. The semi-derivative in the direction is determines the interface Shape calculus for the optimal active set subject to the optimality system of the inner optimization problem

16. Condition for the interface We omit in the outer optimization and determine the interface by Needs an a posteriori-check on feasability:

17. Outline • Introduction • Split method • Bryson-Denham-Dreyfus approach (BDD) • Shape calculus and optimization on vector bundles • Numerics * A.E. Bryson, Jr, W.F. Denham, S.E. Dreyfus: Optimal programming problems with inequality constraints I, AIAA Journal 1(11):2544-2550, 1963. Later extended by Maurer, 1979. *

18. Using the state equation Optimal solution on given by data, but optimization variable Reformulation of the state constraint Transfering the Bryson-Denham-Dreyfus approach

19. First-order necessary condition of set-OCP (split method) (traditional adjoint state and multipliers)

20. First-order necessary condition of set-OCP (BDD method) (new adjoint state and multpliers with higher regularity)

21. First-order necessary condition of set-OCP (BDD method) (new adjoint state and multpliers with higher regularity) BDD approach reveals control law, i.e. “hidden” condition (as for PDAE)

22. Outline • Introduction • Split method • Bryson-Denham-Dreyfus approach (BDD) • Shape calculus and optimization on vector bundles • Numerics

23. Basic considerations with respect to shape calculus Gateaux directional derivative Hadamard directional derivative

24. vector field image holdall perturbation of identity defines curves Basic considerations with respect to shape calculus Delfour, Zolésio, 2011 set (no linear structure, infinite dimensional manifold)

25. Results of basic considerations • Hadamard directional derivative is suitable for nonlinear spaces • Deformation of sets yields „perturbation of identity“ • Metric of function spaces induces metric in has no linear structure is similar to an infinite dimensional manifold • defines curves in • suitable difference quotient

26. Exemplary derivation of directional shape derivative Let implicit explicit set dependence explicit derivative implicit derivative

27. Important results • shape calculus is similar to calculus on manifolds intrinsic nonlinear behaviour • shape (directional) derivative concentrated on boundary and on the normal component of the vector field only

28. Vector bundles Lang, 1995 Choice of predefines the function space : What is the inherent structure of ? diffeomorphism metric of function spaces induces metric in

29. Optimization on vector bundles: shape optimization Typical shape optimization problem Minimize s.t. a BVP for on with Unique solvability implies Minimize s.t.

30. Optimization on vector bundles: set optimal control problem Set optimal control problem Minimize s.t. a BVP for on with Unique solvability implies Minimize s.t.

31. Optimization on vector bundles: set optimal control problem Set optimal control problem Minimize s.t. a BVP for on with Unique solvability implies Minimize s.t.

32. Outline • Introduction • Split method • Bryson-Denham-Dreyfus approach (BDD) • Shape calculus and optimization on vector bundles • Numerics

33. Analysis of necessary conditions Interface by a free BVP Linear PDAE A posterior check orby a nonlinear cond.

34. optimal radius 2nd critical point active set too small active set too big Basic considerations w.r.t. the algorithm • is no (local) minimum of the (unconstrained) , second semi-derivative is not definite at critical points. Hence, steepest decent algorithms not applicable, higher order methods are required. • Solve nonlinear eq. + PDAE by some Newton-type method preserve hierarchy bilevel OP, blockwise solve • Solve free PDAE via Newton iteration (total linearization) equal variables Lagrange approach (one loop) cf. Kari Kärkkainen (PhD, Jyväskylä, 2005) • Relevant questions How does a Newton method look like on manifolds? How to cope with changes in topology? analysis of reduced functional of an analytical example radius of initial guess

35. Towards Newton‘s method on manifolds • Initial guess • loop onstoping criterionNewton equationupdate • end of loop Hessian and gradient require Hilbert spaces

36. Towards Newton‘s method on manifolds • Initial guess • loop onstoping criterionNewton equationupdate • end of loop directional derivatives suitable for linear structure

37. apply chain rule: vector fields may not be constant constant vector fields there is no „constant“ vector field Second directional derivative / second covariant derivative Why are second derivatives more complicated? successive differentiation: term vanishes in linear spaces; does not contain 2nd order information on functional

38. Towards Newton‘s method on manifolds • Initial guess • loop onstoping criterionNewton equationupdate • end of loop sum requires linear structure

39. is a retraction with is zero in and Newton update by retraction Retraction:

40. Newton‘s method on manifolds • Initial guess • loop onstoping criterionNewton equationupdate • end of loop

41. provide initial guess (by formula on data: candidate active set) • loop onstoping criterionidentify (complicated formula)Newton equation: Findprovide retraction: deform (see next transparency) • end of loop • check a posteriori criteria and eventually restart with other initial guess due to strict complementarity: Newton‘s method for set optimal control problems

42. Newton method on manifold and topology changes Newton update by second covariant derivative and retraction online: self-intersection offline: violation of state constraint Michael Frey‘s dissertation, University of Bayreuth,2012: Shape calculus applied to state-constrained elliptic optimal control problems http://opus.ub.uni-bayreuth.de/opus4-ubbayreuth/frontdoor/index/index/docId/996

43. Basic properties of Newton method pro • formulation in infinite dimen. setting mesh independency • no regularization loops better performance than PDAS • feasible approximations of solutions • changes of topology heuristically contra • no convergence analysis • only local convergence (with adaptive smooting for stability) • assumptions on active set

44. The Smiley: construction algorithm can cope with topology changes Construction: Prescribe , choose small, press down . Initial guess: automatically from unconstrained problem I made it! Iter No. 1 3 2 4 5 6 7 8 9

45. The Smiley example: rational initial guess

46. The Smiley example: bad initial guess Algorithm can cope with topology changes to some extent

47. The Smiley example: bad initial guess Adjoint multiplier: continuous on interface, but normal derivatives jump

48. Conclusion 1 Split method • new problem type: set optimal control problem • bilevel formulation geometry-to-solution operator • split of constraint exploitation of structure of multiplier Extended BDD approach for PDEs • differentiation of state constraint control law • higher regularity of multiplier • connection with optimal control and PDAE: index reduction

49. Conclusion 2 Shape calculus • set of admissible active sets has manifold character and thus is intrinsic nonlinear • deformation of sets: perturbation of identity • calculation requires transformation formula Optimization on vector bundles • vector bundles: structure depends upon manifolds • general basis for shape optimization / set optimal control problems • new challenging class of optimization problems

50. Open questions • active sets of measure zero • generalization to parabolic problems and Outlook Conclusion 3 Newton method • several approaches towards solution of new necessary conditions • adaption of Newton method on manifolds • algorithm in function space without regularization • comparable performance to sophisticated PDAS • pays off in case of nonlinear elliptic optimal control problems