1. Heuristic Optimization and Dynamical System Safety Verification Todd W. Neller Knowledge Systems Laboratory Stanford University

2. 1 Outline Motivating Problem Heuristic Optimization Approach Comparative Study of Global Optimization Techniques Information-Based Optimization Recent Research Results

3. 2 Focus Global optimization techniques can be powerfully applied to a class of hybrid system verification problems. When each function evaluation of an optimization is costly, such information should be used intelligently in the course of optimization.

4. 3 Stepper Motor

5. 4 Stepper Motor Safety Verification Given: Bounds on stepper motor system parameters Bounds on initial conditions Verify: No stalls in all possible acceleration scenarios

6. 5 Heuristic Search Landscape Make use of simple knowledge of problem domain to provide landscape helpful to search

7. 6 Verification through Optimization Transform verification problem into an optimization problem with a heuristic measure of relative safety Apply efficient global optimization

8. 7 Comparative Testing Methods: Simulated Annealing: AMEBSA, ASA, SALO Multi Level Single Linkage (MLSL) and variants Random Local Optimization (RANDLO) Test Functions: From optimization literature and method demos Used to gain rough idea of relative strengths

9. 8 Comparative Study Results SALO and RANDLO generally best for functions with many and few minima respectively Local optimization “flattens” and simplifies these search spaces. Local optimization doesn’t always lead to nearest optimum. Minima rarely located at bounds of search space.

10. 9 Global Optimization Results

11. 10 Comparative Study Results (cont.) For test functions STEP1 and STEP2, RANDLO and LMLSL performed best for both constrained local optimization procedures. SALO: ASA did not search the locally optimized search spaces (f ´ ) efficiently. Recent experiments indicate that information-based global optimization performs even better.

12. 11 Global Optimization Results (cont.) CONSTRYURETMIN

13. 12 Information-Based Approach Information-Based Optimization - Previous function evaluations shape probability distribution over possible functions. Most methods waste costly information.

14. 13 Information-Based Local Optimization Choose initial point and search radius Iterate: Evaluate point in sphere where minimum most likely according to information gained thus far If less than initial point, make new point center

15. 14 Multi-Level Local Optimization Each layer of local optimization simplifies search space for the layer above. MLLO-RIQ : Perform random (Monte-Carlo) optimization of: f´´ : Information-based local optimization of:  f´ : Quasi-Newton local optimization of: –f : heuristic function

16. 15 MLLO Example: Rastrigin Function

17. 16 MLLO-RIQ Results For our first set of functions, MLLO-RIQ trial results are very encouraging Local optimization procedure not suited to discontinuous CMMR No startup cost as with MLSL or GA

18. 17 Other Work in Progress Global Information-Based Optimization Information-Based Direction-Set Methods Dynamic Search Tuning Future work: Parallel Information-Based Methods Expert System for Global Optimization Main challenge: Approximating optimal decision procedures

19. 18 Summary Heuristically use domain knowledge to transform an initial safety problem into a global optimization problem Information is costly  Use information well in the course of optimization with information-based approaches