1. CS B553: A LGORITHMS FOR O PTIMIZATION AND L EARNING Global optimization 1

2. A GENDA : G LOBAL O PTIMIZATION Local search, optimization Branch and bound search Online search

3. 3 G LOBAL O PTIMIZATION min f( x ), x in S  R n Want guarantees on a global optimum f x1x1 x2x2 S

4. 4 E XHAUSTIVE S EARCH  S

5. 5 S

6. 6 L OWER - BOUND FUNCTION LowerBound(A): for any set A  S, returns a lower bound on f(x) for all x  A f x1x1 x2x2 S Af L (A)

7. 7 P RUNING THE S EARCH T REE 1 4 Values of LowerBound(A) 2 2.53 f(x) = 3.2

8. 8 B RANCH - AND - BOUND A LGORITHM Let f* be the best value seen so far Init: f* = f(point in S 0 ) Q = {S 0 } While Q not empty, repeat: S = remove an item from Q If LowerBound(S)  f* or |S|< , then discard S f* = min(f*,f(point in S)) Split S and add subregions to FRINGE Pruning step What order?

9. 9 P ERFORMANCE Works well when When LowerBound is relatively tight When n isn’t too large Methods for generating LowerBound Interval arithmetic Solving “relaxed” versions of f Problem-specific ways

10. E XAMPLE : 3D N EEDLE S TEERING F EEDBACK C ONTROLLER Closed-loop feedback rule: 1.Sense current position/orientation of needle 2.Twist at the speed such that the predicted helix path minimizes the distance to target

11. 11 C ONSTANT - TWIST - RATE HELICES

12. 12 R EACHABLE POINTS UNDER CONSTANT - TWIST - RATES

13. 13 F INDING CLOSEST POINT Use cylindrical lower bound Find small initial domain BnB takes < 1ms on average

14. C OLLISION C HECKING 14  Check whether objects overlap

15. H IERARCHICAL C OLLISION C HECKING 15  Enclose objects into bounding volumes (spheres or boxes)  Check the bounding volumes

16. H IERARCHICAL C OLLISION C HECKING 16  Enclose objects into bounding volumes (spheres or boxes)  Check the bounding volumes first  Decompose an object into two

17. H IERARCHICAL C OLLISION C HECKING 17  Enclose objects into bounding volumes (spheres or boxes)  Check the bounding volumes first  Decompose an object into two  Proceed hierarchically

18. H IERARCHICAL C OLLISION C HECKING 18  Enclose objects into bounding volumes (spheres or boxes)  Check the bounding volumes first  Decompose an object into two  Proceed hierarchically

19. B OUNDING V OLUME H IERARCHY (BVH) 19 A BVH (~ balanced binary tree) is pre-computed for each object (obstacle, robot link)

20. BVH OF A 3D T RIANGULATED C AT 20

21. C OLLISION C HECKING B ETWEEN T WO O BJECTS 21 BVH of object 1 A B C A B C BVH of object 2 [Usually, the two trees have different sizes]  Search for a collision

22. S EARCH FOR A C OLLISION 22 A Search tree A A pruning

23. S EARCH FOR A C OLLISION 23 A Search tree A A Heuristic: Break the largest BV

24. S EARCH FOR A C OLLISION 24 A CACABABA Search tree A B C Heuristic: Break the largest BV

25. S EARCH FOR A C OLLISION 25 A CACABABA Search tree C CCBCB B C

26. S EARCH FOR A C OLLISION 26 A CACABABA Search tree C CCBCB B C B If two leaves of the BVH’s overlap (here, C and B) check their content for collision

27. S EARCH S TRATEGY If there is no collision, all paths must eventually be followed down to pruning or a leaf node But if there is collision, one may try to detect it as quickly as possible  Greedy best-first search strategy with f(N) = h(N) = d/(r X +r Y ) [Expand the node XY with largest relative overlap (most likely to contain a collision)] 27 rXrX rYrY d X Y

28. P ERFORMANCE On average, over 10,000 collision checks per second for two 3-D objects each described by 500,000 triangles, on a contemporary PC Checks are much faster when the objects are either neatly separated (  early pruning) or neatly overlapping (  quick detection of collision) 28

29. R EVIEW OF O PTIMIZATION U NIT Descent vs root finding Gradient descent, Newton’s method, Quasi- newton methods Constraints: Lagrange multipliers and KKT conditions Convex optimization LP, QP Interior point methods Metaheuristic methods

30. P UTTING O PTIMIZATION INTO P RACTICE Being able to formulate optimization problems is often just as important as choosing the right algorithm Well conditioned functions Constraints Auxiliary variables Transformations Analysis vs. computation

31. W HAT ELSE IS OUT THERE ? Special techniques for sparse problems Special problem formulations e.g., SDP, SOCP Mixed-integer programming Semi-infinite programming In-depth analysis suitable for “industrial strength” optimization

32. M ODERN T RENDS Learning with big data Combinatorial / hybrid problems Online or distributed optimization Specific function classes: Stochastic functions Robot/biomechanical motions Feedback control policies

33. R EADINGS FOR N EXT C LASS K&F, Chapter 2