Developing a Taylor Model based global optimizer for COSY Infinity

1. Developing a Taylor Model based global optimizer for COSY Infinity Youn-Kyung Kim and Johannes Grote Taylor Model Workshop, Miami Beach, December 2003

2. Primary Goal: • Given a function f:X IR, which is representable as a finite computer code list, over an interval vector X as the domain, find rigorous enclosures of the minimum of the function • This should be done satisfying the requirements of accuracy and speed

3. A recap: the Moore-Skelboe algorithm • „Branch and Bound“-algorithm • Essentially: in each new step bisect the domain further and further, bound f over the new boxes using Interval arithmetic • Throw away boxes with a greater lower bound for the range than an existing upper bound for the range of another box

4. Method is guaranteed to work The resulting enclosure is fully rigorous Rather slow convergence to the desired accuracy How can the domain boxes be analyzed in a favorable order ? Sophisticated listing of new boxes ? Some pros and cons for these types of optimizers:

5. The new Taylor Model based optimizer: • Is still in the development stage, shall be made available to users in the near future • Branch-and-Bound optimizer, but using Taylor models (LDB, QDB, etc) for range bounding. • So far only unconstrained problems are treated. • Also use Taylor Models for other things

6. Other uses of Taylor Models under study: • Reduce boxes by more than ½ (LDB) • Use polynomial information for decision on splitting direction • Perform accelerators on local Taylor models instead of full function • Interval-Newton, preconditioning on Interval-Newton (Kearfott), higher order Interval-Newton

7. Description of the algorithm • Let f:X IR be a function on an interval box B which can be represented in the form of a Taylor Model • Evaluate the bound B of the range of f over X=X(0) using the Taylor Model P+I of f over this box for the range computation, and denote y(0):= B

8. The first step: • Bisect X(0) and obtain the boxes X(1,1) and X(1,2), generate Taylor Models of f over these boxes and obtain bounds B(1,1) and B(1,2) for the ranges. Remark 1: we bisect along the respective longest edges of the box Remark 2: sophisticated techniques for Taylor Model range bounding are available (e.g. LDB) • Obtain y(1,1):=B(1,1) and y(1,2):=B(1,2)

9. The n-th step: • Bisect the existing boxes along the longest edge, thus in the n-th step we generate 2^n boxes X(n,j), where 1 … j … 2^n • Obtain Taylor Model for f over each box, use bounding techniques to obtain bounds B(n,1),…,B(n,2^n), define y(n,j):=B(n,j)

10. Consequence: • The sequence y(n,j) of lower bounds of the minimum of f is monotonously increasing in n, the process is terminated after the accuracy requirement is satisfied,see below

11. Enclosing the minimum: • After each new box X(n,j) is generated and the range B(n,j) is computed, pick the midpoint m(n,j) of the box X(n,j), add a zero remainder bound and evaluate the function value of the objective function using this point interval. • f(m(n,j)) is in f(m(n,j)+[0,0])=[l(n,j),u(n,j)]

12. Enclosing the minimum II: • If we denote f* as the true minimum of f over X, then in any case we have : - u(n,j) >= f* - f* >= y(n,j) • So if u(n,j)-y(n,j) < epsilon (the prespecified accuracy), then an enclosure of f* is provided

13. Discarding boxes: • We want to be able to throw away boxes which cannot contain a minimum. • Recall the midpoint evaluation, for each midpoint m(n,j) an upper bound u(n,j) is computed. • Discard all boxes which have a lower bound of the range of f which is greater than u(n,j).

14. The Box Ordering • The boxes are arranged in a list. • Newly generated boxes are put to the end of the list. • This induces a „chronological order“, the first boxes in the list which have not yet been analyzed are the oldest and have „priority“ • No box is „forgotten“, i.e. it may contain the minimum but is never analyzed, which might happen if one uses a priority order depending on the minimum of the range computation.

15. Current deficiencies and future goals: • Only the enclosures of the minimum are computed, not at which point this minimum as actually assumed. • No acceleration techniques, e.g. gradient-based choice of box splitting, are implemented yet. • Only unconstrained problems can be treated so far. • There probably are better orderings of the boxes than the chronological one. • Suggestions and commentaries are greatly appreciated.