Testing Unconstrained Optimization Software
ACM Transactions on Mathematical Software (TOMS)
| Hi-index | 0.00 |
Taylor models provide enclosures of functional dependencies by a polynomial and an interval remainder bound that scales with a high power of the domain width, allowing a far-reaching suppression of the dependency problem. For the application to range bounding, one observes that the resulting polynomials are more well-behaved than the original function; in fact, merely naively evaluating them in interval arithmetic leads to a quadratic range bounder that is frequently noticeably superior to other second order methods. However, the particular polynomial form allows the use of other techniques. We review the linear dominated bounder (LDB) and the quadratic fast bounder (QFB). LDB often allows an exact bounding of the polynomial part if the function is monotonic. If it does not succeed to provide an optimal bound, it still often provides a reduction of the domain simultaneously in all variables. Near interior minimizers, where the quadratic part of the local Taylor model is positive semidefinite, QFB minimizes the quadratic contribution to the lower bound of the function, avoiding the infamous cluster effect for validated global optimization tasks. Some examples of the performance of the bounders for unconstrained global optimization problems are given, beginning with various common toy problems of the community, and also including a rather challenging Lennard-Jones problem.