Reference alternatives based knockout-tournament procedure for ranking and selection

The knockout-tournament ($\mathrm{KT}$) procedure is an efficient parallel procedure recently developed to solve large-scale ranking and selection (R&S) problems. The procedure adopts a selection structure which is commonly used in many sports tournaments, and eliminates alternatives by conducting “matches” between paired alternatives round-by-round. In this paper, to further improve the procedure’s performance in solving large-scale problems, we propose a major modification of the procedure. Specifically, in each round of the selection, before pairing the surviving alternatives and conducting the matches, we first choose an alternative as the reference alternative and then add the reference alternative to each match. We call the new procedure Procedure $i$-$\mathrm{KT}$, where $i$-$\mathrm{KT}$ stands for “improved knockout-tournament”. We show that by carefully choosing the reference alternative and designing the pairing scheme for the remaining surviving alternatives in each round of the selection, Procedure $i$-$\mathrm{KT}$ can achieve significant improvements on both the average sample size required in each match and the total number of matches required during the entire selection process. In the meantime, we demonstrate that after the modifications, Procedure $i$-$\mathrm{KT}$ still fits parallel computing environments well. We compare Procedure $i$-$\mathrm{KT}$ with various procedures on different test examples and numerically justify our theoretical analysis.