An Enhancement of the Bisection Method Average Performance Preserving Minmax Optimality

Abstract

We identify a class of root-searching methods that surprisingly outperform the bisection method on the average performance while retaining minmax optimality. The improvement on the average applies for any continuous distributional hypothesis. We also pinpoint one specific method within the class and show that under mild initial conditions it can attain an order of convergence of up to 1.618, i.e., the same as the secant method. Hence, we attain both an improved average performance and an improved order of convergence with no cost on the minmax optimality of the bisection method. Numerical experiments show that, on regular functions, the proposed method requires a number of function evaluations similar to current state-of-the-art methods, about 24% to 37% of the evaluations required by the bisection procedure. In problems with non-regular functions, the proposed method performs significantly better than the state-of-the-art, requiring on average 82% of the total evaluations required for the bisection method, while the other methods were outperformed by bisection. In the worst case, while current state-of-the-art commercial solvers required two to three times the number of function evaluations of bisection, our proposed method remained within the minmax bounds of the bisection method.

Efficient solvers for Armijo's backtracking problem

“…The simplest method belonging to the class proposed here, which is based on the traditional bisection algorithm for root-searching, completes the same task with at most ⌈log 2 log 𝛽 𝜖/𝑥 0 ⌉ function evaluations. The same upper bound is provided by another method based on a recent bracketing algorithm for root-searching [9], which requires asymptotically only ∼ log log log 𝜖/𝑥 0 function evaluations in the case of sufficiently smooth functions. Other root-searching bracketing algorithms can be adapted similarly for performing inexact line searches efficiently.…”

“…Proof. Follows immediately from the properties of the ITP method [9]. □ Corollary 3.2 makes use of standard assumptions on the smoothness of 𝑔, under which even faster convergence can be guaranteed.…”

“…This is shown in the following by making explicit the estimated number of iterations when a hybrid technique for the construction of x is used. The exact construction of x is a straightforward application of the ITP root-searching method, described in [9], on the logarithmic scale, and is omitted for brevity. Corollary 3.2.…”

“…The ITP parameters used were of 𝜅 1 = 0.1; 𝜅 2 = 2; and, a slack parameter of 𝑁 0 = 0.99 applied on the rescaled root-searching problem made to satisfy 𝑏 − 𝑎 ≤ 1 in order to benefit of the guarantees of[9].…”

Minmax-optimal list searching with $O(\log_2\log_2 n)$ average cost

“…This problem is ubiquitous in computer science with applications spanning several different fields of computer programming, engineering and mathematics. Variations of (1) include searching unbounded lists [1], tables [2], searching continuous functions for a zero [3], as well as the construction of insertions and deletion procedures in canonical data-structures [4].…”