On the minimum of a conditioned Brownian bridge

TL;DR

This paper investigates the distribution of the minimum and its location in a conditioned Brownian bridge, providing insights useful for comparing optimization algorithms in black-box function scenarios with limited sampling.

Contribution

It characterizes the law of the minimum and its location in a conditioned Brownian bridge, aiding the evaluation of non-adaptive optimization algorithms.

Findings

Derived the distribution of the minimum of a conditioned Brownian bridge.

Analyzed the distribution of the location of the minimum.

Provided a framework for comparing optimization algorithms.

Abstract

We study the law of the minimum of a Brownian bridge, conditioned to take specific values at specific points, and the law of the location of the minimum. They are used to compare some non-adaptive optimisation algorithms for black-box functions for which the Brownian bridge is an appropriate probabilistic model and only a few points can be sampled.