On new methods to construct lower bounds in simplicial branch and bound based on interval arithmetic

B G.-Tóth,E M T Hendrix,F Messine,L G Casado

doi:10.1007/s10898-021-01053-8

Abstract

Branch and Bound (B&B) algorithms in Global Optimization are used to perform an exhaustive search over the feasible area. One choice is to use simplicial partition sets. Obtaining sharp and cheap bounds of the objective function over a simplex is very important in the construction of efficient Global Optimization B&B algorithms. Although enclosing a simplex in a box implies an overestimation, boxes are more natural when dealing with individual coordinate bounds, and bounding ranges with Interval Arithmetic (IA) is computationally cheap. This paper introduces several linear relaxations using gradient information and Affine Arithmetic and experimentally studies their efficiency compared to traditional lower bounds obtained by natural and centered IA forms and their adaption to simplices. A Global Optimization B&B algorithm with monotonicity test over a simplex is used to compare their efficiency over a set of low dimensional test problems with instances that either have a box constrained search region or where the feasible set is a simplex. Numerical results show that it is possible to obtain tight lower bounds over simplicial subsets.

Highlights

A review of simplicial Branch and Bound (B&B) can be found in [15]
If the search region is a simplex, P contains just the initial simplex, and all initial facets are at the boundary, such that all vertices are labelled border
In case the search region is a box, P contains the result of the combinatorial vertex triangulation of the box into n! simplices [24,26]

Summary

Introduction

A review of simplicial Branch and Bound (B&B) can be found in [15]. Recently, there is a renewed interest in generating tight bounds over simplicial partition sets. In [12], focus is on using second derivative enclosures for generating bounds. These works do not take monotonicity considerations over the simplex into account as discussed by [6]. Our research question is how information on the bounds of first derivatives can be used to derive tight bounds and to create new monotonicity tests in simplicial B&B. To investigate this question, we derive bounds based on derivative information and implement them in a B&B algorithm to compare the different techniques

Methods

Results

Conclusion