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.

全局优化中的分支定界 (B&B) 算法用于在可行区域上执行穷举搜索。一种选择是使用简单的分区集。在单纯形上获得目标函数的清晰且廉价的边界对于构建高效的全局优化 B&B 算法非常重要。尽管将单纯形包含在一个框中意味着高估，但在处理单个坐标边界时，框更自然，并且使用区间算术 (IA) 的边界范围在计算上很便宜。本文介绍了几种使用梯度信息和仿射算术的线性松弛，并通过实验研究了它们与通过自然和中心 IA 形式获得的传统下界相比的效率及其对单纯形的适应。全局优化 B& B 算法在单纯形上进行单调性测试，用于将它们在一组低维测试问题上的效率与具有框约束搜索区域或可行集是单纯形的实例进行比较。数值结果表明，可以在简单子集上获得严格的下界。