In this paper we explore the construction of arbitrarily tight α BB relaxations of C 2 general non-linear non-convex functions. We illustrate the theoretical challenges of building such relaxations by deriving conditions under which it is possible for an α BB underestimator to provide exact bounds. We subsequently propose a methodology to build α BB underestimators which may be arbitrarily tight (i.e., the maximum separation distance between the original function and its underestimator is arbitrarily close to 0) in some domains that do not include the global solution (defined in the text as "sub-optimal"), assuming exact eigenvalue calculations are possible. This is achieved using a transformation of the original function into a μ -subenergy function and the derivation of α BB underestimators for the new function. We prove that this transformation results in a number of desirable bounding properties in certain domains. These theoretical results are validated in computational test cases where approximations of the tightest possible μ -subenergy underestimators, derived using sampling, are compared to similarly derived approximations of the tightest possible classical α BB underestimators. Our tests show that μ -subenergy underestimators produce much tighter bounds, and succeed in fathoming nodes which are impossible to fathom using classical α BB.

中文翻译：

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次优域上一般非线性函数的任意紧 α BB 低估量。

在本文中，我们探讨了 C 2 一般非线性非凸函数的任意紧 α BB 松弛的构造。我们通过推导 α BB 低估量可能提供精确界限的条件来说明建立这种松弛的理论挑战。我们随后提出了一种方法来构建 α BB 低估量，该方法在一些不包括全局解决方案的域中可能任意紧（即原始函数与其低估量之间的最大分离距离任意接近 0）作为“次优”），假设精确的特征值计算是可能的。这是通过将原始函数转换为 μ 亚能函数并推导新函数的 α BB 低估量来实现的。我们证明了这种转换会在某些域中产生许多理想的边界属性。这些理论结果在计算测试用例中得到验证，在计算测试用例中，使用采样导出的最严格的 μ 亚能低估量的近似值与最严格的可能经典 α BB 低估量的类似导出的近似值进行了比较。我们的测试表明，μ-亚能量低估产生了更严格的界限，并成功地测透了使用经典 α BB 无法测透的节点。与最严格的可能的经典 α BB 低估量的类似推导的近似值进行比较。我们的测试表明，μ-亚能量低估产生了更严格的界限，并成功地测透了使用经典 α BB 无法测透的节点。与最严格的可能的经典 α BB 低估量的类似推导的近似值进行比较。我们的测试表明，μ-亚能量低估产生了更严格的界限，并成功地测透了使用经典 α BB 无法测透的节点。