In this paper, we devise new relaxations for composite functions, which improve the prevalent factorable relaxations, without introducing additional variables, by exploiting the inner-function structure. We outer-approximate inner-functions using arbitrary under- and over-estimators and then convexify the outer-function over a polytope P , which models the ordering relationships between the inner-functions and their estimators and utilizes bound information on the inner-functions as well as on the estimators. We show that there is a subset Q of P , with significantly simpler combinatorial structure, such that the separation problem of the graph of the outer-function over P is polynomially equivalent, via a fast combinatorial algorithm, to that of its graph over Q . We specialize our study to consider the product of two inner-functions with one non-trivial underestimator for each inner-function. For the corresponding polytope P , we show that there are eight valid inequalities besides the four McCormick inequalities, which improve the factorable relaxation. Finally, we show that our results generalize to simultaneous convexification of a vector of outer-functions.

中文翻译：

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一种在非线性程序中松弛复合函数的新框架

在本文中，我们为复合函数设计了新的松弛，通过利用内部函数结构，在不引入额外变量的情况下改进了普遍的可分解松弛。我们使用任意低估和高估量来外逼近内函数，然后在多面体 P 上凸化外函数，该多面体 P 对内函数与其估计量之间的排序关系进行建模，并利用内函数的边界信息作为以及在估计器上。我们证明了 P 的一个子集 Q ，具有明显更简单的组合结构，使得 P 上的外函数图的分离问题通过快速组合算法在多项式上等价于 Q 上的图的分离问题。我们专门研究了两个内部函数的乘积，每个内部函数都有一个非平凡的低估量。对于相应的多胞体 P ，我们表明除了四个 McCormick 不等式之外还有八个有效的不等式，它们改善了可分解松弛。最后，我们表明我们的结果可以推广到外部函数向量的同时凸化。