The computation of lower bounds via the solution of convex lower bounding problems depicts current state-of-the-art in deterministic global optimization. Typically, the nonlinear convex relaxations are further underestimated through linearizations of the convex underestimators at one or several points resulting in a lower bounding linear optimization problem. The selection of linearization points substantially affects the tightness of the lower bounding linear problem. Established methods for the computation of such linearization points, e.g., the sandwich algorithm, are already available for the auxiliary variable method used in state-of-the-art deterministic global optimization solvers. In contrast, no such methods have been proposed for the (multivariate) McCormick relaxations. The difficulty of determining a good set of linearization points for the McCormick technique lies in the fact that no auxiliary variables are introduced and thus, the linearization points have to be determined in the space of original optimization variables. We propose algorithms for the computation of linearization points for convex relaxations constructed via the (multivariate) McCormick theorems. We discuss alternative approaches based on an adaptation of Kelley’s algorithm; computation of all vertices of an n-simplex; a combination of the two; and random selection. All algorithms provide substantial speed ups when compared to the single point strategy used in our previous works. Moreover, we provide first results on the hybridization of the auxiliary variable method with the McCormick technique benefiting from the presented linearization strategies resulting in additional computational advantages.

通过凸下界问题的解决方案计算下界描述了确定性全局优化中的最新技术。通常，通过在一个或几个点上对凸低估量进行线性化来进一步低估非线性凸弛豫，从而导致较低边界线性优化问题。线性化点的选择实质上会影响下界线性问题的紧密度。用于此类线性化点的计算的既定方法（例如，三明治算法）已可用于最新的确定性全局优化求解器中的辅助变量方法。相比之下，还没有针对（多元）McCormick松弛提出这样的方法。为McCormick技术确定一组好的线性化点的困难在于，没有引入辅助变量，因此必须在原始优化变量的空间中确定线性化点。我们提出了通过（多变量）McCormick定理构造的凸松弛线性化点的计算算法。我们讨论了基于Kelley算法的替代方法。计算一个顶点的所有顶点 我们讨论了基于Kelley算法的替代方法。计算一个顶点的所有顶点 我们讨论了基于Kelley算法的替代方法。计算一个顶点的所有顶点n-简单 两者的结合；和随机选择。与我们以前的工作中使用的单点策略相比，所有算法都可以大幅提高速度。此外，我们提供了辅助变量方法与McCormick技术混合的第一个结果，这得益于所提出的线性化策略，从而带来了额外的计算优势。