Convergence analysis of multivariate McCormick relaxations

Abstract

The convergence rate is analyzed for McCormick relaxations of compositions of the form $F \circ f$, where F is a multivariate function, as established by Tsoukalas and Mitsos (J Glob Optim 59:633–662, 2014). Convergence order in the Hausdorff metric and pointwise convergence order are analyzed. Similar to the convergence order propagation of McCormick univariate composition functions, Bompadre and Mitsos (J Glob Optim 52(1):1–28, 2012), the convergence order of the multivariate composition is determined by the minimum of the orders of the inclusion functions of the inner functions and the convergence order of the multivariate outer function. The convergence order in the Hausdorff metric additionally depends on the enclosure order of the image of the inner functions introduced in this work. The result established holds for any composition and can be further specialized for specific compositions. In some cases this specialization results in the bounds established by Bompadre and Mitsos. Examples of important functions, e.g., binary product of functions and minimum of functions show that the convergence rate of the relaxations based on multivariate composition theorem results in a higher convergence rate than the convergence rate of univariate McCormick relaxations. Refined bounds, employing also the range order, similar to those determined by Bompadre et al. (J Glob Optim 57(1):75–114, 2013), on the convergence order of McCormick relaxations of univariate and multivariate composite functions are developed.

Appendix

Proof of the equivalence of Definition 10 and the definition used in [7, 8, 23]:

Definition

(Hausdorff convergence) Let $Z \subset \mathbb {R}^n$ and $f:Z \rightarrow \mathbb {R}$ be a continuous function, and let $H_f$ be an inclusion function of f on Z. The inclusion function $H_f$ has Hausdorff convergence of order $\beta >0$ if there exists a constant $\tau >0$ such that, for any interval $Y \in \mathbb {I}Z$,

$$\begin{aligned} q(\bar{f}(Y), H_f(Y)) \le \tau w(Y)^{\beta }. \end{aligned}$$

Proof

Let $Z\subset \mathbb {R}^n$, $f:Z\rightarrow \mathbb {R}$, $H_f$ be its inclusion function, $Y\in \mathbb {I}Z$ and $\tau ,\beta >0$ be given as in Definition above with

$$\begin{aligned} q(\bar{f}(Y),H_{f}(Y))\le \tau w(Y)^\beta . \end{aligned}$$

Since $\bar{f}(Y) \subset H_f(Y)$, the equivalence directly follows from

$$\begin{aligned} w(H(Y))-w(\bar{f}(Y))\le 2q(\bar{f}(Y),H_{f}(Y))\le 2\tau w(Y)^\beta = \hat{\tau }w(Y)^\beta . \end{aligned}$$

$\square$