Convergence-order analysis for differential-inequalities-based bounds and relaxations of the solutions of ODEs

Abstract

For the performance of global optimization algorithms, the rate of convergence of convex relaxations to the objective and constraint functions is critical. We extend results from Bompadre and Mitsos (J Glob Optim 52(1):1–28, 2012) to characterize the convergence rate of parametric bounds and relaxations of the solutions of ordinary differential equations (ODEs). Such bounds and relaxations are used for global dynamic optimization and are computed using auxiliary ODE systems that use interval arithmetic and McCormick relaxations. Two ODE relaxation methods (Scott et al. in Optim Control Appl Methods 34(2):145–163, 2013; Scott and Barton in J Glob Optim 57:143–176, 2013) are shown to give second-order convergence, yet they can behave very differently from each other in practice. As time progresses, the prefactor in the convergence-order bound tends to grow much more slowly for one of these methods, and can even decrease over time, yielding global optimization procedures that require significantly less computation time.

A Supporting lemmas and proofs

1.1 A.1 Proof of Lemma 2.2

Proof

If $\mu = 0$, the result is trivial. Now consider $\mu \ne 0$. Define

$$\begin{aligned} v(s) \equiv \exp (\mu (t_0 - s)) \int _{t_0}^s \mu x(r) {\mathrm {d}}r,\quad \forall s \in I. \end{aligned}$$

(A.1)

Differentiating gives

$$\begin{aligned} v'(s) = \mu \exp (\mu (t_0-s))\left( x(s) - \int _{t_0}^s \mu x(r) {\mathrm {d}}r \right) , \quad \forall s \in I. \end{aligned}$$

Since $\mu \ne 0$,

$$\begin{aligned} \frac{v'(s)}{\mu } = \underbrace{\exp (\mu (t_0-s))}_{\ge 0} \underbrace{\left( x(s) - \int _{t_0}^s \mu x(r) {\mathrm {d}}r\right) }_{\le \lambda _0 + \lambda _1(s-t_0)}, \quad \forall s \in I, \end{aligned}$$

where the bound on the second term comes from (2.1). Therefore,

$$\begin{aligned} \frac{v'(s)}{\mu } \le \exp (\mu (t_0-s))(\lambda _0 + \lambda _1(s-t_0)),\quad \forall s \in I. \end{aligned}$$

Note that $v(t_0) = 0$ and integrate:

$$\begin{aligned} \frac{v(t)}{\mu }&= \int _{t_0}^t \frac{v'(s)}{\mu } {\mathrm {d}}s \le \int _{t_0}^t \exp (\mu (t_0-s))(\lambda _0 + \lambda _1(s-t_0)) {\mathrm {d}}s,\\&= \frac{\lambda _0 \mu + \lambda _1 - \exp (\mu (t_0 - t))(\lambda _0 \mu + \lambda _1 + \lambda _1 \mu (t-t_0))}{\mu ^2},\quad \forall t \in I. \end{aligned}$$

Substitute in the definition for v from (A.1):

$$\begin{aligned} \exp (\mu (t_0 - t)) \int _{t_0}^t x(s) {\mathrm {d}}s \le \frac{\lambda _0 \mu + \lambda _1 - \exp (\mu (t_0 - t))(\lambda _0 \mu + \lambda _1 + \lambda _1 \mu (t-t_0))}{\mu ^2},\quad \forall t \in I. \end{aligned}$$

Multiply by $\exp (\mu (t-t_0))$:

$$\begin{aligned} \int _{t_0}^t x(s) {\mathrm {d}}s \le \frac{(\lambda _0 \mu + \lambda _1)\exp (\mu (t-t_0)) - (\lambda _0 \mu + \lambda _1 + \lambda _1 \mu (t-t_0))}{\mu ^2},\quad \forall t \in I. \end{aligned}$$

Differentiate the above inequality to obtain

$$\begin{aligned} x(t)&\le \left( \lambda _0 + \frac{\lambda _1}{\mu }\right) \exp (\mu (t-t_0))-\frac{\lambda _1}{\mu },\quad \forall t \in I. \end{aligned}$$

$\square$

1.2 A.2 Proof of Theorem 4.14

The following argument follows a similar line of reasoning to [47, Corollary 3.3.6] and [47, Proof of Theorem 3.3.2], but is sufficiently different that we prove it in full.

Proof

Fix any ${\widehat{P}}\in {\mathbb {I}P}$. Since solutions to (4.13) and (4.14) are assumed to exist,

$$\begin{aligned}&{{\mathbf {v}}}(t,{\widehat{P}}), {{\mathbf {w}}}(t,{\widehat{P}}), \widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}), \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}}) \in D,\quad \forall t\in I. \end{aligned}$$

Since, in addition, the intervals in Hypotheses 1 and 2 of Theorem 4.14 are proper,

$$\begin{aligned}&{{\mathbf {v}}}(t,{\widehat{P}}) \le {{\mathbf {w}}}(t,{\widehat{P}})\quad \text {and} \quad \widetilde{{{\mathbf {v}}}}(t,{\widehat{P}})\le \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}}), \quad \forall t \in I. \end{aligned}$$

We need to prove that $\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}) \le {{\mathbf {v}}}(t,{\widehat{P}})$ and ${{\mathbf {w}}}(t,{\widehat{P}}) \le \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}})$, $\forall t \in I$, which holds at $t_0$ by Hypothesis 1. Suppose (to arrive at a contradiction) $\exists t \in I$ such that either $v_i(t,{\widehat{P}}) < \widetilde{v}_i(t,{\widehat{P}})$ or $w_i(t,{\widehat{P}}) > \widetilde{w}_i(t,{\widehat{P}})$ for at least one $i \in \{1,\ldots ,n_x\}$ and define

$$\begin{aligned} t_1 \equiv \inf \{t \in I: v_i(t,{\widehat{P}}) < \widetilde{v}_i(t,{\widehat{P}}) \text { or } w_i(t,{\widehat{P}}) > \widetilde{w}_i(t,{\widehat{P}}) \text {, for at least one } i\}. \end{aligned}$$

Applying [47, Lemma 3.3.5] with the definition $\varvec{\delta }(t) \equiv (\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}) - {{\mathbf {v}}}(t,{\widehat{P}}), {{\mathbf {w}}}(t,{\widehat{P}}) - \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}}))$, we obtain the following fact:

• For any $t_4 \in (t_1, t_f]$, $\varepsilon > 0$, and $\beta \in \mathbb {R}_+$, there exists $j \in \{1, \ldots , n_x\}$, an absolutely continuous and non-decreasing function $\rho :[t_1,t_4]\rightarrow \mathbb {R}$, and numbers $t_2, t_3 \in [t_1, t_4]$ with $t_2 < t_3$ such that

  $$\begin{aligned} 0 < \rho (t)&\le \varepsilon ,\quad \forall t \in [t_1,t_4]\quad \text {and} \quad \dot{\rho }(t) > \beta \rho (t),\quad \text {a.e.}\ t\in [t_1,t_4], \end{aligned}$$

  (A.2)
  $$\begin{aligned} \widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}) - {\mathbf {1}}\rho (t)&< {{\mathbf {v}}}(t,{\widehat{P}}) \quad \text {and} \quad {{\mathbf {w}}}(t,{\widehat{P}}) < \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}}) + {\mathbf {1}}\rho (t),\quad \forall t \in [t_2, t_3), \end{aligned}$$

  (A.3)

  where ${\mathbf {1}}$ is a vector with all components equal to 1, and

  $$\begin{aligned}&v_{j}(t_2,{\widehat{P}}) = \widetilde{v}_{j}(t_2,{\widehat{P}}),\quad v_{j}(t_3,{\widehat{P}}) = \widetilde{v}_{j}(t_3,{\widehat{P}}) - \rho (t_3), \nonumber \\&\quad \text {and}\quad v_{j}(t,{\widehat{P}}) < \widetilde{v}_{j}(t,{\widehat{P}}), \quad \forall t \in (t_2,t_3) \end{aligned}$$

  (A.4)
  $$\begin{aligned}&\Big (\text {or}\quad w_{j}(t_2,{\widehat{P}}) = \widetilde{w}_{j}(t_2,{\widehat{P}}),\quad w_{j}(t_3,{\widehat{P}}) = \widetilde{w}_{j}(t_3,{\widehat{P}}) + \rho (t_3), \nonumber \\&\quad \text {and}\quad w_{j}(t,{\widehat{P}}) > \widetilde{w}_{j}(t,{\widehat{P}}), \quad \forall t \in (t_2,t_3) \Big ). \end{aligned}$$

  (A.5)

To apply this fact, we choose $\varepsilon >0$ small enough that

$$\begin{aligned} K \equiv [\widetilde{{{\mathbf {v}}}}(t_1,{\widehat{P}}),\widetilde{{{\mathbf {w}}}}(t_1,{\widehat{P}})]+2\varepsilon [-{\mathbf {1}},{\mathbf {1}}] \subset D, \end{aligned}$$

(A.6)

which is possible because D is open and $[\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}), \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}})] \subset D$, $\forall t \in I$, by the existence of a solution to (4.14). Next, we choose $\beta =L$, where L is the larger of the two Lipschitz constants for $\widetilde{{{\mathbf {u}}}}$ and $\widetilde{{{\mathbf {o}}}}$ on $I \times \mathbb {I}K\times \mathbb {I}{\widehat{P}}$. Such an L exists by Hypothesis 4 (see also Remark 4.2). Finally, we choose $t_4$ sufficiently small that

$$\begin{aligned} {[}\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}), \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}})] \subset [\widetilde{{{\mathbf {v}}}}(t_1,{\widehat{P}}),\widetilde{{{\mathbf {w}}}}(t_1,{\widehat{P}})]+\varepsilon [- {\mathbf {1}},{\mathbf {1}}],\quad \forall t \in (t_1,t_4]. \end{aligned}$$

(A.7)

Now, suppose (A.4) holds (the proof is analogous if instead (A.5) holds). We know from (A.3) that

$$\begin{aligned} {[}{{\mathbf {v}}}(t,{\widehat{P}}), {{\mathbf {w}}}(t,{\widehat{P}})] \subset [\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}})-\rho (t){\mathbf {1}}, \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}})+\rho (t){\mathbf {1}}] ,\quad \forall t \in [t_2, t_3).\end{aligned}$$

By (4.17), (4.18), and the inclusion above, we have

$$\begin{aligned} u_j(t, [{{\mathbf {v}}}(t,{\widehat{P}}), {{\mathbf {w}}}(t,{\widehat{P}})], {\widehat{P}})&\ge \widetilde{u}_j(t, [{{\mathbf {v}}}(t,{\widehat{P}}), {{\mathbf {w}}}(t,{\widehat{P}})], {\widehat{P}}), \nonumber \\&\ge \widetilde{u}_j(t, [\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}) - \rho (t){\mathbf {1}}, \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}}) + \rho (t){\mathbf {1}}], {\widehat{P}}),\quad \mathrm {a.e.}\ t \in [t_2,t_3). \end{aligned}$$

(A.8)

Above, $[\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}) - \rho (t){\mathbf {1}}, \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}}) + \rho (t){\mathbf {1}}]$ is guaranteed to be a subset of K, and hence of D, by (A.2), (A.7), and (A.6). Thus, Lipschitz continuity of $\widetilde{u}_j$ on $I\times {\mathbb {I}}K\times {\mathbb {I}}{\widehat{P}}$ gives

$$\begin{aligned} \widetilde{u}_j(t, [\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}) - \rho (t){\mathbf {1}}, \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}}) + \rho (t){\mathbf {1}}], {\widehat{P}}) \ge \widetilde{u}_j(t, [\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}), \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}})], {\widehat{P}}) - L \rho (t). \end{aligned}$$

(A.9)

Combining (A.8) and (A.9),

$$\begin{aligned} u_j(t, [{{\mathbf {v}}}(t,{\widehat{P}}), {{\mathbf {w}}}(t,{\widehat{P}})], {\widehat{P}}) \ge \widetilde{u}_j(t, [\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}), \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}})], {\widehat{P}}) - L \rho (t),\quad \text {a.e.}\ t \in [t_2, t_3]. \end{aligned}$$

Adding $\dot{\rho }(t)$ to both sides,

$$\begin{aligned} u_j(t, [{{\mathbf {v}}}(t,{\widehat{P}}), {{\mathbf {w}}}(t,{\widehat{P}})], {\widehat{P}}) + \dot{\rho }(t)&\ge \widetilde{u}_j(t, [\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}), \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}})], {\widehat{P}}) - L \rho (t) + \dot{\rho }(t),\\&> \widetilde{u}_j(t, [\widetilde{{{\mathbf {v}}}}(t,{\widehat{P}}), \widetilde{{{\mathbf {w}}}}(t,{\widehat{P}})], {\widehat{P}}),\quad \text {a.e.}\ t \in [t_2, t_3], \end{aligned}$$

where the second inequality follows from (A.2) with $\beta = L$. By [47, Theorem 3.3.3], this implies that $(\widetilde{v}_j(\cdot ,{\widehat{P}}) - v_j(\cdot ,{\widehat{P}}) - \rho )$ is non-increasing on $[t_2, t_3]$, so that

$$\begin{aligned} \widetilde{v}_j(t_3,{\widehat{P}}) - v_j(t_3,{\widehat{P}}) - \rho (t_3) \le \widetilde{v}_j(t_2,{\widehat{P}}) - v_j(t_2,{\widehat{P}}) - \rho (t_2). \end{aligned}$$

But, by (A.4), this implies that $0 \le - \rho (t_2)$, which contradicts (A.2). $\square$

1.3 A.3 (1,2)-Convergence of natural McCormick extensions

In this section, we establish the (1, 2)-convergence of natural McCormick extensions [47, Definition 2.4.31] as required by Assumption 5.17. To begin, we show that (1, 2)-convergence is composable.

Theorem A.1

Let $D_x\subset \mathbb {R}^{n}$ and let $\mathcal {F}:\mathbb {M}D_x \rightarrow \mathbb {MR}^m$ have the form $\mathcal {F}(\mathcal {X})=(F^B(X^B),F^C(\mathcal {X}))$, $\forall \mathcal {X}=(X^B,X^C)\in \mathbb {M}D_x$. Moreover, let $F^B(X^B)\subset D_y\subset \mathbb {R}^m$ for all $X^B\in \mathbb {I}D_x$, and let $\mathcal {G}:\mathbb {M}D_y \rightarrow \mathbb {MR}^q$ have the form $\mathcal {G}(\mathcal {Y})=(G^B(Y^B),G^C(\mathcal {Y}))$. If $\mathcal {F}$ and $\mathcal {G}$ have (1, 2)-convergence on $\mathbb {M}D_x$ and $\mathbb {M}D_y$, respectively, then $\mathcal {G}\circ \mathcal {F}$ has (1, 2)-convergence on $\mathbb {M}D_x$.

Proof

Applying (1, 2)-convergence for $\mathcal {G}$ and $\mathcal {F}$ sequentially, we obtain for any $\mathcal {X}\in \mathbb {M}D_x$,

$$\begin{aligned} w(G^B\circ F^B(X^B))&\le \tau ^G_{BB}w(F^B(X^B)) \end{aligned}$$

(A.10)

$$\begin{aligned}&\le \tau ^G_{BB}\tau ^F_{BB}w(X^B), \end{aligned}$$

(A.11)

$$\begin{aligned} w(G^C\circ \mathcal {F}(\mathcal {X}))&\le \tau ^G_{CC}w(F^C(\mathcal {X}))+\tau ^G_{CB}w(F^B(X^B))^{2}, \end{aligned}$$

(A.12)

$$\begin{aligned}&\le \tau ^G_{CC}(\tau ^F_{CC}w(X^C)+\tau ^F_{CB}w(X^B)^2)+\tau ^G_{CB}(\tau ^F_{BB}w(X^B))^{2}, \end{aligned}$$

(A.13)

$$\begin{aligned}&\le \tau ^G_{CC}\tau ^F_{CC}w(X^C)+(\tau ^G_{CC}\tau ^F_{CB}+\tau ^G_{CB}(\tau ^F_{BB})^2)w(X^B)^{2}. \end{aligned}$$

(A.14)

$\square$

Recall that a function is factorable if it can be written as a finite recursive sequence of basic operations including addition, multiplication, and composition with univariate functions from a standard library (e.g., $\frac{1}{x}$, $e^x$, $x^n$, etc.) Natural McCormick extensions are constructed for such functions by recursively applying simple relaxation rules for each basic operation in this sequence (see [47] for a precise definition). Specifically, these rules are relaxation functions for the basic operations. In light of Theorem A.1, it follows that, in order to establish (1, 2)-convergence of natural McCormick extensions, it suffices to establish (1, 2)-convergence of the relaxation functions for addition, multiplication, and univariate composition.

1.3.1 A.3.1 Addition

Definition A.2

McCormick addition $+:\mathbb {MR}\times \mathbb {MR}\rightarrow \mathbb {MR}$ is defined by

$$\begin{aligned} +(\mathcal {X},\mathcal {Y})&= \mathcal {X}+\mathcal {Y} = (X^B+Y^B,(X^B\cap X^C)+(Y^B\cap Y^C)). \end{aligned}$$

(A.15)

Lemma A.3

McCormick addition has (1,2)-convergence on $\mathbb {MR}\times \mathbb {MR}$.

Proof

Choose any $(\mathcal {X},\mathcal {Y})\in \mathbb {MR}\times \mathbb {MR}$ and let $\mathcal {Z}=+(\mathcal {X},\mathcal {Y})$. Then,

$$\begin{aligned} w(Z^B)&= w(X^B)+w(Y^B) \le 2\max (w(X^B),w(Y^B)) =2w(X^B\times Y^B) \end{aligned}$$

(A.16)

and

$$\begin{aligned} w(Z^C) = w(X^B\cap X^C)+w(Y^B\cap Y^C)&\le w(X^C)+w(Y^C), \nonumber \\&\le 2 \max \left( w(X^C),w(Y^C)\right) , \nonumber \\&= 2w(X^C\times Y^C). \end{aligned}$$

(A.17)

$\square$

1.3.2 A.3.2 Multiplication

The following definition of McCormick’s multiplication rule is nonstandard but facilitates our convergence arguments. It is shown in [47, §2.4.2] that this definition is equivalent except that the $\mathrm {Cut}$ function is not applied to $\mathcal {X}$ and $\mathcal {Y}$ in McCormick’s original work [30].

Definition A.4

McCormick multiplication $\times :\mathbb {MR}\times \mathbb {MR}\rightarrow \mathbb {MR}$ is defined by

$$\begin{aligned} \times (\mathcal {X},\mathcal {Y})&= \mathcal {XY} = (X^BY^B,[z^{cv},z^{cc}]), \end{aligned}$$

where $X^BY^B$ denotes standard interval multiplication [33] and

$$\begin{aligned} z^{cv}&= \max \left( \left[ y^L\bar{X}^C+x^L\bar{Y}^C-x^Ly^L\right] ^L,\left[ y^U\bar{X}^C+x^U\bar{Y}^C-x^Uy^U\right] ^L\right) , \\ z^{cc}&= \min \left( \left[ y^L\bar{X}^C+x^U\bar{Y}^C-y^Lx^U\right] ^U,\left[ y^U\bar{X}^C+x^L\bar{Y}^C-y^Ux^L\right] ^U\right) . \end{aligned}$$

Above, $\bar{\mathcal {X}}=\mathrm {Cut}(\mathcal {X})$, $\bar{\mathcal {Y}}=\mathrm {Cut}(\mathcal {Y})$, and the notations $[\cdot ]^L$ and $[\cdot ]^U$ refer to the lower and upper bounds of the interval-valued quantity in brackets, respectively.

Lemma A.5

McCormick multiplication is (1,2)-convergent on $\mathbb {M}K$ for any compact $K\subset \mathbb {R}\times \mathbb {R}$.

Proof

Choose any compact $K\subset \mathbb {R}\times \mathbb {R}$. The existence of $\tau _{BB}\ge 0$ such that $w(X^BY^B)\le \tau _{BB}w(X^B\times Y^B)$ for all $(X^B,Y^B)\in {\mathbb {I}}K$ is well known [33]. Choose any $(\mathcal {X},\mathcal {Y}) \in \mathbb {M}K$ and let $\mathcal {Z}=\times (\mathcal {X},\mathcal {Y})$. We have $w(Z^C) = z^{cc}-z^{cv}$. There are four cases to consider. For the first case,

$$\begin{aligned} w(Z^C)&= \left[ y^L\bar{X}^C+x^U\bar{Y}^C-y^Lx^U\right] ^U-\left[ y^L\bar{X}^C+x^L\bar{Y}^C-x^Ly^L\right] ^L. \end{aligned}$$

Writing $r^U=w(R)+r^L$ for $R=\left[ y^L\bar{X}^C+x^U\bar{Y}^C-y^Lx^U\right]$ on the right,

$$\begin{aligned} w(Z^C)&= w\left( \left[ y^L\bar{X}^C+x^U\bar{Y}^C-y^Lx^U\right] \right) \\&\quad +\left[ y^L\bar{X}^C+x^U\bar{Y}^C-y^Lx^U\right] ^L-\left[ y^L\bar{X}^C+x^L\bar{Y}^C-x^Ly^L\right] ^L, \\&= w\left( \left[ y^L\bar{X}^C+x^U\bar{Y}^C\right] \right) \\&\quad +\left[ y^L\bar{X}^C\right] ^L+\left[ x^U\bar{Y}^C\right] ^L-y^Lx^U-\left[ y^L\bar{X}^C\right] ^L-\left[ x^L\bar{Y}^C\right] ^L+x^Ly^L, \\&\le |y^L|w(\bar{X}^C)+|x^U|w(\bar{Y}^C)+\left[ x^U\bar{Y}^C\right] ^L-y^Lx^U-\left[ x^L\bar{Y}^C\right] ^L+x^Ly^L, \\&= |y^L|w(\bar{X}^C)+|x^U|w(\bar{Y}^C)+\left[ x^U\bar{Y}^C-y^Lx^U\right] ^L-\left[ x^L\bar{Y}^C-x^Ly^L\right] ^L, \\&\le |y^L|w(X^C)+|x^U|w(Y^C)+\left[ x^U(\bar{Y}^C-y^L)\right] ^L-\left[ x^L(\bar{Y}^C-y^L)\right] ^L. \end{aligned}$$

Noting that $\bar{Y}^C\subset Y^B$, it follows that every element of $\bar{Y}^C-y^L$ is nonnegative and bounded above by $w(Y^B)$. Thus, $\exists q_1,q_2\in (\bar{Y}^C-y^L)$, both bounded between 0 and $w(Y^B)$, satisfying

$$\begin{aligned} w(Z^C)&\le |y^L|w(X^C)+|x^U|w(Y^C)+x^Uq_1-x^Lq_2, \\&= |y^L|w(X^C)+|x^U|w(Y^C)+(x^L+w(X^B))q_1-x^Lq_2, \\&= |y^L|w(X^C)+|x^U|w(Y^C)+x^L(q_1-q_2)+w(X^B)q_1, \\&\le |y^L|w(X^C)+|x^U|w(Y^C)+|x^L|w(\bar{Y}^C-y^L)+w(X^B)w(Y^B), \\&\le |y^L|w(X^C)+(|x^U|+|x^L|)w(Y^C)+w(X^B)w(Y^B), \\&\le (|y^L|+|x^U|+|x^L|)\max (w(X^C),w(Y^C))+\max (w(X^B)^2,w(Y^B)^2), \\&\le 3\max \{|x^L|,|x^U|,|y^L|,|y^U|\}w(X^C\times Y^C)+w(X^B\times Y^B)^2, \\&\le 3(\max _{x\in K}|x|)w(X^C\times Y^C)+w(X^B\times Y^B)^2. \end{aligned}$$

Similar arguments hold in the remaining three cases. $\square$

1.3.3 A.3.3 Univariate functions

Let $\mathcal {L}$ be a library of univariate functions $u:B\subset \mathbb {R}\rightarrow \mathbb {R}$ that are permissible in the definition of a factorable function. The construction of the natural McCormick extension requires the following assumption.

Assumption A.6

For every $u:B\subset \mathbb {R}\rightarrow \mathbb {R}$ in $\mathcal {L}$, the following objects are available:

• 1. An inclusion function $U^B:\mathbb {I}B\rightarrow \mathbb {IR}$.

• 2. A scheme of estimators $(u^{cv},u^{cc}):\mathbb {I}B\times B\rightarrow \mathbb {R}\times \mathbb {R}$.

• 3. Functions $x^{\min },x^{\max }:\mathbb {I}B\rightarrow \mathbb {R}$ such that $x^{\min }(X)$ and $x^{\max }(X)$ are a minimum of $u^{cv}(X,\cdot )$ and a maximum of $u^{cc}(X,\cdot )$ on X, respectively.

Definition A.7

For every $u:B\subset \mathbb {R}\rightarrow \mathbb {R}$ in $\mathcal {L}$, the McCormick univariate composition rule $\mathcal {U}:\mathbb {M}B\rightarrow \mathbb {MR}$ is defined by

$$\begin{aligned} \mathcal {U}(\mathcal {X}) = \big (U^B(X^B),\big [u^{cv}(X^B,&\,\mathrm {mid}(x^{cv},x^{cc},x^{\min }(X^B))), u^{cc}(X^B,\mathrm {mid}(x^{cv},x^{cc},x^{\max }(X^B)))\big ]\big ), \end{aligned}$$

where the $\mathrm {mid}$ function selects the middle value of its arguments.

Note that $\mathcal {X}\in \mathbb {M}B$ implies that either $x^{cv}\in X^B$ or $x^{cc}\in X^B$, or both. By definition, $x^{\min }(X^B)$ and $x^{\max }(X^B)$ are both also in $X^B$, so that, in both uses of the $\mathrm {mid}$ function above, at least two of the three arguments lie in $X^B$. Thus, $\mathrm {mid}$ chooses an element of $X^B$, and hence of B, so that $\mathcal {U}(\mathcal {X})$ is defined.

The following assumptions are required to establish (1, 2)-convergence of $\mathcal {U}$ and are standard in the convergence literature (see Theorem 8 in [4]).

Assumption A.8

For every $u:B\subset \mathbb {R}\rightarrow \mathbb {R}$ in $\mathcal {L}$ and any compact $K\subset B$:

• 1. u is Lipschitz continuous on K.

• 2. The inclusion function $U^B$ converges in diameter in K with order at least one.

• 3. The scheme of estimators $(u^{cv},u^{cc})$ converges pointwise in K with order at least 2.

Lemma A.9

$\mathcal {U}:\mathbb {M}B\rightarrow \mathbb {MR}$ is (1,2)-convergent on $\mathbb {M}K$ for any compact $K\subset B$.

Proof

Choose any compact $K\subset B$. By Assumption A.8, there exists $\tau _{BB}\ge 0$ such that $w(U^B(X^B))\le \tau _{BB}w(X^B)$, $\forall X^B\in \mathbb {I}K$. Let $L\in \mathbb {R}_+$ be a Lipschitz constant for u on K and let $\tau \in \mathbb {R}_+$ be the pointwise convergence-order prefactor for $\mathcal {U}$ in K as per Assumption A.8. Choose any $\mathcal {X}\in \mathbb {M}K$. Since both of the points $\mathrm {mid}(x^{cv},x^{cc},x^{\min }(X^B)))$ and $\mathrm {mid}(x^{cv},x^{cc},x^{\max }(X^B)))$ are in $X^C$, it follows that

$$\begin{aligned} |\mathrm {mid}(x^{cv},x^{cc},x^{\max }(X^B))-\mathrm {mid}(x^{cv},x^{cc},x^{\min }(X^B))|\le w(X^C). \end{aligned}$$

(A.18)

Then,

$$\begin{aligned} w(U^C(\mathcal {X}))&= |u^{cc}(X^B,\mathrm {mid}(x^{cv},x^{cc},x^{\max }(X^B))) - u^{cv}(X^B,\mathrm {mid}(x^{cv},x^{cc},x^{\min }(X^B)))|, \nonumber \\&\le |u^{cc}(X^B,\mathrm {mid}(x^{cv},x^{cc},x^{\max }(X^B))) - u(\mathrm {mid}(x^{cv},x^{cc},x^{\max }(X^B)))| \nonumber \\&\quad +|u(\mathrm {mid}(x^{cv},x^{cc},x^{\max }(X^B)))-u(\mathrm {mid}(x^{cv},x^{cc},x^{\min }(X^B)))| \nonumber \\&\quad +|u(X^B,\mathrm {mid}(x^{cv},x^{cc},x^{\min }(X^B)))-u^{cv}(X^B,\mathrm {mid}(x^{cv},x^{cc},x^{\min }(X^B)))|, \nonumber \\&\le \tau w(X^B)^2 +Lw(X^C) +\tau w(X^B)^2. \end{aligned}$$

(A.19)

$\square$

Combining the results of Appendix A.3.1–A.3.3 with the Composition Theorem A.1, it is straightforward to show that the natural McCormick extension $\mathcal {F}:\mathbb {MD}\rightarrow \mathbb {MR}^m$ of a factorable function ${{\mathbf {f}}}:D\rightarrow \mathbb {R}^m$ has (1, 2)-convergence on any compact $K\subset D$, as required by Assumption 5.17.