这是一份 Imperial帝国理工大学 MATH97117作业代写的成功案例
Consider a compact $\Omega$ with the origin $x=0$ belonging to it; since $f$ is $\mathscr{C}^{2}$, each $z_{j}$ can be bounded in $\Omega$ by two affine functions:
$$
z_{j}^{0}(x) \leq z_{j}(x) \leq z_{j}^{1}(x)
$$
where
$$
z_{j}^{1}(x)=a_{1}^{j} v_{j}^{T} x+b_{1}^{j}, \quad z_{j}^{0}(x)=a_{0}^{j} v_{j}^{T} x+b_{0}^{j},
$$
being $a_{i}^{j}, b_{i}^{j}$ scalars, and $v_{j}^{T}$ row vectors which constitute arbitrarily tight linear bounds on $z_{j}(\boldsymbol{x})$. With available, we have
$$
z_{j}(x)=\sum_{i=0}^{1} w_{i}^{j}(x)\left(a_{i}^{j} v_{j}^{T} x+b_{i}^{j}\right),
$$
with weights given by the well-known interpolation expression:
$$
w_{0}^{j}\left(z_{j}\right):=\frac{z_{j}^{1}(\boldsymbol{x})-z_{j}(\boldsymbol{x})}{z_{j}^{1}(\boldsymbol{x})-z_{j}^{0}(\boldsymbol{x})}, \quad w_{1}^{j}\left(z_{j}\right):=1-w_{0}^{j}\left(z_{j}\right) .
$$
MATH97117 COURSE NOTES ：
$$
\sin x_{3}=x_{3}-\frac{x_{3}^{3}}{3 !}+\frac{x_{3}^{5}}{5 !}-\frac{x_{3}^{7}}{7 !}+\cdots
$$
If only the first-degree term is used to rewrite $\sin x_{3}$ as a convex expression, it yields the same outcome as if sector nonlinearity was used, i.e. the terms
$$
\xi_{1}^{1}\left(z_{1}\right)=0, \quad T_{1}^{1}\left(z_{1}\right)=\frac{\sin z_{1}-0}{z_{1}},
$$
where $N=1, z_{1}=x_{3}$, allow bounding $T_{1}^{1}\left(z_{1}\right)$ by $\psi_{1}^{1}=1$ and $\psi_{1}^{2}=0.8415$ in $\left|x_{3}\right| \leq$ 1. Thus, the corresponding weights are defined as follows:
$$
w_{0}^{1}\left(z_{1}\right)=\frac{T_{1}^{1}\left(z_{1}\right)-0.8415}{1-0.8415}, \quad w_{1}^{1}\left(z_{1}\right)=1-w_{0}^{1}\left(z_{1}\right) .
$$