Constructs the constraint coefficient arrays of a market impact term in the
compact form. The constraint is generated during the transformation of the
objective function.
This market impact term in the objective function takes this form:
\[
\sum_{j=1}^n\left(m_j |x_j|^{\frac{3}{2}}\right)\leq t_2.
\]
Let \(\bar{x}=|x|\), the market impact term is transformed into:
\[
\sum_{j=1}^n\left(
m_{j}\bar{x}_{j}^{\frac{3}{2}}\right)\leq t_2
\]
By introducing variables \(y=x+w^{0}\), \(\alpha\) and \(\beta\), the above
inequality is equivalent to:
\[
\sum\beta_{j}\leq t_{2}\\
|y_{j}-w_{j}^{0}|\leq\bar{x}_{j}\;j=1,\cdots,n\\
m_{j}\bar{x}_{j}^{\frac{3}{2}}\leq
\beta_{j}\;j=1,\cdots,n\\
\]
These constraints can be further transformed into:
\[
\sum_{j=1}^{n}\beta_{j}\leq t_{2}\\
y_{j}-w_{j}^{0}\leq\bar{x}_{j},\; j =1,\cdots, n\\
-y_{j}+w_{j}^{0}\leq \bar{x}_{j},\; j=1,\cdots,n\\
\bar{x}_{j}^{\frac{3}{2}}\leq \frac{\beta_{j}}{m_{j}},\; j=1,\cdots,n\\
\]
Therefore we have:
\[
||0||_{2}\leq t_{2}-\sum_{j=1}^{n}\beta_{j}\\
||0||_{2}\leq\bar{x}_{j}-(y_{j}-w_{j}^{0}),\; j =1,\cdots, n\\
||0||_{2}\leq \bar{x}_{j}-(-y_{j}+w_{j}^{0}),\; j=1,\cdots,n\\
\bar{x}_{j}^{\frac{3}{2}}\leq \frac{\beta_{j}}{m_{j}},\;j=1,\cdots,n.
\]
The last term, \(\bar{x}_{j}^{\frac{3}{2}}\leq \frac{\beta_{j}}{m_{j}}\), is
equivalent to:
\[
\bar{x}_{j}^{\frac{3}{2}}\leq \frac{\beta_{j}}{m_{j}},\;\bar{x}_{j}\geq 0\Longleftrightarrow
\bar{x}_{j}^2\leq s_{j} \frac{\beta_{j}}{m_{j}},\; s_{j}\leq
\sqrt{\bar{x}_{j}},\;\bar{x}_{j}\geq0,\;
s_{j}\geq 0\;, \frac{\beta_{j}}{m_{j}}\geq0 \Longleftrightarrow\\
\bar{x}_{j}^2\leq s_{j} \frac{\beta_{j}}{m_{j}},\; s_{j}^{2} \leq
\bar{x}_{j},\;\bar{x}_{j}\geq0,\;
s_{j}\geq 0\;, \frac{\beta_{j}}{m_{j}}\geq 0 \Longleftrightarrow\\
\bar{x}_{j}^2+\left(\frac{\beta_{j}}{2m_{j}}-\frac{s_{j}}{2}\right)^2\leq
\left(\frac{\beta_{j}}{2m_{j}}+\frac{s_{j}}{2}\right)^2,\;s_{j}^{2}+\left(\frac{1-\bar{x}_{j}}{2}\right)^2\leq
\left(\frac{1+\bar{x}_{j}}{2}\right)^2,\;\bar{x}_{j}\geq0,\;s_{j}\geq 0\;,
\frac{\beta_{j}}{m_{j}}\geq 0.
\]
Because \(\bar{x}_{j}\geq0,\;s_{j}\geq 0\;, \frac{\beta_{j}}{m_{j}}\geq 0\)
can be deduced from other constraints, they can be deleted from the system of
constraints. The constraints deduced from
\(\bar{x}_{j}^{\frac{3}{2}}\leq \frac{\beta_{j}}{m_{j}}\) can be written as:
\[
||\left(\begin{array}{c}\bar{x}_{j}\\\frac{\beta_{j}}{2m_{j}}-\frac{s_{j}}{2}\end{array}\right)||_{2}\leq\frac{\beta_{j}}{2m_{j}}+\frac{s_{j}}{2},j=1,\cdots,n,\\
||\left(\begin{array}{c}s_{j}\\\frac{1-\bar{x}_{j}}{2}\end{array}\right)||_{2}\leq\frac{1+\bar{x}_{j}}{2},j=1,\cdots,n.
\]
Combine all the constraints together, the system of constraints for market
impact is:
\[
||0||_{2}\leq t_{2}-\sum_{j=1}^{n}\beta_{j},\\
||0||_{2}\leq\bar{x}_{j}-(y_{j}-w_{j}^{0}), \;j =1,\cdots, n,\\
||0||_{2}\leq \bar{x}_{j}-(-y_{j}+w_{j}^{0}),\; j=1,\cdots,n,\\
||\left(\begin{array}{c}\bar{x}_{j}\\\frac{\beta_{j}}{m_{j}}-\frac{s_{j}}{2}\end{array}\right)||_{2}\leq\frac{\beta_{j}}{m_{j}}+\frac{s_{j}}{2},\;j=1,\cdots,n,\\
||\left(\begin{array}{c}s_{j}\\\frac{1-\bar{x}_{j}}{2}\end{array}\right)||_{2}\leq\frac{1+\bar{x}_{j}}{2},\;j=1,\cdots,n.
\]
The standard SOCP form of the constraints are:
\[
||0||_{2}\leq t_{2}-\sum_{j=1}^{n}\beta_{j} \Longleftrightarrow ||A_{1}^{\top}z+C_{1}||_{1}\leq
b^{\top}_{1}z+d_{1}\\
A_{1}^{\top}=0_{1\times n},\;
C_{1}=0,\;
b_{1}=\left(\begin{array}{c}-1_{n\times 1}\\1\end{array}\right),\;
d_{1}=0,\;
z=\left(\begin{array}{c}\beta\\t_{2}\end{array}\right).
\]
\[
||0||_{2}\leq\bar{x}_{j}-(y_{j}-w_{j}^{0}) \Longleftrightarrow
||A_{2,j}^{\top}z+C_{2,j}||_{2}\leq b^{\top}_{2,j}z+d_{2,j},\quad j=1,\cdots,n\\
A_{2,j}^{\top}=0_{1\times 2n},\;
C_{2,j}=0,\;
b_{2,j}=\left(\begin{array}{c}-e_{j}\\e_{j}\end{array}\right),\;
d_{2,j}=w_{j}^{0},\;
z=\left(\begin{array}{c}y\\\bar{x}\end{array}\right),
\]
where \(e_{j}\) is a \(n\) dimensional vector whose \(j\)-th entry is \(1\)
and all the other entries are \(0\).
\[
||0||_{2}\leq\bar{x}_{j}-(-y_{j}+w_{j}^{0}) \Longleftrightarrow
||A_{3,j}^{\top}z+C_{3,j}||_{2}\leq b^{\top}_{3,j}z+d_{3,j},\quad j=1,\cdots,n\\
A_{3,j}^{\top}=0_{1\times 2n},\;
C_{3,j}=0,\;
b_{3,j}=\left(\begin{array}{c}e_{j}\\e_{j}\end{array}\right),\;
d_{3,j}=-w_{j}^{0},\;
z=\left(\begin{array}{c}y\\\bar{x}\end{array}\right).
\]
\[
||\left(\begin{array}{c}\bar{x}_{j}\\\frac{\beta_{j}}{2m_{j}}-\frac{s_{j}}{2}\end{array}\right)||_{2}\leq\frac{\beta_{j}}{2m_{j}}+\frac{s_{j}}{2}\Longleftrightarrow
||A_{4,j}^{\top}z+C_{4,j}||_{2}\leq b^{\top}_{4,j}z+d_{4,j},\quad j=1,\cdots,n\\
A_{4,j}^{\top}=\left(\begin{array}{ccc}e_{j}^{\top}&0_{1\times n}&0_{1\times n}\\0_{1\times
n}&\frac{1}{2m_{j}}e_{j}^{\top}& -\frac{1}{2}e_{j}^{\top}\end{array}\right),\;
C_{4,j}=0_{2 \times 1},\;
b_{4,j}=\left(\begin{array}{c}0_{n\times 1}
\\\frac{1}{2m_{j}}e_{j}\\\frac{1}{2}e_{j}\end{array}\right),\;
d_{4,j}=0,\;
z=\left(\begin{array}{c}\bar{x}\\\beta_{j}\\s\end{array}\right).
\]
\[
||\left(\begin{array}{c}s_{j}\\\frac{1-\bar{x}_{j}}{2}\end{array}\right)||_{2}\leq\frac{1+\bar{x}_{j}}{2}\Longleftrightarrow
||A_{5,j}^{\top}z+C_{5,j}||_{2}\leq b^{\top}_{5,j}z+d_{5,j},\quad j=1,\cdots,n\\
A_{5,j}^{\top}=\left(\begin{array}{ccc}0_{1\times
n}&e_{j}^{\top}\\-\frac{1}{2}e_{j}^{\top}&0_{1\times n}\end{array}\right),\;
C_{5,j}=\left(\begin{array}{c}0\\\frac{1}{2}\end{array}\right),\;
b_{5,j}=\left(\begin{array}{c}\frac{1}{2}e_{j}\\0_{n\times 1}\end{array}\right),\;
d_{5,j}=\frac{1}{2},\;
z=\left(\begin{array}{c}\bar{x}\\s\end{array}\right).
\]