Formatted Formula View

The Amplify SDK formulas are automatically rendered in LaTeX in IPython front-end environments that support LaTeX formula display, such as Jupyter Notebook and Visual Studio Code.

Polynomials, polynomial arrays, coefficient matrices, constraints, and models are displayed as mathematical expressions as follows.

Variable array

import amplify

gen = amplify.VariableGenerator()
q = gen.array("Binary", shape=(4, 4))
q

\[\begin{split}\displaystyle \begin{aligned}&\left[\begin{matrix}q_{0,0}& q_{0,1}& q_{0,2}& q_{0,3}\\q_{1,0}& q_{1,1}& q_{1,2}& q_{1,3}\\q_{2,0}& q_{2,1}& q_{2,2}& q_{2,3}\\q_{3,0}& q_{3,1}& q_{3,2}& q_{3,3}\end{matrix}\right]\end{aligned}\end{split}\]

Polynomial

p = 2 * (q[0] * q[1]).sum()
p

\[\displaystyle 2 q_{0,0} q_{1,0} + 2 q_{0,1} q_{1,1} + 2 q_{0,2} q_{1,2} + 2 q_{0,3} q_{1,3}\]

Polynomial array

q[0:2] + q[2:]

\[\begin{split}\displaystyle \begin{aligned}&\left[\begin{matrix}q_{0,0} + q_{2,0}& q_{0,1} + q_{2,1}& q_{0,2} + q_{2,2}& q_{0,3} + q_{2,3}\\q_{1,0} + q_{3,0}& q_{1,1} + q_{3,1}& q_{1,2} + q_{3,2}& q_{1,3} + q_{3,3}\end{matrix}\right]\end{aligned}\end{split}\]

Constraint

c1 = amplify.one_hot(q, axis=1)
c1

\[\begin{split}\displaystyle \begin{array}{l}q_{0,0} + q_{0,1} + q_{0,2} + q_{0,3} = 1\ (\text{weight}\colon\ 1)\\q_{1,0} + q_{1,1} + q_{1,2} + q_{1,3} = 1\ (\text{weight}\colon\ 1)\\q_{2,0} + q_{2,1} + q_{2,2} + q_{2,3} = 1\ (\text{weight}\colon\ 1)\\q_{3,0} + q_{3,1} + q_{3,2} + q_{3,3} = 1\ (\text{weight}\colon\ 1)\end{array}\end{split}\]

Coefficient matrix

import numpy as np

m = gen.matrix("Binary", 4)
m.quadratic = np.array([[1, 2, 3, 4],
                        [0, 5, 6, 7],
                        [0, 0, 8, 9],
                        [0, 0, 0, 10]])
m.linear = np.array([1, 2, 3, 4])
c2 = amplify.one_hot(m.variable_array)
m

\[\begin{split}\displaystyle \begin{array}{l}x^\top Qx + p^\top x + c\\\text{where:}\\\quad\begin{aligned}x=&\left[\begin{matrix}q'_0& q'_1& q'_2& q'_3\end{matrix}\right],\\Q=&\left[\begin{matrix} 1.& 2.& 3.& 4.\\ 0.& 5.& 6.& 7.\\ 0.& 0.& 8.& 9.\\ 0.& 0.& 0.& 10.\end{matrix}\right],\\p=&\left[\begin{matrix} 1.& 2.& 3.& 4.\end{matrix}\right],\\c=&\ 0\end{aligned}\end{array}\end{split}\]

Model

model = p + c1
model

\[\begin{split}\displaystyle \begin{array}{l}\text{minimize:}\\\quad 2 q_{0,0} q_{1,0} + 2 q_{0,1} q_{1,1} + 2 q_{0,2} q_{1,2} + 2 q_{0,3} q_{1,3}\\\text{subject to:}\\\quad \begin{array}{l}q_{0,0} + q_{0,1} + q_{0,2} + q_{0,3} = 1\ (\text{weight}\colon\ 1)\\q_{1,0} + q_{1,1} + q_{1,2} + q_{1,3} = 1\ (\text{weight}\colon\ 1)\\q_{2,0} + q_{2,1} + q_{2,2} + q_{2,3} = 1\ (\text{weight}\colon\ 1)\\q_{3,0} + q_{3,1} + q_{3,2} + q_{3,3} = 1\ (\text{weight}\colon\ 1)\end{array}\end{array}\end{split}\]

model = m + c2
model

\[\begin{split}\displaystyle \begin{array}{l}\text{minimize:}\\\quad \begin{array}{l}x^\top Qx + p^\top x + c\\\text{where:}\\\quad\begin{aligned}x=&\left[\begin{matrix}q'_0& q'_1& q'_2& q'_3\end{matrix}\right],\\Q=&\left[\begin{matrix} 1.& 2.& 3.& 4.\\ 0.& 5.& 6.& 7.\\ 0.& 0.& 8.& 9.\\ 0.& 0.& 0.& 10.\end{matrix}\right],\\p=&\left[\begin{matrix} 1.& 2.& 3.& 4.\end{matrix}\right],\\c=&\ 0\end{aligned}\end{array}\\\text{subject to:}\\\quad \begin{array}{l}q'_0 + q'_1 + q'_2 + q'_3 = 1\ (\text{weight}\colon\ 1)\end{array}\end{array}\end{split}\]