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We are trying to solve the following optimization problem using ADMM:

$$ \begin{aligned} & \min _{\left\{y_{i j}^{m}\right\}} \sum_{i \in I_{m}} f_{i j}^{m}\left(y_{i j}^{m}\right)+\sum_{j \in J} \Phi^{m}\left(z_{i j}^{m}\right) \\ & \text { subject to } y_{i j}^{m}-z_{i j}^{m}=0 \quad \forall i, j, m \end{aligned} $$

Where $\Phi^{m}\left(z_{i j}^{m}\right)$ is an indicator function such that, if the value of $z_{i j}^{m}$ is greater than 0, the value of the function is equal to 1, otherwise, 0.


I am trying to solve this optimization function using the Qiskit Ecosystem:

# construct model using docplex
mdl = Model("ex6")

v = mdl.binary_var(name="v")
w = mdl.binary_var(name="w")
t = mdl.binary_var(name="t")
u = mdl.continuous_var(name="u")

mdl.minimize(v + w + t + 5 * (u - 2) ** 2)
mdl.add_constraint(v + 2 * w + t + u <= 3, "cons1")
mdl.add_constraint(v + w + t >= 1, "cons2")
mdl.add_constraint(v + w == 1, "cons3")

# load quadratic program from docplex model
qp = from_docplex_mp(mdl)
print(qp.prettyprint())

In the above code, I am unable to understand how to define the mdl.minimize() function, given that the indicator function $\Phi^{m}\left(z_{i j}^{m}\right)$ is a step function.

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