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Update README.md
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pedromxavier authored Sep 12, 2023
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Expand Up @@ -27,23 +27,27 @@ Recently, significant advances in computing systems and algorithms specialized f

These novel tools include Quantum Annealing, Quantum Gate-Circuit Optimization Algorithms (Quantum Optimization Alternating Ansatz, Variational Quantum Eigensolver), other hardware-accelerated platforms, such as Coherent Ising Machines and Simulated Bifurcation Machines, not to mention traditional methods such as Simulated Annealing and Parallel Tempering.

<details>
<summary>Show Math</summary>
### Mathematical Model

Mathematically speaking, this kind of optimization problem is defined by models of the form
Conceptually speaking, it is an optimization model with a **quadratic objective function** on **binary variables** and **no constraints**.
Despite being very simple, these models are capable of representing other nonconvex global optimization problems.

$$
```math
\begin{array}{rl}
\min & \alpha \left[\mathbf{x}' Q\,\mathbf{x} + \mathbf{\ell}' \mathbf{x} + \beta \right] \\
\textrm{s.t.} & \mathbf{x} \in \lbrace{0, 1}\rbrace^{n}
\min & \alpha \left[\mathbf{x}' \mathbf{Q}\,\mathbf{x} + \mathbf{\ell}' \mathbf{x} + \beta \right] \\
\textrm{s.t.} & \mathbf{x} \in \lbrace{0, 1}\rbrace^{n}
\end{array}
$$
```

where $\min$ and $\max$ are interchangeable under sign inversion and spin variables $\mathbf{s} \in \lbrace{\pm1}\rbrace^{n}$ may be employed by taking $s = 2x - 1$ as convention.
<details>
<summary><strong>Show Description</strong></summary>

In other words, it is an optimization model with a **quadratic objective function** on **binary variables** and **no constraints**.
Analizing the model attentively, let $\mathbf{x}$ be a **vector of boolean (zero-one) variables**.
Take also the **vector of linear terms** $\mathbf{\ell} \in \mathbb{R}^{n}$ and the **strictly upper triangular matrix of quadratic terms** $\mathbf{Q} \in \mathbb{R}^{n \times n}$.
Last but not least, consider $\alpha, \beta \in \mathbb{R}$ as the **scaling** and **offset** parameters, respectively.

Despite being very simple, these models are capable of representing other nonconvex global optimization problems.
Note that in this kind of problem, $\min$ and $\max$ are interchangeable under sign inversion.
Also, spin variables $\mathbf{s} \in \lbrace{\pm 1}\rbrace^{n}$ may be employed instead, assuming that $s = 2x - 1$ by convention.

</details>

Expand Down Expand Up @@ -72,7 +76,7 @@ Given the following binary Knapsack Problem
one could write a simple [JuMP](https://jump.dev) model and have its constraint automatically encoded by [ToQUBO.jl](https://github.com/psrenergy/ToQUBO.jl).

<details>
<summary>Show Code</summary>
<summary><strong>Show Code</strong></summary>

```julia
using JuMP
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