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English suggestions for Integer Programming #1189

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55 changes: 22 additions & 33 deletions applications/optimization/integer_linear_programming/integer_linear_programming.ipynb
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{
"cell_type": "markdown",
"id": "9d91ea6c-9f4c-4c9e-9e33-f4333482eee5",
"metadata": {
"jp-MarkdownHeadingCollapsed": true
},
"metadata": {},
"source": [
"# Integer Linear Programming"
]
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"id": "877ae3cf-10cc-4e51-a428-eb57687d8d18",
"metadata": {},
"source": [
"## Introduction\n",
"\n",
"In Integer Linear Programming (ILP), we seek to find a vector of integer numbers that maximizes (or minimizes) a linear cost function under a set of linear equality or inequality constraints [[1]](#ILP). In other words, it is an optimization problem where the cost function to be optimized and all the constraints are linear and the decision variables are integer.\n",
"\n",
"Integer Linear Programming (ILP) seeks a vector of integer numbers that maximizes (or minimizes) a linear cost function under a set of linear equality or inequality constraints [[1]](#ILP). In other words, it is an optimization problem where the cost function to optimize and all the constraints are linear and the decision variables are integers.\n",
"\n",
"\n",
"## Mathematical Formulation\n",
"The ILP problem can be formulated as follows:\n",
"\n",
"Given an $n$-dimensional vector $\\vec{c} = (c_1, c_2, \\ldots, c_n)$, an $m \\times n$ matrix $A = (a_{ij})$ with $i=1,\\ldots,m$ and $j=1,\\ldots,n$, and an $m$-dimensional vector $\\vec{b} = (b_1, b_2, \\ldots, b_m)$, find an $n$-dimensional vector $\\vec{x} = (x_1, x_2, \\ldots, x_n)$ with integer entries that maximizes (or minimizes) the cost function:\n",
"The ILP problem can be formulated as follows: given an $n$-dimensional vector $\\vec{c} = (c_1, c_2, \\ldots, c_n)$, an $m \\times n$ matrix $A = (a_{ij})$ with $i=1,\\ldots,m$ and $j=1,\\ldots,n$, and an $m$-dimensional vector $\\vec{b} = (b_1, b_2, \\ldots, b_m)$, find an $n$-dimensional vector $\\vec{x} = (x_1, x_2, \\ldots, x_n)$ with integer entries that maximizes (or minimizes) the cost function:\n",
"\n",
"\\begin{align*}\n",
"\\vec{c} \\cdot \\vec{x} = c_1x_1 + c_2x_2 + \\ldots + c_nx_n\n",
"\\end{align*}\n",
"\n",
"subject to the constraints:\n",
"subject to these constraints:\n",
"\n",
"\\begin{align*}\n",
"A \\vec{x} & \\leq \\vec{b} \\\\\n",
"x_j & \\geq 0, \\quad j = 1, 2, \\ldots, n \\\\\n",
"x_j & \\in \\mathbb{Z}, \\quad j = 1, 2, \\ldots, n\n",
"\\end{align*}\n",
"\n",
"\n",
"\n",
"# Solving with the Classiq platform\n",
"\n",
"We go through the steps of solving the problem with the Classiq platform, using QAOA algorithm [[2](#QAOA)]. The solution is based on defining a Pyomo model for the optimization problem we would like to solve."
"This tutorial guides you through the steps of solving the problem with the Classiq platform, using QAOA [[2](#QAOA)]. The solution is based on defining a Pyomo model for the optimization problem to solve."
]
},
{
"cell_type": "markdown",
"id": "8517ef73-faf6-4fd8-9db8-4088551398e0",
"metadata": {},
"source": [
"## Building the Pyomo model from a graph input\n",
"## Building the Pyomo Model from a Graph Input\n",
"\n",
"We proceed by defining the Pyomo model that will be used on the Classiq platform, using the mathematical formulation defined above:"
"Define the Pyomo model to use on the Classiq platform, using the mathematical formulation defined above:"
]
},
{
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"source": [
"## Setting Up the Classiq Problem Instance\n",
"\n",
"In order to solve the Pyomo model defined above, we use the `CombinatorialProblem` quantum object. Under the hood it tranlastes the Pyomo model to a quantum model of the QAOA algorithm, with cost hamiltonian translated from the Pyomo model. We can choose the number of layers for the QAOA ansatz using the argument `num_layers`, and the `penalty_factor`, which will be the coefficient of the constraints term in the cost hamiltonian."
"To solve the Pyomo model defined above, use the `CombinatorialProblem` quantum object. Under the hood it translates the Pyomo model to a quantum model of QAOA, with the cost Hamiltonian translated from the Pyomo model. Choose the number of layers for the QAOA ansatz using the `num_layers` argument. The `penalty_factor` is the coefficient of the constraints term in the cost Hamiltonian."
]
},
{
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"source": [
"## Synthesizing the QAOA Circuit and Solving the Problem\n",
"\n",
"We can now synthesize and view the QAOA circuit (ansatz) used to solve the optimization problem:"
"Synthesize and view the QAOA circuit (ansatz) used to solve the optimization problem:"
]
},
{
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"id": "b119464b-9d46-4ea0-ba4a-1734f3e0e3e5",
"metadata": {},
"source": [
"We also set the quantum backend we want to execute on:"
"Set the quantum backend on which to execute:"
]
},
{
Expand All @@ -270,7 +259,7 @@
"id": "07621d7c-0e54-4adb-9c47-8fd99a346e29",
"metadata": {},
"source": [
"We now solve the problem by calling the `optimize` method of the `CombinatorialProblem` object. For the classical optimization part of the QAOA algorithm we define the maximum number of classical iterations (`maxiter`) and the $\\alpha$-parameter (`quantile`) for running CVaR-QAOA, an improved variation of the QAOA algorithm [[3](#cvar)]:"
"Solve the problem by calling the `optimize` method of the `CombinatorialProblem` object. For the classical optimization part of QAOA, define the maximum number of classical iterations (`maxiter`) and the $\\alpha$-parameter (`quantile`) for running CVaR-QAOA, an improved variation of QAOA [[3](#cvar)]:"
]
},
{
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"id": "620ea6a0-cd05-41a9-a2ed-9631c680d2e6",
"metadata": {},
"source": [
"We can check the convergence of the run:"
"Check the convergence of the run:"
]
},
{
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"tags": []
},
"source": [
"# Optimization Results"
"## Optimization Results"
]
},
{
"cell_type": "markdown",
"id": "670eddd3-2da7-4a88-b571-7884ef24f60c",
"metadata": {},
"source": [
"We can also examine the statistics of the algorithm. The optimization is always defined as a minimzation problem, so the positive maximization objective was tranlated to a negative minimization one by the Pyomo to qmod translator."
"Examine the statistics of the algorithm. The optimization is always defined as a minimization problem, so the positive maximization objective is translated to negative minimization by the Pyomo-to-Qmod translator."
]
},
{
"cell_type": "markdown",
"id": "c99a7e3e-5203-4893-b970-dc23739f6df2",
"metadata": {},
"source": [
"In order to get samples with the optimized parameters, we call the `sample` method:"
"To get samples with the optimized parameters, call the `sample` method:"
]
},
{
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"id": "f08c8085-b50a-41a1-9359-46413f60739c",
"metadata": {},
"source": [
"We also want to compare the optimized results to uniformly sampled results:"
"Compare the optimized results to uniformly sampled results:"
]
},
{
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"id": "a3a890a1-c5d4-409d-b9a3-d7ffd4fdd6c0",
"metadata": {},
"source": [
"Let us plot the solution:"
"Plot the solution:"
]
},
{
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"id": "149932e1-bfa8-4c27-b5f9-037e74eba400",
"metadata": {},
"source": [
"## Comparison to a classical solver"
"## Comparing to a Classical Solver"
]
},
{
"cell_type": "markdown",
"id": "dde1905d-aeff-4297-a9d3-ad14910e6161",
"metadata": {},
"source": [
"Lastly, we can compare to the classical solution of the problem:"
"Compare to the classical solution of the problem:"
]
},
{
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"\n",
"## References\n",
"\n",
"<a id='MVC'>[1]</a>: [Integer Programming (Wikipedia).](https://en.wikipedia.org/wiki/Integer_programming)\n",
"<a id='MVC'>[1]</a> [Integer Programming (Wikipedia).](https://en.wikipedia.org/wiki/Integer_programming)\n",
"\n",
"<a id='QAOA'>[2]</a>: [Farhi, Edward, Jeffrey Goldstone, and Sam Gutmann. \"A quantum approximate optimization algorithm.\" arXiv preprint arXiv:1411.4028 (2014).](https://arxiv.org/abs/1411.4028)\n",
"<a id='QAOA'>[2]</a> [Farhi, Edward, Jeffrey Goldstone, and Sam Gutmann. (2014). \"A quantum approximate optimization algorithm.\" arXiv preprint arXiv:1411.4028.](https://arxiv.org/abs/1411.4028)\n",
"\n",
"<a id='cvar'>[3]</a>: [Barkoutsos, Panagiotis Kl, et al. \"Improving variational quantum optimization using CVaR.\" Quantum 4 (2020): 256.](https://arxiv.org/abs/1907.04769)\n"
"<a id='cvar'>[3]</a> [Barkoutsos, Panagiotis Kl, et al. (2020). \"Improving variational quantum optimization using CVaR.\" Quantum 4: 256.](https://arxiv.org/abs/1907.04769)\n"
]
}
],
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