English suggestions for Integer Programming

Author: AnnePicus

applications/optimization/integer_linear_programming/integer_linear_programming.ipynb

@@ -3,9 +3,7 @@
   {
    "cell_type": "markdown",
    "id": "0",
-   "metadata": {
-    "jp-MarkdownHeadingCollapsed": true
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+   "metadata": {},
    "source": [
     "# Integer Linear Programming"
    ]
@@ -15,44 +13,35 @@
    "id": "1",
    "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",
+    "## Mathematical Formulation\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": "2",
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    "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:"
    ]
   },
   {
@@ -160,7 +149,7 @@
    "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."
    ]
   },
   {
@@ -195,7 +184,7 @@
    "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:"
    ]
   },
   {
@@ -222,7 +211,7 @@
    "id": "11",
    "metadata": {},
    "source": [
-    "We also set the quantum backend we want to execute on:"
+    "Set the quantum backend on which to execute:"
    ]
   },
   {
@@ -244,7 +233,7 @@
    "id": "13",
    "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)]:"
    ]
   },
   {
@@ -270,7 +259,7 @@
    "id": "15",
    "metadata": {},
    "source": [
-    "We can check the convergence of the run:"
+    "Check the convergence of the run:"
    ]
   },
   {
@@ -322,15 +311,15 @@
    "id": "18",
    "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": "19",
    "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:"
    ]
   },
   {
@@ -431,7 +420,7 @@
    "id": "21",
    "metadata": {},
    "source": [
-    "We also want to compare the optimized results to uniformly sampled results:"
+    "Compare the optimized results to uniformly sampled results:"
    ]
   },
   {
@@ -497,7 +486,7 @@
    "id": "25",
    "metadata": {},
    "source": [
-    "Let us plot the solution:"
+    "Plot the solution:"
    ]
   },
   {
@@ -529,15 +518,15 @@
    "id": "27",
    "metadata": {},
    "source": [
-    "## Comparison to a classical solver"
+    "## Comparing to a Classical Solver"
    ]
   },
   {
    "cell_type": "markdown",
    "id": "28",
    "metadata": {},
    "source": [
-    "Lastly, we can compare to the classical solution of the problem:"
+    "Compare to the classical solution of the problem:"
    ]
   },
   {
@@ -571,11 +560,11 @@
     "\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"
    ]
   }
  ],