|
| 1 | +{ |
| 2 | + "cells": [ |
| 3 | + { |
| 4 | + "cell_type": "raw", |
| 5 | + "metadata": { |
| 6 | + "editable": true, |
| 7 | + "raw_mimetype": "text/restructuredtext", |
| 8 | + "slideshow": { |
| 9 | + "slide_type": "" |
| 10 | + }, |
| 11 | + "tags": [] |
| 12 | + }, |
| 13 | + "source": [ |
| 14 | + ".. _nb_matrix_inversion:" |
| 15 | + ] |
| 16 | + }, |
| 17 | + { |
| 18 | + "cell_type": "markdown", |
| 19 | + "metadata": { |
| 20 | + "editable": true, |
| 21 | + "slideshow": { |
| 22 | + "slide_type": "" |
| 23 | + }, |
| 24 | + "tags": [] |
| 25 | + }, |
| 26 | + "source": [ |
| 27 | + "## Matrix Inversion\n", |
| 28 | + "\n", |
| 29 | + "In this case study, the optimization of a matrix shall be illustrated. Of course, we all know that there are very efficient algorithms for calculating an inverse of a matrix. However, for the sake of illustration, a small example shall show that pymoo can also be used to optimize matrices or even tensors.\n", |
| 30 | + "\n", |
| 31 | + "Assuming matrix `A` has a size of `n x n`, the problem can be defined by optimizing a vector consisting of `n**2` variables. During evaluation the vector `x`, is reshaped to inversion of the matrix to be found (and also stored as the attribute `A_inv` to be retrieved later)." |
| 32 | + ] |
| 33 | + }, |
| 34 | + { |
| 35 | + "cell_type": "code", |
| 36 | + "execution_count": 1, |
| 37 | + "metadata": { |
| 38 | + "editable": true, |
| 39 | + "slideshow": { |
| 40 | + "slide_type": "" |
| 41 | + }, |
| 42 | + "tags": [] |
| 43 | + }, |
| 44 | + "outputs": [], |
| 45 | + "source": [ |
| 46 | + "from pymoo.core.problem import ElementwiseProblem\n", |
| 47 | + "\n", |
| 48 | + "\n", |
| 49 | + "class MatrixInversionProblem(ElementwiseProblem):\n", |
| 50 | + "\n", |
| 51 | + " def __init__(self, A, **kwargs):\n", |
| 52 | + " self.A = A\n", |
| 53 | + " self.n = len(A)\n", |
| 54 | + " super().__init__(n_var=self.n**2, n_obj=1, xl=-100.0, xu=+100.0, **kwargs)\n", |
| 55 | + "\n", |
| 56 | + "\n", |
| 57 | + " def _evaluate(self, x, out, *args, **kwargs):\n", |
| 58 | + " A_inv = x.reshape((self.n, self.n))\n", |
| 59 | + " out[\"A_inv\"] = A_inv\n", |
| 60 | + "\n", |
| 61 | + " I = np.eye(self.n)\n", |
| 62 | + " out[\"F\"] = ((I - (A @ A_inv)) ** 2).sum()" |
| 63 | + ] |
| 64 | + }, |
| 65 | + { |
| 66 | + "cell_type": "markdown", |
| 67 | + "metadata": { |
| 68 | + "editable": true, |
| 69 | + "slideshow": { |
| 70 | + "slide_type": "" |
| 71 | + }, |
| 72 | + "tags": [] |
| 73 | + }, |
| 74 | + "source": [ |
| 75 | + "Now let us see what solution is found to be optimal" |
| 76 | + ] |
| 77 | + }, |
| 78 | + { |
| 79 | + "cell_type": "code", |
| 80 | + "execution_count": 2, |
| 81 | + "metadata": { |
| 82 | + "editable": true, |
| 83 | + "slideshow": { |
| 84 | + "slide_type": "" |
| 85 | + }, |
| 86 | + "tags": [] |
| 87 | + }, |
| 88 | + "outputs": [], |
| 89 | + "source": [ |
| 90 | + "import numpy as np\n", |
| 91 | + "from pymoo.algorithms.soo.nonconvex.de import DE\n", |
| 92 | + "from pymoo.optimize import minimize\n", |
| 93 | + "\n", |
| 94 | + "np.random.seed(1)\n", |
| 95 | + "A = np.random.random((2, 2))\n", |
| 96 | + "\n", |
| 97 | + "problem = MatrixInversionProblem(A)\n", |
| 98 | + "\n", |
| 99 | + "algorithm = DE()\n", |
| 100 | + "\n", |
| 101 | + "res = minimize(problem,\n", |
| 102 | + " algorithm,\n", |
| 103 | + " seed=1,\n", |
| 104 | + " verbose=False)\n", |
| 105 | + "\n", |
| 106 | + "opt = res.opt[0]" |
| 107 | + ] |
| 108 | + }, |
| 109 | + { |
| 110 | + "cell_type": "markdown", |
| 111 | + "metadata": { |
| 112 | + "editable": true, |
| 113 | + "slideshow": { |
| 114 | + "slide_type": "" |
| 115 | + }, |
| 116 | + "tags": [] |
| 117 | + }, |
| 118 | + "source": [ |
| 119 | + "In this case the true optimum is actually known. It is" |
| 120 | + ] |
| 121 | + }, |
| 122 | + { |
| 123 | + "cell_type": "code", |
| 124 | + "execution_count": 3, |
| 125 | + "metadata": { |
| 126 | + "editable": true, |
| 127 | + "slideshow": { |
| 128 | + "slide_type": "" |
| 129 | + }, |
| 130 | + "tags": [] |
| 131 | + }, |
| 132 | + "outputs": [ |
| 133 | + { |
| 134 | + "data": { |
| 135 | + "text/plain": [ |
| 136 | + "array([[ 2.39952297e+00, -5.71699951e+00],\n", |
| 137 | + " [-9.07758630e-04, 3.30977861e+00]])" |
| 138 | + ] |
| 139 | + }, |
| 140 | + "execution_count": 3, |
| 141 | + "metadata": {}, |
| 142 | + "output_type": "execute_result" |
| 143 | + } |
| 144 | + ], |
| 145 | + "source": [ |
| 146 | + "np.linalg.inv(A)" |
| 147 | + ] |
| 148 | + }, |
| 149 | + { |
| 150 | + "cell_type": "markdown", |
| 151 | + "metadata": { |
| 152 | + "editable": true, |
| 153 | + "slideshow": { |
| 154 | + "slide_type": "" |
| 155 | + }, |
| 156 | + "tags": [] |
| 157 | + }, |
| 158 | + "source": [ |
| 159 | + "Let us see if the black-box optimization algorithm has found something similar" |
| 160 | + ] |
| 161 | + }, |
| 162 | + { |
| 163 | + "cell_type": "code", |
| 164 | + "execution_count": 4, |
| 165 | + "metadata": { |
| 166 | + "editable": true, |
| 167 | + "slideshow": { |
| 168 | + "slide_type": "" |
| 169 | + }, |
| 170 | + "tags": [] |
| 171 | + }, |
| 172 | + "outputs": [ |
| 173 | + { |
| 174 | + "data": { |
| 175 | + "text/plain": [ |
| 176 | + "array([[ 2.39916052e+00, -5.71656622e+00],\n", |
| 177 | + " [-8.41267527e-04, 3.30978797e+00]])" |
| 178 | + ] |
| 179 | + }, |
| 180 | + "execution_count": 4, |
| 181 | + "metadata": {}, |
| 182 | + "output_type": "execute_result" |
| 183 | + } |
| 184 | + ], |
| 185 | + "source": [ |
| 186 | + "opt.get(\"A_inv\")" |
| 187 | + ] |
| 188 | + }, |
| 189 | + { |
| 190 | + "cell_type": "markdown", |
| 191 | + "metadata": { |
| 192 | + "editable": true, |
| 193 | + "slideshow": { |
| 194 | + "slide_type": "" |
| 195 | + }, |
| 196 | + "tags": [] |
| 197 | + }, |
| 198 | + "source": [ |
| 199 | + "This small example shall have illustrated how a matrix can be optimized. In fact, this is implemented by optimizing a vector of variables that are reshaped during evaluation." |
| 200 | + ] |
| 201 | + } |
| 202 | + ], |
| 203 | + "metadata": { |
| 204 | + "kernelspec": { |
| 205 | + "display_name": "Python 3 (ipykernel)", |
| 206 | + "language": "python", |
| 207 | + "name": "python3" |
| 208 | + }, |
| 209 | + "language_info": { |
| 210 | + "codemirror_mode": { |
| 211 | + "name": "ipython", |
| 212 | + "version": 3 |
| 213 | + }, |
| 214 | + "file_extension": ".py", |
| 215 | + "mimetype": "text/x-python", |
| 216 | + "name": "python", |
| 217 | + "nbconvert_exporter": "python", |
| 218 | + "pygments_lexer": "ipython3", |
| 219 | + "version": "3.12.1" |
| 220 | + } |
| 221 | + }, |
| 222 | + "nbformat": 4, |
| 223 | + "nbformat_minor": 4 |
| 224 | +} |
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