Aula 8 - Programação Não Linear.ipynb

{"cells":[{"cell_type":"markdown","source":["---\n","---\n","# UNIVERSIDADE FEDERAL FLUMINENSE\n","---\n","---\n","## TEP00118 - PESQUISA OPERACIONAL II\n","## Prof.: Diogo Ferreira de Lima Silva\n","---\n","### Monitor: Rodrigo Celso de Lima Porto\n","### Matrícula: 319042042\n","---"],"metadata":{"id":"GZnzduvrTaHr"}},{"cell_type":"markdown","metadata":{"id":"HDm5fsKVifYJ"},"source":["---\n","---\n","# PROGRAMAÇÃO NÃO LINEAR\n","---\n","---\n","\n","<p align=justify> &emsp; Os problemas de Programação Não-Linear (PNL) dizem respeito aos problemas de otimização que atendem a pelo menos um dos requisitos abaixo:\n","<ul>\n","<li>A função objetivo não é uma função linear, ou seja, não é expressa como $\\text{Max ou Min }z=a_{1}x_{1}+a_{2}x_{2}+\\cdot \\cdot \\cdot \\ + a_{n}x_{n}$;</li>\n","<li>Haja ao menos uma restrição que não seja expressa como uma equação ou inequação linear.</li>\n","</ul>\n","&emsp; E existem vários métodos para resolver um problema de PNL, sendo o mais abrangente as condições de <a href=\"https://pt.wikipedia.org/wiki/Condi%C3%A7%C3%B5es_de_Karush-Kuhn-Tucker\" target=\"self\">Karush-Kuhn-Tucker</a> (ou simplesmente KKT) pelo fato de este método conseguir resolver PNL tanto com restrições de igualdade quanto com os de desigualdade.\n","</p>"]},{"cell_type":"markdown","source":["---\n","---\n","# INSTÂNCIA DE UM PROBLEMA\n","---\n","---\n","\n","Exemplo retirado em: [MÉTODO DOS MULTIPLICADORES DE LAGRANGE 📚 AULA 1](https://www.youtube.com/watch?v=iPyMGocdQyU)\n","\n","---\n","\n","> <p align=justify>\n","&emsp; Uma indústria de papel está criando um novo modelo de folha com o intuito de atender o pedido de um cliente, que está requisitando um tipo não padronizado de folha para o seu novo projeto de caderno. No entanto, o cliente não informou quais seriam as dimensões desta folha de papel. Quando questionado, ele respondeu que as dimensões podem ser de quaisquer valores, desde que o perímetro da folha de papel seja igual a 200 cm, e de preferência que possua a maior área possível. Com o intuito de atender a preferência do cliente, deseja-se saber quais são as dimensões do papel que maximizem a sua área.\n","</p>\n","<p align=justify>\n","&emsp; O problema é um clássico exemplo de PNL, pois a função objetivo, que é a área da folha de papel, é uma função-não linear por se tratar do produto entre duas variáveis de decisão, neste caso as dimensões $x$ e $y$ da folha de papel.\n","<br>\n","<br>\n","$$\n","\\text{Max } Área(x,y)=xy \\\\\n","\\begin{equation}\n","  \\begin{array}{rll}\n","    \\text{Sujeito a: } & & \\\\\n","    & x + y = 100 & (1) \\\\\n","    & x, y > 0  & (2)\n","  \\end{array}\n","\\end{equation}\n","$$\n","<br>\n","<br>\n","&emsp; Onde:\n","<ul align=justify>\n","<li>As variáveis $x$ e $y$ respresentarão as dimensões da folha de papel dadas em cm;</li>\n","<li>A restrição $(1)$ é a simplificação da restrição do perímetro dado pelo cliente (Como o perímetro da folha dever ser de 200 cm, significa que $2x+2y=200 → x+y=100$);</li>\n","<li>A restrição $(2)$ garante a positividade das variáveis $x$ e $y$.</li>\n","</ul>\n","</p>\n"],"metadata":{"id":"O8PqCB7zNsGd"}},{"cell_type":"markdown","metadata":{"id":"dZo1u4QPZJv0"},"source":["---\n","---\n","# MODELANDO COM PYOMO\n","---\n","---\n","\n","Recomendações:\n","* Site [Pyomo.org](https://www.pyomo.org)\n","* Site [ND Pyomo Cookbook](https://jckantor.github.io/ND-Pyomo-Cookbook/)\n","* Vídeo [Noções básicas de Pyomo - Parte 1](https://www.youtube.com/watch?v=YDqalQH2b9w)\n","* Vídeo [Noções básicas de Pyomo - Parte 2](https://www.youtube.com/watch?v=nao-zTTUQPQ)\n","\n","---\n","<p align=justify>\n","&emsp; Para esta aula, usaremos o solver COIN-OR Ipopt, que é um otimizador de ponto interior de código aberto para otimização não linear em larga escala (<b>I</b>nterior <b>P</b>oint <b>Opt</b>imizer) disponível sob a Eclipse Public License (EPL), sendo mais adequado para resolver problemas de PNL sem restrições de número inteiro ou binário.\n","<br>\n","&emsp; Caso não tenha instalado ainda o Pyomo e o Ipopt, siga os passos abaixo:\n","\n","<ol>\n","<li>Baixe e instale o <a href='https://www.anaconda.com/'>Anaconda</a></li>\n","<li>Clique no ícone do Windows e pesquise por \"Anaconda Prompt\"</li>\n","<li>Escreva no prompt <code>conda install -c conda-forge pyomo</code></li>\n","<li>Aguarde o término da instalação, e quando o prompt te perguntar <code>Proceed ([y]/n)?</code>, tecle \"y\"</li>\n","<li>Agora escreva no prompt <code>conda install -c conda-forge ipopt</code></li>\n","<li>Aguarde o término da instalação, e quando o prompt te perguntar <code>Proceed ([y]/n)?</code>, tecle \"y\"</li>\n","<li>Pronto!</li>\n","</ol>\n","\n","&emsp; Para maiores detalhes, assista este <a href='https://www.youtube.com/watch?v=lzMpvlKNQ-Y&t=294s'>vídeo</a>.\n","</p>\n","\n","---\n","\n","<p align=justify>\n","&emsp; Caso esteja no Google Colab, execute o bloco de códigos a seguir:\n","</p>"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"MWGu1HG6y9Ci"},"outputs":[],"source":["# Execute este bloco caso esteja no Google Colab\n","!pip install -q pyomo # Instala o Pyomo no ambiente de execução\n","!wget -N -q \"https://ampl.com/dl/open/ipopt/ipopt-linux64.zip\" # Pega o zip contendo o solver Ipopt\n","!unzip -o -q ipopt-linux64 # Descomprime o documento zip do Ipopt"]},{"cell_type":"markdown","metadata":{"id":"kUL_tepHd3-X"},"source":["---\n","## Resolução\n","\n","---"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"CFZrgzGqzukb"},"outputs":[],"source":["import pyomo.environ as pyo # Importando o Pyomo\n","\n","# MODELO\n","modelo = pyo.ConcreteModel() # Criando o modelo concreto para o nosso problema\n","\n","# VARIÁVEIS\n","modelo.x = pyo.Var(domain= PositiveReals) # Definindo a variável x\n","modelo.y = pyo.Var(domain= PositiveReals) # Definindo a variável y\n","x = modelo.x # \"Renomeando\" a variável x\n","y = modelo.y # \"Renomeando\" a variável y\n","\n","# FUNÇÃO OBJETIVA\n","modelo.Area = pyo.Objective(expr= x*y, sense= maximize) # Definindo a função objetivo\n","Area = modelo.Area # \"Renomeando\" a função Área\n","\n","# RESTRIÇÃO\n","modelo.Perimetro = pyo.Constraint(expr= x+y==100) # Definindo a restrição simplificada do perímetro\n","\n","# SOLUCIONANDO O MODELO\n","ipopt = pyo.SolverFactory('ipopt', executable='/content/ipopt') # Construindo o solver Ipopt\n","resultado = ipopt.solve(modelo) # Mandando o Ipopt resolver o modelo"]},{"cell_type":"markdown","metadata":{"id":"Y0qKJeQOVA98"},"source":["<p align=justify>\n","&emsp; E o resultado do nosso problema é dado no bloco a seguir:\n","</p>"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"executionInfo":{"elapsed":5,"status":"ok","timestamp":1660437437337,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"},"user_tz":180},"id":"DNrspE1DVIgI","outputId":"ef436361-7d9e-4dae-f3af-fad59f7d3101"},"outputs":[{"output_type":"stream","name":"stdout","text":["Área máxima = 2500.00 cm²\n","x = 50.00 cm\n","y = 50.00 cm\n"]}],"source":["# Se o status do resultado for 'ok', exibe as informações\n","if str(resultado.Solver.status) == 'ok':\n","  print(f\"Área máxima = {value(Area):.2f} cm²\")\n","  print(f\"x = {value(x):.2f} cm\\ny = {value(y):.2f} cm\")\n","else: # Caso contrário...\n","  print(\"Nenhuma solução válida foi encontrada.\")"]},{"cell_type":"markdown","source":["---\n","---\n","# EXERCÍCIO DE FIXAÇÃO\n","---\n","---\n","\n","> <p align=justify>\n","&emsp; Um galpão retangular deve ser construído em um terreno com a forma de um triângulo, conforme a figura abaixo. Utilize o Pyomo para descobrir qual a área máxima possível para o galpão.\n","</p>\n","\n","<center>\n","\n","![terreno_triangular.png](data:image/png;base64,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)\n","\n","\n","</center>\n","\n","---\n","Fonte: [MÉTODO DOS MULTIPLICADORES DE LAGRANGE 📚 AULA 1](https://www.youtube.com/watch?v=iPyMGocdQyU)\n","\n","---"],"metadata":{"id":"ZbIwSwMxq1bQ"}},{"cell_type":"code","source":["import pyomo.environ as pyo\n","\n"],"metadata":{"id":"aR51owcztCTU"},"execution_count":null,"outputs":[]},{"cell_type":"markdown","source":["---\n","---\n","# BREVE INTRODUÇÃO AO SYMPY\n","---\n","---\n","\n","<p align=justify>\n","&emsp; O SymPy (<b>Sym</b>bolic <b>Py</b>thon) é uma biblioteca Python para matemática simbólica que visa se tornar um sistema de álgebra computacional (ou <i>Computer Algebric System</i>, ou ainda, CAS) completo, mantendo o código o mais simples possível para ser compreensível e facilmente extensível.<br>\n","&emsp; Um CAS, por sua vez, pode ser utilizado para cálculos com expressões matemáticas complicadas, resolução de equações, simulações, dentre outras aplicações. Há sistemas gratuitos como o SymPy e o Octave, e comerciais como Maple, MATLAB e Mathematica, cuja decisão de preferir um ou outro depende dos aspectos técnicos e da análise custo vs benefício do seu projeto. Para mais informações sobre a biblioteca, visite <a href='https://www.sympy.org/en/index.html'>o site do projeto</a>.<br>\n","&emsp; Por se tratar de um CAS, o SymPy, ao contrário de bibliotecas como Math cujos cálculos são aproximações (o que pode levar a erros e acumulos de erros nas contas), representa números e operações simbolicamente, o que permite que o resultado das contas seja obtida de forma <b>EXATA</b> (ou <b>analiticamente</b>, como é formalmente dito).<br>\n","&emsp; Um breve exemplo disso é como mostrado a seguir, onde queremos calcular $\\sqrt{3}$.\n","</p>"],"metadata":{"id":"XguJfjGt2gVj"}},{"cell_type":"code","source":["import math\n","math.sqrt(3)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":37},"id":"NaQwmd3FBhGO","executionInfo":{"status":"ok","timestamp":1671329424015,"user_tz":180,"elapsed":13,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"14d75348-112e-4c94-fdc5-e3a3dd869527"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["1.7320508075688772"],"image/png":"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\n","text/latex":"$\\displaystyle 1.73205080756888$"},"metadata":{},"execution_count":122}]},{"cell_type":"markdown","source":["<p align=justify>\n","&emsp; Aí, sabe-se que elevando $\\sqrt{3}$ ao quadrado obtemos $3$ de volta, certo?\n","</p>"],"metadata":{"id":"_50w9xNPBydr"}},{"cell_type":"code","source":["math.sqrt(3)**2"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":37},"id":"2Nsrzf2QBtMy","executionInfo":{"status":"ok","timestamp":1671329425902,"user_tz":180,"elapsed":8,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"4d9d0f32-510a-49aa-e3d1-e217821888d1"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["2.9999999999999996"],"image/png":"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\n","text/latex":"$\\displaystyle 3.0$"},"metadata":{},"execution_count":123}]},{"cell_type":"markdown","source":["<p align=justify>\n","&emsp; E aí fica claro a limitação da biblioteca Math, pois o valor obtido por <code>math.sqrt(3)</code> só corresponde a uma aproximação da $\\sqrt{3}$ até a 16ª casa decimal, uma vez que o nível de precisão do Python vai até $1 \\cdot 10^{-16}$. E o SymPy consegue driblar esse limite, uma vez que ele não lê os números como <code>float</code> ou <code>int</code>, mas sim como símbolos. Tanto que ao calcular a $\\sqrt{3}$ ele nos simplesmente retorna a própria $\\sqrt{3}$.\n","</p>"],"metadata":{"id":"2rWaeKyfCZUT"}},{"cell_type":"code","source":["import sympy as sp\n","sp.sqrt(3)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":39},"id":"u75QqzotD18i","executionInfo":{"status":"ok","timestamp":1671329432901,"user_tz":180,"elapsed":8,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"8992e677-6371-449d-8277-5384e8fd0e46"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["√3"],"image/png":"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\n","text/latex":"$\\displaystyle \\sqrt{3}$"},"metadata":{},"execution_count":124}]},{"cell_type":"markdown","source":["<p align=justify>\n","&emsp; Para pedir ao Sympy calcular a $\\sqrt{3}$, com $n$ casas decimais, usa-se o método <code>.evalf(n)</code>. No exemplo a seguir, queremos o valor da $\\sqrt{3}$ até a 100ª casa decimal.\n","</p>"],"metadata":{"id":"JDoo9vp6Ej_3"}},{"cell_type":"code","source":["sp.sqrt(3).evalf(100)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":37},"id":"RV8VwkUjEe26","executionInfo":{"status":"ok","timestamp":1671329435039,"user_tz":180,"elapsed":11,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"2ae46a6e-f2f2-4804-ed11-35c3c8a538cc"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["1.7320508075688772935274463415058723669428052538103806280558069794519330169088\n","00037081146186757248576"],"image/png":"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\n","text/latex":"$\\displaystyle 1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576$"},"metadata":{},"execution_count":125}]},{"cell_type":"markdown","source":["<p align=justify>\n","&emsp; E se for feito $\\sqrt{3}^{2}$ pelo SymPy, obteremos o valor exato $3$ novamente, sem mais nem menos.\n","</p>"],"metadata":{"id":"o5rPSGbaCoDP"}},{"cell_type":"code","source":["sp.sqrt(3)**2"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":37},"id":"HgPJC2seDB1X","executionInfo":{"status":"ok","timestamp":1671329438718,"user_tz":180,"elapsed":8,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"057b5594-30e2-425c-c8be-6c9ce77a72a0"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["3"],"image/png":"iVBORw0KGgoAAAANSUhEUgAAAA4AAAASCAYAAABrXO8xAAAABHNCSVQICAgIfAhkiAAAAP9JREFUOI2d0rErhVEYx/HPvdkIRUhZhMVERmUzKIu/wGKRRVlYvIsw8ieQxWJV2GRVNxkskoWESQyU4bynXq/3eq/7W36d0/Pt/M7zPJUkSTSjau68jTPc4x0vuMQ6uv4Cl9GKE+zgAJ9IUMNALGzJge34KEi2gTWsYrHoxSIIDlMfrhe1nmZTr8WLfNSoFbShAxOYTKGtRsDezPkY83gqi9qHSupzGBTGMl4GRj3iCNPCHPcaBaPucI1RdP8HhP7Uv/LgiNDFvKrCAvTgAq/87OoMNnGOWzwLnZ0SmvOAhVicBU8xJMxsDJ14ww32sSss/S/wCktlH83mb0pNg98rUiwuFBKrLAAAAABJRU5ErkJggg==\n","text/latex":"$\\displaystyle 3$"},"metadata":{},"execution_count":126}]},{"cell_type":"markdown","source":["<p align=justify>\n","&emsp; Por causa disso, o Sympy é útil para realização de vários problemas de matemática, englobando resolução de expressões algébricas, cálculo de limite, derivada, integral, equações diferenciais, transformadas de Laplace, cálculo vetorial, matrizes etc.<br>\n","&emsp; Apenas para dar um gostinho do que o Sympy é capaz de fazer, segue alguns exemplos.\n","</p>"],"metadata":{"id":"iPIOixm4FRqA"}},{"cell_type":"code","source":["import sympy as sp\n","\n","x = sp.Symbol('x') # Aqui declaramos o símbolo 'x'\n","\n","# Cálculo de limites\n","f = 1/x**2\n","L1 = sp.limit(f, x, 0, \"+\") # x se aproxima de 0 pela direita.\n","L1"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":37},"id":"jLmnQUOGGNQv","executionInfo":{"status":"ok","timestamp":1671329452042,"user_tz":180,"elapsed":410,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"94111adf-7a5a-4002-a376-aa14d631d86b"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["∞"],"image/png":"iVBORw0KGgoAAAANSUhEUgAAABEAAAALCAYAAACZIGYHAAAABHNCSVQICAgIfAhkiAAAAPBJREFUKJGl0bErhlEUBvDfi0EWC6vJhj9ASSmjkMngD/hGBuY3ZWFgsn0bi5IJg7wlE0lKUgZlM5DR9jF8562v68rgqTuc557znOc+tyjL0n/RleGW8IJP7KEv+Cnc4AOnGPpNZAw7WMUcxnGEERyHyDyecIgCehKRFazjIOpZ3OI8RBrBX+Aq3FWpk9FoqPGAJgawlvRehsNsJl9J/YxuDGb6ipzIIyY66n4s4w2b9VBgMpz+yGQbZzH0ig3tn2poZ9DELhbQiyrn5C42b+EE7zFwj5nYfo1pLNZPT53AfpwUFYZjcavzIhfsX2ilxDc70jLsX6uPPgAAAABJRU5ErkJggg==\n","text/latex":"$\\displaystyle \\infty$"},"metadata":{},"execution_count":127}]},{"cell_type":"code","source":["f = 1/x\n","L2 = sp.limit(f, x, 0, \"-\") # x se aproxima de 0 pela esquerda.\n","L2"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":37},"id":"MfKy2lgTH24W","executionInfo":{"status":"ok","timestamp":1669838617861,"user_tz":180,"elapsed":365,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"26bf9061-69ac-4343-faa6-02f372992715"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["-oo"],"text/latex":"$\\displaystyle -\\infty$"},"metadata":{},"execution_count":15}]},{"cell_type":"code","source":["f  = sp.sin(x)/x\n","L3 = sp.limit(f, x, 0) # x se aproxima de 0, não importa para qual lado.\n","L3"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":37},"id":"0O3JWh2FH8w0","executionInfo":{"status":"ok","timestamp":1671329457641,"user_tz":180,"elapsed":8,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"d5765a72-c1cc-4c43-ac46-e6c98d1aa553"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["1"],"image/png":"iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAIdJREFUKJHt0SEOwmAMBeBvBD2NJAiCG5ZbIPDTSJJJxH8HLoDgHghOQoJEEfxmftVkyzBTNHlpX9OX17RFSsmvMQv8gAse+KDFLYrmgZ9R4YsXNmOcTlijxLFvveh07xscchoVf9HkovjcfQYsct7hmus3mijaog69VQY80cT1EooBLJnyEB2yexAji7I9mAAAAABJRU5ErkJggg==\n","text/latex":"$\\displaystyle 1$"},"metadata":{},"execution_count":128}]},{"cell_type":"code","source":["f  = (1 + 1/x)**x\n","L4 = sp.limit(f, x, sp.oo) # oo é infinito.\n","L4"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":38},"id":"EFVkTowaIA8s","executionInfo":{"status":"ok","timestamp":1671329459373,"user_tz":180,"elapsed":13,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"0ca0b5f7-63b4-4cfd-ab61-3b93794b7d47"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["ℯ"],"image/png":"iVBORw0KGgoAAAANSUhEUgAAAA4AAAAPCAYAAADUFP50AAAABHNCSVQICAgIfAhkiAAAAOhJREFUKJG10rErxWEUxvHPvZe6fwCTMt2F7mZQN8XkD7gMLHZKWSSm36KsFqMyKa7BICWTwWKzmkwWi4VyJYNXTrffi9R9lrfec77nnOe8b6UoCv9RNXM/h3M84hV32EItB9ZwiA4aOMYe3rGN/Ry4iwXsYAzLWEMT11jCeC84iRWcYhNvIdbFQcgzEIKrqOAZRYnvZpwygrPpXCyBou4jWMcwrjD9C/jdNo0IQ3+BIviCW58ba2dyp4R3jB7XcYYTXKZCVYxgAoMYLQMv0MJGqj6DJzykQkexfQThBvM/eftS7q/2D/wAO78kutI+uVIAAAAASUVORK5CYII=\n","text/latex":"$\\displaystyle e$"},"metadata":{},"execution_count":129}]},{"cell_type":"code","source":["f  = (2*x**2 + 5*x - 3)/(2*x**2 - 5*x + 2)\n","L5 = sp.limit(f, x, sp.Rational(1/2))\n","L5"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":51},"id":"dDIiLgKiIIJl","executionInfo":{"status":"ok","timestamp":1669838842024,"user_tz":180,"elapsed":5,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"4d1bc7bd-463d-43d9-b6f7-b0486b6eee81"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["-7/3"],"text/latex":"$\\displaystyle - \\frac{7}{3}$"},"metadata":{},"execution_count":24}]},{"cell_type":"code","source":["f  = (sp.sqrt(x+2) + sp.sqrt(x+6) - sp.sqrt(6) - sp.sqrt(2))/x\n","L6 = sp.limit(f, x, 0)\n","L6"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":55},"id":"UWUm_bsiIMj9","executionInfo":{"status":"ok","timestamp":1669838843466,"user_tz":180,"elapsed":14,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"70ba292f-0921-48ec-9cf1-51d9ded98701"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["sqrt(6)/12 + sqrt(2)/4"],"text/latex":"$\\displaystyle \\frac{\\sqrt{6}}{12} + \\frac{\\sqrt{2}}{4}$"},"metadata":{},"execution_count":25}]},{"cell_type":"code","source":["from sympy import Symbol, Function, cos, sin, ln, sqrt, diff, Eq, dsolve\n","\n","x = sp.Symbol('x')\n","y = sp.Symbol('y')\n","\n","#Cálculo de derivadas\n","f = 3*x**2 + sin(3*sqrt(x))\n","D1 = diff(f, x) # Derivada em relação a x\n","D1"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":58},"id":"C8qf0lqVGdBV","executionInfo":{"status":"ok","timestamp":1669937869562,"user_tz":180,"elapsed":321,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"8cfd097a-cacd-462d-8ae5-311d301b9c99"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["6*x + 3*cos(3*sqrt(x))/(2*sqrt(x))"],"text/latex":"$\\displaystyle 6 x + \\frac{3 \\cos{\\left(3 \\sqrt{x} \\right)}}{2 \\sqrt{x}}$"},"metadata":{},"execution_count":5}]},{"cell_type":"code","source":["f = cos(x**3) + sin(x-1)**2\n","D2 = diff(f, x, x) # Derivada em relação a x, duas vezes.\n","D2"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":38},"id":"SC3ybbyhJub2","executionInfo":{"status":"ok","timestamp":1669937872397,"user_tz":180,"elapsed":309,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"515afd83-aa47-4bd8-95fe-3bfe56391a7a"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["-9*x**4*cos(x**3) - 6*x*sin(x**3) - 2*sin(x - 1)**2 + 2*cos(x - 1)**2"],"text/latex":"$\\displaystyle - 9 x^{4} \\cos{\\left(x^{3} \\right)} - 6 x \\sin{\\left(x^{3} \\right)} - 2 \\sin^{2}{\\left(x - 1 \\right)} + 2 \\cos^{2}{\\left(x - 1 \\right)}$"},"metadata":{},"execution_count":6}]},{"cell_type":"code","source":["f = 4*x*y + cos(x*y**2) - ln(x**y)\n","D3 = diff(f, x, y) # Derivada parcial em relação a x, e depois a y.\n","D3"],"metadata":{"id":"6dm9npXkSuae","colab":{"base_uri":"https://localhost:8080/","height":51},"executionInfo":{"status":"ok","timestamp":1669937874117,"user_tz":180,"elapsed":344,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"1539bb7e-9bf1-48a3-9cd9-9cb1e6380ce7"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["-2*x*y**3*cos(x*y**2) - 2*y*sin(x*y**2) + 4 - 1/x"],"text/latex":"$\\displaystyle - 2 x y^{3} \\cos{\\left(x y^{2} \\right)} - 2 y \\sin{\\left(x y^{2} \\right)} + 4 - \\frac{1}{x}$"},"metadata":{},"execution_count":7}]},{"cell_type":"code","source":["f  = ln(x**y**3)\n","D4 = diff(f, y, y, x) # Deriva duas vezes em relação a y, depois uma vez a x.\n","D4.subs({y:1, x:2})   # Substitui y por 1 e x por 2 na expressão obtida."],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":37},"id":"JVSs6HjLKe4_","executionInfo":{"status":"ok","timestamp":1669937875273,"user_tz":180,"elapsed":237,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"c2bd92ee-c557-42d4-b8cd-daf9167eb57f"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["3"],"text/latex":"$\\displaystyle 3$"},"metadata":{},"execution_count":8}]},{"cell_type":"code","source":["f = Function('f')  # Definindo um função\n","df = diff(f(x), x) # Derivada de f em x\n","eq_dif = Eq(f(x) + df, 0) # Equação diferencial\n","dsolve(eq_dif, f(x)) # resolve a equação diferencial"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":38},"id":"C8zYMEY2M-iO","executionInfo":{"status":"ok","timestamp":1671329476215,"user_tz":180,"elapsed":409,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"b1121891-f884-41c7-c8b2-5b3861448d80"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["           -x\n","f(x) = C₁⋅ℯ  "],"image/png":"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\n","text/latex":"$\\displaystyle f{\\left(x \\right)} = C_{1} e^{- x}$"},"metadata":{},"execution_count":130}]},{"cell_type":"code","source":["from sympy import Symbol, Function, integrate, exp, sqrt, oo\n","\n","f = Function('f')\n","x = Symbol('x')\n","y = Symbol('y')\n","z = Symbol('z')\n","\n","# Cálculo de integrais\n","\n","f = exp(x**2)*x\n","I1 = integrate(f, x) # Integral indefinida.\n","I1"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":57},"id":"QskwhXCaMiV2","executionInfo":{"status":"ok","timestamp":1671329490718,"user_tz":180,"elapsed":8,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"9951a20f-a3b9-4262-8f42-2a2909c46563"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":[" ⎛ 2⎞\n"," ⎝x ⎠\n","ℯ    \n","─────\n","  2  "],"image/png":"iVBORw0KGgoAAAANSUhEUgAAADEAAAAdCAYAAAAD8oRRAAAABHNCSVQICAgIfAhkiAAAAxtJREFUWIXt2FuIVVUYB/DfOGY+RA+WGURKT5EoPhgIFWVKopjV8YL6kD75UKQIFUNDwaGS5iXKl/RB1KynYioRtQbxVigownRBgoqsoMmwoIcu6nh5+Na23Z49nnHOPmOIfzisfdb69rfW/7uuc9rq9boW4k68g9vQj1fwftWbjK5aYQH9WIte3I5j2IU/q9yk1ST60gd+wSmMUzGJUVUqa4DpaMdPVStutScyjMM2rGqF8pHwxI34CF041IoNmiWxGKcxKTe3Ht9hAtqwFXtFlWoJmiXRjS/xYvr+HJZjLk7ifizFE6JC9WJqk3sOQLM5cQGd2Cms34nZ+Catf2bohpqAn/EW6qhhviB9B84Ig21Jn/PZi1Ukdg+O4lUsSM/DweOC8AdYgg2iPO/Dj4LkQmzCvCRzgWpIzMI0Ef8nm9BTw284mHQ9Jjx8PifTiSNYJAh1M7irF2G3aE5nRHh0ijqfxzR8iNWiAr02TAI3C2PswDlRCHYUCBANc2N6nplNFj3RjnexDN+Ke85p4b51uBsrk+wkQfR1bBYW+iIp33+FJB7FGGGQRjibxv5souiJ9YlAF+7BU+LuM0XU+BWYLJrXx8JaL6d3vxKkh+ONmriK9DSQG53OIO1/aTLDDDyN7Xih8PJZvI37ktwWQbKIpUM9dQ5jRUnejX8ayHYJg+7CJ9lknsRqkVB/iRJXxJQ0Vt3lH8FNGofSGjyLr/FkfiFPYk4alzdQ9sMVHHAoWCiKx87LyDwjQv246EO/5xczEmMxXpS3hyo+5OXQLnrLXvwxiMxavCFybjZ+LQpkodGWxlurPWNDPIhbDB5KHYJALx5WQoB/SfwtyuNk4d4yPGBgn2gWNdELtpesvSQS+ZjwwKnBlORz4nkRl93YI0iNEveW6bgBEys4eIY2cTE8bGCnXylK9zl8KpK6iBPihvwfEj2ihHYIq88UcdonSL1Xzdkv4V7xR8KbJWt3pbFd5EQZDighQVzeFjd/viGhlsayfKgrL/OlGMnf2EXU8Dm+b1bRSP3GLkNZxx8WrqYnKsN1Ev8XXBMkLgJOkp9JIinMyQAAAABJRU5ErkJggg==\n","text/latex":"$\\displaystyle \\frac{e^{x^{2}}}{2}$"},"metadata":{},"execution_count":131}]},{"cell_type":"code","source":["f = sqrt(1-x**2)\n","I2 = integrate(f, (x, -1, 1)) # Integral definida\n","I2"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":47},"id":"5mUBwQXZEhYl","executionInfo":{"status":"ok","timestamp":1671329493047,"user_tz":180,"elapsed":381,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"bb32fdca-06ef-4b75-aae9-7188d757c102"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["π\n","─\n","2"],"image/png":"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\n","text/latex":"$\\displaystyle \\frac{\\pi}{2}$"},"metadata":{},"execution_count":132}]},{"cell_type":"code","source":["f = exp(-x)\n","I3 = integrate(f, (x, 0, oo)) # Integral imprópria\n","I3"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":37},"id":"FcJyNgfIFScs","executionInfo":{"status":"ok","timestamp":1669938440695,"user_tz":180,"elapsed":11,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"3ebf6271-af39-4663-bdc6-d7fee09d5ddb"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["1"],"text/latex":"$\\displaystyle 1$"},"metadata":{},"execution_count":21}]},{"cell_type":"code","source":["f = 1/x**2\n","I4 = integrate(f, (x, -1, 1))\n","I4"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":37},"id":"0WntrAcLFo8j","executionInfo":{"status":"ok","timestamp":1669938491010,"user_tz":180,"elapsed":371,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"686998b6-c104-41de-fe87-39447187602b"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["oo"],"text/latex":"$\\displaystyle \\infty$"},"metadata":{},"execution_count":22}]},{"cell_type":"code","source":["f = x**2 * y**3\n","I5 = integrate(f, (y, x**2, x), (x, 0, 1)) # Integral com duas variáveis\n","I5"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":51},"id":"esuwzyHtHqe7","executionInfo":{"status":"ok","timestamp":1669939515587,"user_tz":180,"elapsed":11,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"a64e0519-5a35-414f-a2ad-33b61c8aaddc"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["1/77"],"text/latex":"$\\displaystyle \\frac{1}{77}$"},"metadata":{},"execution_count":30}]},{"cell_type":"code","source":["f = exp(x + y + z)\n","I6 = integrate(f, (x, 0, 1), (y, 0, 1), (z, 0, 1))\n","I6"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":38},"id":"p-j3UI4wJ0bY","executionInfo":{"status":"ok","timestamp":1670103084474,"user_tz":180,"elapsed":672,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"76ca4355-0507-47a1-dce7-82f73160c575"},"execution_count":null,"outputs":[{"output_type":"execute_result","data":{"text/plain":["-exp(2) - 1 + 2*E + E*(-2*E + 1 + exp(2))"],"text/latex":"$\\displaystyle - e^{2} - 1 + 2 e + e \\left(- 2 e + 1 + e^{2}\\right)$"},"metadata":{},"execution_count":3}]},{"cell_type":"markdown","source":["---\n","---\n","# USANDO SYMPY PARA VERIFICAR CONVEXIDADE DE FUNÇÕES\n","---\n","---\n","\n","<p align=justify>\n","&emsp; Uma coisa interessante de ver em PNL é se uma dada função é estritamente convexa, convexa, côncava, estritamente côncava ou nem côncava nem convexa. E a forma mais geral de verificar a convexidade de uma função é a partir da seguinte regra:<br>\n","&emsp; Sendo $f(X = (x_1, x_2,..., x_n)):\\mathbb{R}^{n} \\rightarrow \\mathbb{R}$, a sua matriz Hessiana $H[f(X)]$ será dada por:\n","<br>\n","<br>\n","$$\n","H[f(X)] = D^{2}f(X) = \\begin{bmatrix}\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1^{2}} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1 \\partial x_2} &\n","  \\dots &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1\\partial x_n} \\\\\n","  \\frac{\\partial^{2} f(X)}{\\partial x_2\\partial x_1} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_2^{2}} &\n","  \\dots &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1\\partial x_n} \\\\\n","  \\vdots & \\vdots & \\ddots & \\vdots \\\\\n","  \\frac{\\partial^{2} f(X)}{\\partial x_n\\partial x_1} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_n\\partial x_2} &\n","  \\dots &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_n^{2}}\n","\\end{bmatrix} \n","$$\n","<br>\n","<br>\n","&emsp; Por sua vez, os menores principais líderes de $H[f(X)]$ serão definidos por:\n","<br>\n","<br>\n","$$\n","\\begin{array}{lcl}\n","H_1 & = & \\begin{bmatrix}\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1^{2}}\n","\\end{bmatrix} \\\\\n","H_2 & = & \\begin{bmatrix}\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1^{2}} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1 \\partial x_2} \\\\\n","  \\frac{\\partial^{2} f(X)}{\\partial x_2\\partial x_1} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_2^{2}}\n","\\end{bmatrix} \\\\\n","H_3 & = & \\begin{bmatrix}\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1^{2}} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1 \\partial x_2} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1 \\partial x_3}\\\\\n","  \\frac{\\partial^{2} f(X)}{\\partial x_2\\partial x_1} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_2^{2}} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_2\\partial x_3} \\\\\n","  \\frac{\\partial^{2} f(X)}{\\partial x_3\\partial x_1} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_3\\partial x_2} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_3^{2}}\n","\\end{bmatrix} \\\\\n","& \\vdots & \\\\\n","H_{k \\le n} & = & \\begin{bmatrix}\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1^{2}} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1 \\partial x_2} &\n","  \\dots &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1\\partial x_k} \\\\\n","  \\frac{\\partial^{2} f(X)}{\\partial x_2\\partial x_1} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_2^{2}} &\n","  \\dots &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_1\\partial x_k} \\\\\n","  \\vdots & \\vdots & \\ddots & \\vdots \\\\\n","  \\frac{\\partial^{2} f(X)}{\\partial x_k\\partial x_1} &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_k\\partial x_2} &\n","  \\dots &\n","  \\frac{\\partial^{2} f(X)}{\\partial x_k^{2}}\n","\\end{bmatrix}\n","\\end{array}\n","$$\n","<br>\n","<br>\n","&emsp; E as determinantes $|H_1|, |H_2|, \\dots, |H_n|$ dos menores principais líderes determinarão se a matriz $H[f(X)]$ é positiva definida, positiva semidefinida, negativa definida, negativa semidefinida ou indefinida segundo os seguintes casos:\n","<ol align=justify>\n","<li>$H[f(X)]$ é <b>positiva definida</b> se, e somente se, as determinantes de todos os seus menores principais líderes forem positivas.</li>\n","<li>$H[f(X)]$ é <b>positiva semidefinida</b> se, e somente se, as determinantes de todos os seus menores principais líderes forem positivas ou nulas.</li>\n","<li>$H[f(X)]$ é <b>negativa definida</b> se, e somente se, as determinantes de todos os seus menores principais líderes de ordem ímpar forem negativas e as de ordem par forem positivas.</li>\n","<li>$H[f(X)]$ é <b>negativa semidefinida</b> se, e somente se, as determinantes de todos os seus menores principais líderes de ordem ímpar forem negativas ou nulas e as de ordem par forem positivas ou nulas.</li>\n","<li>$H[f(X)]$ é <b>indefinida</b> se não cair em nenhum dos casos anteriores.</li>\n","</ol>\n","&emsp; Finalmente, dada todas as definições até aqui, a convexidade de $f(X^{*})$, num dado ponto crítico $X^{*}=(x_1^*, x_2^*, \\dots, x_n^*)$, será dada conforme os casos a seguir:\n","<ol align=justify>\n","<li>$f(X^{*})$ será estritamente convexa se, e somente se, $H[f(X^{*})]$ for positiva definida.</li>\n","<li>$f(X^{*})$ será convexa se, e somente se, $H[f(X^{*})]$ for positiva semidefinida.</li>\n","<li>$f(X^{*})$ será estritamente côncava se, e somente se, $H[f(X^{*})]$ for negativa definida.</li>\n","<li>$f(X^{*})$ será côncava se, e somente se, $H[f(X^{*})]$ for negativa semidefinida.</li>\n","<li>$f(X^{*})$ não será côncava nem convexa se, e somente se, $H[f(X^{*})]$ for indefinida.</li>\n","</ol>\n","&emsp; A tabela a seguir resume todos os casos:\n","</p>\n","\n","<center>\n","\n","|Condição|Matriz $H[f(X^{*})]$|$f(X^{*})$|\n","|:-------|:-------------------|:---------|\n","|$|H_1|>0, |H_2|>0, |H_3|>0, \\dots$|Positiva definida|Estritamente convexa|\n","|$|H_1| \\ge 0, |H_2| \\ge 0, |H_3|\\ge 0, \\dots$|Positiva semidefinida|Convexa|\n","|$|H_1|<0, |H_2|>0, |H_3|<0, \\dots$|Negativa definida|Estritamente Côncava|\n","|$|H_1| \\le 0, |H_2| \\ge 0, |H_3| \\le 0, \\dots$|Negativa semidefinida|Côncava|\n","|Nenhuma das condições anteriores é satisfeita|Indefinida|Nem côncava nem convexa|\n","\n","</center>\n","\n","<p align=justify>\n","&emsp; E a partir dos conhecimentos apresentados até aqui, utilizando a biblioteca SymPy, vamos construir uma função em Python que determinará a convexidade de uma expressão <i>f</i> a partir de sua matriz Hessiana e do cálculo da determinante dos seus menores principais líderes, utilizando a função <code>hessian()</code> e o método <code>.det()</code>, respectivamente.\n","</p>"],"metadata":{"id":"Ypc6WzWfKdmZ"}},{"cell_type":"code","source":["def convexidade(funcao, ponto: dict): # NOTA: Essa function só funciona se a determinante tiver APENAS CONSTANTES.\n","\n","  from sympy import hessian, pprint, Function # Importando a função para calcular a matriz Hessiana\n","  f = Function('f')\n","  f = funcao\n","\n","  vars = [var for var in f.free_symbols] # Lista de variáveis de f.\n","  H = hessian(f, vars) # Matriz Hessiana\n","  print(\"Matriz Hessiana\")\n","  pprint(H)\n","\n","  Dets = [] # Array que armazenará as determinantes dos menores principais líderes\n","  for i in range(1, len(vars)+1):\n","    Dets.append(H[:i,:i].det().subs(ponto))\n","\n","  estritamente_convexa = [Dets[i] >  0 for i in range(len(Dets))]\n","  convexa              = [Dets[i] >= 0 for i in range(len(Dets))]\n","  estritamente_concava = [Dets[i] <  0 if i%2==0 else Dets[i] >  0 for i in range(len(Dets))]\n","  concava              = [Dets[i] <= 0 if i%2==0 else Dets[i] >= 0 for i in range(len(Dets))]\n","\n","  if all(estritamente_convexa):\n","    return f\"A função é estritamente convexa em {ponto}.\"\n","  elif all(convexa):\n","    return f\"A função é convexa em {ponto}.\"\n","  elif all(estritamente_concava):\n","    return f\"A função é estritamente côncava em {ponto}.\"\n","  elif all(concava):\n","    return f\"A função é côncava em {ponto}.\"\n","  else:\n","    return f\"A função não é convexa nem côncava em {ponto}.\""],"metadata":{"id":"Zr2-qRLLEi02"},"execution_count":null,"outputs":[]},{"cell_type":"code","source":["from sympy import symbols\n","\n","x, y, z, w = symbols('x, y, z, w')\n","expressao = x**2 + 4*y**3 + x*y + 4*x*z + y*z/10 + 3*z**2 + w*2 + 19*w*z + 2*w*y\n","ponto_critico = {'x': 0, 'y': 0, 'z': 1, 'w': 1}\n","\n","convexidade(f, ponto_critico)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":174},"id":"ok0gGwURs7QD","executionInfo":{"status":"ok","timestamp":1670117745439,"user_tz":180,"elapsed":259,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"859e0e18-b12b-4f84-cabd-397632c9df84"},"execution_count":null,"outputs":[{"output_type":"stream","name":"stdout","text":["Matriz Hessiana\n","⎡0    19    2    0⎤\n","⎢                 ⎥\n","⎢19   6    1/10  4⎥\n","⎢                 ⎥\n","⎢2   1/10   8    1⎥\n","⎢                 ⎥\n","⎣0    4     1    2⎦\n"]},{"output_type":"execute_result","data":{"text/plain":["\"A função não é convexa nem côncava em {'x': 0, 'y': 0, 'z': 1, 'w': 1}.\""],"application/vnd.google.colaboratory.intrinsic+json":{"type":"string"}},"metadata":{},"execution_count":36}]},{"cell_type":"markdown","source":["---\n","---\n","# USANDO SYMPY PARA RESOLVER CONDIÇÕES DE KKT\n","---\n","---\n","\n","<p align=justify>\n","&emsp; As condições de KKT representam o método mais abrangente de resolver problemas de PNL por poderem lidar com restrições de igualdade e de desigualdade, cuja formulação é dada a seguir:\n","<br>\n","&emsp; Seja $f(x = (x_1, x_2,\\dots, x_n)):\\mathbb{R}^{n} \\to \\mathbb{R}$ uma função escalar de classe $C^1$ cujo problema de otimização é dado por:\n","<br>\n","<br>\n","$$\n","\\text{Max } z=f(x)\n","$$\n","<br>\n","$$\n","\\begin{equation}\n","  \\begin{array}{ll}\n","    \\text{Sujeito a: } & \\\\\n","    & H = \\left\\{\\begin{array}{l}\n","          h_1(x)=0 \\\\\n","          h_2(x)=0 \\\\\n","          \\vdots \\\\\n","          h_m(x)=0\n","          \\end{array}\\right. \\\\\n","    & G = \\left\\{\\begin{array}{l}\n","          g_1(x) \\le 0 \\\\\n","          g_2(x) \\le 0 \\\\\n","          \\vdots \\\\\n","          g_k(x) \\le 0\n","          \\end{array}\\right.\n","  \\end{array}\n","\\end{equation}\n","$$\n","<br>\n","<br>\n","&emsp; Onde os conjuntos $H$ e $G$ são restrições de igualdade e desigualdade, respectivamente, definidas por funções escalares $h, g: \\mathbb{R}^n \\to \\mathbb{R}$, taambém de classe $C^1$, que cumprem as codições de regularidade em todos os pontos, isto é, a jacobiana do conjunto de restrições ativas $R=H \\cup G$ é linearmente indenpendente. Em termos mais formais, o posto de $J_R(x)$ é igual a $m+k$.\n","<br>\n","<br>\n","$$\n","J_R(x)=\\begin{bmatrix}\n","  \\nabla h_1(x) \\\\\n","  \\nabla h_2(x) \\\\\n","  \\vdots \\\\\n","  \\nabla h_m(x) \\\\\n","  \\nabla g_1(x) \\\\\n","  \\nabla g_2(x) \\\\\n","  \\vdots \\\\\n","  \\nabla g_k(x) \\\\\n","\\end{bmatrix}\n","=\\begin{bmatrix}\n","  \\frac{\\partial h_1(x)}{\\partial x_1} &\n","  \\frac{\\partial h_1(x)}{\\partial x_2} &\n","  \\dots &\n","  \\frac{\\partial h_1(x)}{\\partial x_n} \\\\\n","  \\frac{\\partial h_2(x)}{\\partial x_1} &\n","  \\frac{\\partial h_2(x)}{\\partial x_2} &\n","  \\dots &\n","  \\frac{\\partial h_2(x)}{\\partial x_n} \\\\\n","  \\vdots & \\vdots & \\ddots & \\vdots \\\\\n","  \\frac{\\partial h_m(x)}{\\partial x_1} &\n","  \\frac{\\partial h_m(x)}{\\partial x_2} &\n","  \\dots &\n","  \\frac{\\partial h_m(x)}{\\partial x_n} \\\\\n","  \\frac{\\partial g_1(x)}{\\partial x_1} &\n","  \\frac{\\partial g_1(x)}{\\partial x_2} &\n","  \\dots &\n","  \\frac{\\partial g_1(x)}{\\partial x_n} \\\\\n","  \\frac{\\partial g_2(x)}{\\partial x_1} &\n","  \\frac{\\partial g_2(x)}{\\partial x_2} &\n","  \\dots &\n","  \\frac{\\partial g_2(x)}{\\partial x_n} \\\\\n","  \\vdots & \\vdots & \\ddots & \\vdots \\\\\n","  \\frac{\\partial g_k(x)}{\\partial x_1} &\n","  \\frac{\\partial g_k(x)}{\\partial x_2} &\n","  \\dots &\n","  \\frac{\\partial g_k(x)}{\\partial x_n}\n","\\end{bmatrix}\n","\\ne \\overrightarrow{0}, \\forall  X \\in R\n","$$\n","<br>\n","<br>\n","&emsp; Então existem constantes $\\lambda_1, \\lambda_2, \\dots, \\lambda_m$ e $\\mu_1, \\mu_2, \\dots, \\mu_k$, chamados de multiplicadores de KKT, que satisfazem um conjunto de restrições que permitem encontrar os pontos críticos do Lagrangeano $L(x_1, \\dots, x_n, \\lambda_1, \\dots, \\lambda_m, \\mu_1, \\dots, \\mu_k): \\mathbb{R}^{n+m_+k} \\to \\mathbb{R}$, que são os candidatos a maximizarem $f(x)$.\n","<br>\n","&emsp; Tal Lagrangeano é dado por:\n","<br>\n","$$\n","L = f(x) - \\sum_{i=1}^{m} \\lambda_{i} h_{i}(x)-\\sum_{j=1}^{k} \\mu_{j} g_{j}(x)\n","$$\n","<br>\n","&emsp; Cujo conjunto de restrições, conhecido por <b>condições de KKT</b>, as quais permitem encontrar os pontos candidatos a maximizarem f(x), é dado por.\n","<br>\n","<br>\n","$$\n","\\left\\{\\begin{array}{lll}\n","  \\nabla f(x) = \\sum_{i=1}^{m} \\lambda_{i} \\nabla h_i(x) + \\sum_{j=1}^{k} \\mu_{j} \\nabla g_j(x) & & (1) \\\\\n","  h_i(x) = 0, & \\forall i=1,\\dots,m & (2) \\\\\n","  g_j(x) \\le 0, & \\forall j=1, \\dots, k & (3) \\\\\n","  \\mu_j \\cdot g_j(x)=0, & \\forall j=1,\\dots,k & (4) \\\\\n","  \\mu_j \\ge 0, & \\forall j=1, \\dots, k & (5)\n","  \\end{array}\\right.\n","$$\n","<br>\n","&emsp; Note-se que as restrições $(1)$ e $(2)$ são obtidas ao se calcular $\\frac{\\partial L(x, \\lambda, \\mu)}{\\partial x}$ e $\\frac{\\partial L(x, \\lambda, \\mu)}{\\partial \\lambda}$, respectivamente. E que caso $k=0$, isto é, não houvesse restrições de desigualdade, as condições KKT se transformam nas condições de Lagrange e os multiplicadores KKT são chamados multiplicadores de Lagrange.\n","<br>\n","&emsp; No intuito de resolver os problemas de otimização não linear, foi implementado a função <code><font color='#DCDCAA'>KKT</font>()</code>, utilizando a biblioteca SymPy já apresentada, em que são passadas como argumentos para essa função <i>strings</i> para a função $f(x)$ a ser otimizada e as restrições de igualdade e desigualdade já no formato $h(x)=0$ e $g(x) \\le 0$, ambas passadas como duas listas de <i>strings</i>. Caso se busque minimizar ao invés de maximizar a função, o parâmetro booleano <code>max</code> deve ser definida como <code><font color='#569CD6'>False</font></code>. Ao final, a função <code><font color='#DCDCAA'>KKT</font>()</code> retorna uma lista cujo primeiro item corresponde ao valor ótimo encontrado para $f(x)$ e os itens restantes são os pontos $p$ encontrados que maximizam $f(x)$.\n","<br>\n","&emsp; Além disso, foi criado a função <code><font color='#DCDCAA'>print_solucao_KKT</font>()</code> para dar uma melhor visualização da solução encontrada pela função <code><font color='#DCDCAA'>KKT</font>()</code> quando imprimá-la na tela.\n","</p>"],"metadata":{"id":"ht2wzoLKx0Ac"}},{"cell_type":"code","source":["def KKT(f: str, h: list = [], g: list = [], max=True) -> list:\n","\n","  # NOTA: As expressões contidas em h e g devem ser tais que h=0 e g<=0\n","\n","  import sympy as sp # Importando a biblioteca SymPy\n","  m = len(h) # Quantidade de restrições de igualdade\n","  k = len(g) # Quantidade de restrições de desigualdade\n","\n","  if max == True:\n","    f = sp.parse_expr(f) # Construindo a função f pela string passada como parâmetro\n","  else:\n","    f = sp.parse_expr(f'-({f})')\n","  \n","  vars = f.free_symbols # Obtendo o conjunto de variáveis\n","  pars = list(vars) # Lista de parâmetros\n","\n","  if m > 0:\n","    for i in range(m):\n","      exec(f'h{i+1} = sp.Symbol(\"h_{i+1}\")')\n","      exec(f'h{i+1} = sp.parse_expr(h[{i}])') # Cria a restrição de igualdade h_i\n","      exec(f'l{i+1} = sp.Symbol(\"\\\\lambda_{i+1}\")') # Cria o parâmetro lambda_i\n","      exec(f'pars.append(l{i+1})') # Adiciona lambda_i à lista de parâmetros\n","\n","  if k > 0:\n","    for j in range(k):\n","      exec(f'g{j+1} = sp.Symbol(\"g_{j+1}\")')\n","      exec(f'g{j+1} = sp.parse_expr(g[{j}])') # Cria a restrição de desigualdade g_i\n","      exec(f'u{j+1} = sp.Symbol(\"\\\\mu_{j+1}\", negative=False)') # Cria o parâmetro mu_i\n","      exec(f'p{j+1} = sp.Symbol(\"p_{j+1}\", negative=False)') # Cria a folga p_i\n","      exec(f'pars.append(u{j+1})') # Adiciona mu_i à lista de parâmetros\n","      exec(f'pars.append(p{j+1})') # Adiciona p_i à lista de parâmetros\n","\n","  # Criando o Lagrangeano\n","  L = sp.Symbol('L')\n","  L = f\n","\n","  if m > 0:\n","    for i in range(m):\n","      locais_dict = dict() # Dicionário de variáveis locais a ser passado para exec()\n","      exec(f'L = L - l{i+1}*h{i+1}',\n","           {'L': L,\n","            f'l{i+1}': sp.Symbol(f'\\\\lambda_{i+1}'),\n","            f'h{i+1}': sp.parse_expr(h[i])\n","           },\n","           locais_dict\n","          )\n","      L = locais_dict['L'] # Adicionando a parcela lambda_j * h_j ao Lagrangeano\n","\n","  if k > 0:\n","    for j in range(k):\n","      locais_dict = dict() # Dicionário de variáveis locais a ser passado para exec()\n","      exec(f'L = L - u{j+1}*g{j+1}',\n","           {'L': L,\n","            f'u{j+1}': sp.Symbol(f'\\\\mu_{j+1}', negative=False),\n","            f'g{j+1}': sp.parse_expr(g[j])\n","           },\n","           locais_dict\n","          )\n","      L = locais_dict['L'] # Adicionando a parcela mu_i * g_i ao Lagrangeano\n","\n","  # Criando as condições de KKT a partir do Lagrangeano\n","  R = [] # Conjunto de restrições\n","\n","  for var in vars:\n","    R.append(sp.diff(L, var)) # Adiciona a derivada do Lagrangeano em relação à variável 'var'\n","\n","  if m > 0:\n","    for i in range(m):\n","      exec(f'R.append(h{i+1})')\n","  \n","  if k > 0:\n","    for j in range(k):\n","      exec(f'R.append(u{j+1} * g{j+1})')\n","      exec(f'R.append(g{j+1} + p{j+1})')\n","\n","  # Resolvendo as condições de KKT\n","  crit_lagr = sp.solve(R, pars, dict=True) # Pontos críticos do Lagrangeano\n","  pontos_f = [{var: ponto.get(var) for var in ponto if var in vars} for ponto in crit_lagr] # pontos no domínio de f\n","\n","  # Buscando o(s) ponto(s) que maximiza(m) f\n","\n","  maximo = -float('inf') # infinito negativo\n","  solucao = list()\n","\n","  for ponto in pontos_f:\n","    if f.subs(ponto) > maximo:\n","      maximo = f.subs(ponto)\n","  \n","  if max:\n","    solucao.append({'valor ótimo': maximo})\n","  else:\n","    solucao.append({'valor ótimo': -maximo})\n","  \n","  for ponto in pontos_f:\n","    if f.subs(ponto) == maximo:\n","      solucao.append(ponto)\n","\n","  return solucao\n","\n","\n","\n","def print_solucao_KKT(solucao: list, vars: list):\n","\n","  from IPython.display import display, Latex\n","  from sympy import latex, Symbol\n","\n","  valor_otimo = latex(solucao[0]['valor ótimo'])\n","\n","  display(Latex('\\\\text{Valor ótimo: } f(p^*)=' + valor_otimo))\n","  print('- '*50)\n","  display(Latex('\\\\text{Ponto(s)}'))\n","\n","  for i in range(1, len(solucao)):\n","\n","    ponto = f\"$p_{i}({vars[0]}\"\n","    for j in range(1, len(vars)):\n","      ponto += f\",{vars[j]}\"\n","\n","    ponto += f\")=({latex(solucao[i][Symbol(vars[0])])}\"\n","    for j in range(1, len(vars)):     \n","      ponto += f\",{latex(solucao[i][Symbol(vars[j])])}\"\n","    ponto += \")$\"\n","\n","    display(Latex(ponto))"],"metadata":{"id":"qkx2NoYAAKNc"},"execution_count":null,"outputs":[]},{"cell_type":"markdown","source":["<p align=justify>\n","&emsp; Com a função <code><font color='#DCDCAA'>KKT</font>()</code> implementada, vamos resolver alguns exemplos de PNL para demonstrar a sua aplicação.\n","<br>\n","<br>\n","<hr/>\n","&emsp; $\\text{Exemplo 1}$\n","<hr/>\n","<br>\n","$$\n","\\text{Max } f(x_1,x_2)=24x_1 - x_1^2 + 10x_2 - 2x_2^2\n","$$\n","<br>\n","$$\n","\\begin{array}{ll}\n","  \\text{Sujeito a: } & \\\\\n","    & x_1 \\le 8 \\\\\n","    & x_2 \\le 7 \\\\\n","    & x_1 \\ge 0 \\\\\n","    & x_2 \\ge 0\n","\\end{array}\n","$$\n","</p>"],"metadata":{"id":"d2kg4OazWgFR"}},{"cell_type":"code","source":["solucao1 = KKT('24*x_1 - x_1**2 + 10*x_2 - 2*x_2**2',\n","              g=['x_1 - 8',\n","                 'x_2 - 7',\n","                 '-x_1',\n","                 '-x_2'\n","                ]\n","              )\n","\n","print_solucao_KKT(solucao1, ['x_1', 'x_2'])"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":97},"id":"acv7_66SA4P1","executionInfo":{"status":"ok","timestamp":1671328983537,"user_tz":180,"elapsed":4,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"d4b1e3a1-416e-4e34-8f01-a3ce5537d6f6"},"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Valor ótimo: } f(p^*)=\\frac{281}{2}"},"metadata":{}},{"output_type":"stream","name":"stdout","text":["- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \n"]},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Ponto(s)}"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_1(x_1,x_2)=(8,\\frac{5}{2})$"},"metadata":{}}]},{"cell_type":"markdown","source":["<p align=justify>\n","<hr/>\n","&emsp; $\\text{Exemplo 2}$\n","<hr/>\n","<br>\n","$$\n","\\text{Max } f(x_1,x_2)=x_1^2x_2\n","$$\n","<br>\n","$$\n","\\begin{array}{ll}\n","  \\text{Sujeito a: } & \\\\\n","    & 2x_1^2 + x_2^2 = 3\n","\\end{array}\n","$$\n","</p>"],"metadata":{"id":"PinrqKuzftri"}},{"cell_type":"code","source":["solucao2 = KKT('x_1**2 * x_2',\n","               h=['2*x_1**2 + x_2**2 - 3']\n","              )\n","\n","print_solucao_KKT(solucao2, ['x_1', 'x_2'])"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":119},"id":"GuvmOjd4fvN5","executionInfo":{"status":"ok","timestamp":1671328979619,"user_tz":180,"elapsed":6,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"33a023de-378b-465b-c5d8-caa336444041"},"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Valor ótimo: } f(p^*)=1"},"metadata":{}},{"output_type":"stream","name":"stdout","text":["- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \n"]},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Ponto(s)}"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_1(x_1,x_2)=(-1,1)$"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_2(x_1,x_2)=(1,1)$"},"metadata":{}}]},{"cell_type":"markdown","source":["<p align=justify>\n","<hr/>\n","&emsp; $\\text{Exemplo 3}$\n","<hr/>\n","<br>\n","$$\n","\\text{Max } f(x_1,x_2)=2x_1 + 3x_2 - x_1^2 - x_2^2\n","$$\n","<br>\n","$$\n","\\begin{array}{ll}\n","  \\text{Sujeito a: } & \\\\\n","    & x_1 + x_2 \\le 2 \\\\\n","    & x_1 \\ge 0 \\\\\n","    & x_2 \\ge 0\n","\\end{array}\n","$$\n","</p>"],"metadata":{"id":"vxTyS5w9aw4g"}},{"cell_type":"code","source":["solucao3 = KKT('2*x_1 + 3*x_2 - x_1**2 - x_2**2',\n","               g=['x_1 + x_2 - 2',\n","                  '-x_1',\n","                  '-x_2'\n","                 ]\n","               )\n","\n","print_solucao_KKT(solucao3, ['x_1', 'x_2'])"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":97},"id":"2n5vJP2vRudw","executionInfo":{"status":"ok","timestamp":1671328797047,"user_tz":180,"elapsed":501,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"c43959c1-6bf2-42d6-aa71-90cdeafd99cc"},"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Valor ótimo: } f(p^*)=\\frac{25}{8}"},"metadata":{}},{"output_type":"stream","name":"stdout","text":["- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \n"]},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Ponto(s)}"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_1(x_1,x_2)=(\\frac{3}{4},\\frac{5}{4})$"},"metadata":{}}]},{"cell_type":"markdown","source":["<p align=justify>\n","<hr/>\n","&emsp; $\\text{Exemplo 4}$\n","<hr/>\n","<br>\n","$$\n","\\text{Max } f(x,y)=x^2+y^2\n","$$\n","<br>\n","$$\n","\\begin{array}{ll}\n","  \\text{Sujeito a: } & \\\\\n","    & xy = 1\n","\\end{array}\n","$$\n","</p>"],"metadata":{"id":"_1mD8T_7cAzC"}},{"cell_type":"code","source":["solucao4 = KKT('x**2 + y**2',\n","               h=['x*y - 1']\n","              )\n","\n","print_solucao_KKT(solucao4, ['x', 'y'])"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":119},"id":"P7TMudjWT28b","executionInfo":{"status":"ok","timestamp":1671328812761,"user_tz":180,"elapsed":6,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"a72d3b86-b5fd-42a7-b409-ac80f109cf72"},"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Valor ótimo: } f(p^*)=2"},"metadata":{}},{"output_type":"stream","name":"stdout","text":["- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \n"]},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Ponto(s)}"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_1(x,y)=(-1,-1)$"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_2(x,y)=(1,1)$"},"metadata":{}}]},{"cell_type":"markdown","source":["<p align=justify>\n","<hr/>\n","&emsp; $\\text{Exemplo 5}$\n","<hr/>\n","<br>\n","$$\n","\\text{Max } f(x,y)=3x+4y\n","$$\n","<br>\n","$$\n","\\begin{array}{ll}\n","  \\text{Sujeito a: } & \\\\\n","    & x^2 + y^2 \\le 4 \\\\\n","    & x \\ge 1\n","\\end{array}\n","$$\n","</p>"],"metadata":{"id":"mA4pn-bTh3KD"}},{"cell_type":"code","source":["solucao5 = KKT('3*x + 4*y',\n","               g=['x**2 + y**2 - 4', '1 - x']\n","              )\n","\n","print_solucao_KKT(solucao5, ['x', 'y'])"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":97},"id":"r1IXAxlliGC6","executionInfo":{"status":"ok","timestamp":1671328829597,"user_tz":180,"elapsed":5,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"78df43f0-9090-464a-cfb0-01b563554427"},"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Valor ótimo: } f(p^*)=10"},"metadata":{}},{"output_type":"stream","name":"stdout","text":["- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \n"]},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Ponto(s)}"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_1(x,y)=(\\frac{6}{5},\\frac{8}{5})$"},"metadata":{}}]},{"cell_type":"markdown","source":["<p align=justify>\n","<hr/>\n","&emsp; $\\text{Exemplo 6}$\n","<hr/>\n","<br>\n","$$\n","\\text{Max } f(x,y,z)=xyz\n","$$\n","<br>\n","$$\n","\\begin{array}{ll}\n","  \\text{Sujeito a: } & \\\\\n","    & x^2 + y^2 = 1 \\\\\n","    & x + z = 1\n","\\end{array}\n","$$\n","</p>"],"metadata":{"id":"CTXHiyTZdZPY"}},{"cell_type":"code","source":["solucao6 = KKT('x*y*z',\n","               h=['x**2 + y**2 - 1', 'x + z - 1']\n","              )\n","\n","print_solucao_KKT(solucao6, ['x', 'y', 'z'])"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":118},"id":"cTXBSz-ZeHLK","executionInfo":{"status":"ok","timestamp":1671328849566,"user_tz":180,"elapsed":915,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"0042f08b-f6fd-4f13-82c9-921a66b12f7e"},"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Valor ótimo: } f(p^*)=\\frac{\\left(1 - \\sqrt{13}\\right) \\left(- \\frac{\\sqrt{13}}{6} - \\frac{1}{6}\\right) \\left(\\frac{\\sqrt{13}}{6} + \\frac{7}{6}\\right) \\sqrt{\\sqrt{13} + 16}}{18}"},"metadata":{}},{"output_type":"stream","name":"stdout","text":["- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \n"]},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Ponto(s)}"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_1(x,y,z)=(- \\frac{\\sqrt{13}}{6} - \\frac{1}{6},\\frac{\\left(1 - \\sqrt{13}\\right) \\sqrt{\\sqrt{13} + 16}}{18},\\frac{\\sqrt{13}}{6} + \\frac{7}{6})$"},"metadata":{}}]},{"cell_type":"markdown","source":["<p align=justify>\n","<hr/>\n","&emsp; $\\text{Exemplo 7}$\n","<hr/>\n","<br>\n","$$\n","\\text{Max } f(x,y,z)=8xyz\n","$$\n","<br>\n","$$\n","\\begin{array}{ll}\n","  \\text{Sujeito a: } & \\\\\n","    & \\frac{x^2}{9} + \\frac{y^2}{4} + z^2 = 1\n","\\end{array}\n","$$\n","</p>"],"metadata":{"id":"9qLOQOTNbaLp"}},{"cell_type":"code","source":["solucao7 = KKT('8*x*y*z',\n","               h=['x**2/9 + y**2/4 + z**2 - 1']\n","              )\n","\n","print_solucao_KKT(solucao7, ['x', 'y', 'z'])"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":174},"id":"32w1mvYOSkgx","executionInfo":{"status":"ok","timestamp":1671328892345,"user_tz":180,"elapsed":471,"user":{"displayName":"Rodrigo Celso de Lima Porto","userId":"00317348114914252656"}},"outputId":"5014b787-025c-4181-c555-22233c12b16a"},"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Valor ótimo: } f(p^*)=\\frac{16 \\sqrt{3}}{3}"},"metadata":{}},{"output_type":"stream","name":"stdout","text":["- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \n"]},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"\\text{Ponto(s)}"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_1(x,y,z)=(- \\sqrt{3},- \\frac{2 \\sqrt{3}}{3},\\frac{\\sqrt{3}}{3})$"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_2(x,y,z)=(- \\sqrt{3},\\frac{2 \\sqrt{3}}{3},- \\frac{\\sqrt{3}}{3})$"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_3(x,y,z)=(\\sqrt{3},- \\frac{2 \\sqrt{3}}{3},- \\frac{\\sqrt{3}}{3})$"},"metadata":{}},{"output_type":"display_data","data":{"text/plain":["<IPython.core.display.Latex object>"],"text/latex":"$p_4(x,y,z)=(\\sqrt{3},\\frac{2 \\sqrt{3}}{3},\\frac{\\sqrt{3}}{3})$"},"metadata":{}}]}],"metadata":{"colab":{"provenance":[],"collapsed_sections":["O8PqCB7zNsGd"]},"kernelspec":{"display_name":"Python 3 (ipykernel)","language":"python","name":"python3"},"language_info":{"codemirror_mode":{"name":"ipython","version":3},"file_extension":".py","mimetype":"text/x-python","name":"python","nbconvert_exporter":"python","pygments_lexer":"ipython3","version":"3.9.7"}},"nbformat":4,"nbformat_minor":0}