{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# First steps with the nullspace optimizer\n",
    "\n",
    "This notebook proposes you to familiar you with the nullspace optimization algorithm, which in the case of equality constrained optimization, reduces to the flow proposed by Yamashita in \n",
    "\n",
    "*H. Yamashita, A differential equation approach to nonlinear programming. Math. Program.18(1980) 155–168.*\n",
    "\n",
    "The flow reads \n",
    "\n",
    "\\begin{equation}\n",
    "\\newcommand{\\DD}{\\mathrm{D}}\n",
    "\\renewcommand{\\dagger}{T}\n",
    "\\renewcommand{\\x}{x}\n",
    "\\renewcommand{\\I}{I}\n",
    "\\renewcommand{\\g}{g}\n",
    "    \\label{eqn:flow}\n",
    "    \\left\\{\\begin{aligned}\n",
    "            \\dot\\x&={ -\\alpha_J(\\I-\\DD\\g^\\dagger(\\DD\\g\\DD\\g^ \\dagger)^{-1}\\DD\\g(x))\\nabla\n",
    "            J(x) }{ -\\alpha_C\\DD\\g^ \\dagger(\\DD\\g\\DD\\g^ \\dagger)^{-1}\\g(x) }\\\\\n",
    "    \\x(0)&=x_0\n",
    "        \\end{aligned}\\right.\n",
    "\\end{equation}\n",
    "\n",
    "where $J$ is the cost function and $g$ the equality constraint:\n",
    "\\begin{equation}\n",
    "    \\label{eqn:nlsvg}\n",
    "\\begin{aligned}\n",
    "\t\\min_{\\x\\in \\mathbb{R}^n}&  \\quad J(\\x)\\\\\n",
    "\t\\textrm{s.t.} & \\left.\\begin{aligned}\n",
    " \\g(\\x)&=0\n",
    "\t\t\\end{aligned}\\right.\n",
    "\\end{aligned}\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. A first optimization program"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Write an optimization program to solve the constrained minimization problem on the hyperbola:\n",
    "            $$\n",
    "            \\begin{aligned}\n",
    "                \\min_{(x_1,x_2)\\in\\mathbb{R}^{2}} & \\qquad x_1+x_2\\\\\n",
    "                s.t. & \\left.\\begin{aligned}\n",
    "  x_1x_2 &= 1. \n",
    "                \\end{aligned}\\right.\n",
    "            \\end{aligned}\n",
    "$$\n",
    "\n",
    "In order to solve the optimization problem, we use the function `nlspace_solve` which solves the above optimization program and whose prototype is \n",
    "```python\n",
    "from nullspace_optimizer import nlspace_solve\n",
    "\n",
    "# Define problem\n",
    "\n",
    "results = nlspace_solve(problem: Optimizable, params=None, results=None)\n",
    "```\n",
    "The input variables are \n",
    "-  `problem` : an `Optimizable` object described below. This variable contains all the information about the optimization problem to solve (objective and constraint functions, derivatives...)\n",
    "- `params`  : (optional) a dictionary containing algorithm parameters.\n",
    "\n",
    "- `results` : (optional) a previous output of the `nlspace_solve` function.         The optimization will then keep going from the last input of           the dictionary `results['x'][-1]`. This is useful when one needs to restart an optimization after an interruption.\n",
    "\n",
    "The optimization routine `nlspace_solve` returns the dictionary `opt_results` which contains various information about the optimization path, including the values of the optimization variables `results['x']`.\n",
    "    \n",
    "    \n",
    " In our particular optimization test case in $\\mathbb R^n$ with $n=2$, we use the  `EuclideanOptimizable` class which inherits `Optimizable` which simplifies the definition of the optimization program (the inner product is specified by default). \n",
    "We allow the user to specify the initialization in the constructor `__init__`.\n",
    "\n",
    "Fill in the definition below with the right values for solving the above optimization program.\n",
    "   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from nullspace_optimizer import nlspace_solve, EuclideanOptimizable\n",
    "import numpy as np\n",
    "\n",
    "class problem1(EuclideanOptimizable):\n",
    "    def __init__(self,x0):\n",
    "        super().__init__(2)\n",
    "        self.xinit = x0\n",
    "        self.nconstraints = 1\n",
    "        self.nineqconstraints = 0\n",
    "\n",
    "    def x0(self):\n",
    "        return self.xinit\n",
    "\n",
    "    def J(self, x):\n",
    "        return x[0]+x[1]\n",
    "\n",
    "    def dJ(self, x):\n",
    "        return [1, 1]\n",
    "\n",
    "    def G(self, x):\n",
    "        return [x[0]*x[1]-1]\n",
    "\n",
    "    def dG(self, x):\n",
    "        return [[x[1],x[0]]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then write a routine to solve the problem several time with several initializations. Adapt the code below to specify the entries (0.1,0.1), (4.0,0.25), (4,1). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def run_problems():\n",
    "    xinits = ([0.1,0.1], [4.0, 0.25],  [4, 1])\n",
    "    # Write xinits\n",
    "    problems = [problem1(x0=x0) for x0 in xinits]\n",
    "    params = {'dt': 0.1,  'alphaJ':2, 'alphaC':1, 'debug': -1}\n",
    "    return [nlspace_solve(pb, params) for pb in problems]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function nlspace_solve calls the null space optimizer on this problem. Get the result by calling the function run_problems:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "\n",
      "Optimization completed.\n",
      "\u001b[38;5;4m14. J=2 G=[-9.741e-08] H=[]\u001b[0m\n",
      "\n",
      "\n",
      "Optimization completed.\n",
      "\u001b[38;5;4m65. J=2 G=[-5.8e-13] H=[]\u001b[0m\n",
      "\n",
      "\n",
      "Optimization completed.\n",
      "\u001b[38;5;4m50. J=2 G=[-4.101e-13] H=[]\u001b[0m\n"
     ]
    }
   ],
   "source": [
    "results = run_problems()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The variable `results` contains various data about the optimization trajectories, e.g.:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'it': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14],\n",
       " 'x': [[0.1, 0.1],\n",
       "  array([0.2, 0.2]),\n",
       "  array([0.3, 0.3]),\n",
       "  array([0.4, 0.4]),\n",
       "  array([0.5, 0.5]),\n",
       "  array([0.6, 0.6]),\n",
       "  array([0.69999998, 0.69999998]),\n",
       "  array([0.79999998, 0.79999998]),\n",
       "  array([0.89999996, 0.89999996]),\n",
       "  array([0.995, 0.995]),\n",
       "  array([0.99951131, 0.99951131]),\n",
       "  array([0.99995128, 0.99995128]),\n",
       "  array([0.99999513, 0.99999513]),\n",
       "  array([0.99999951, 0.99999951]),\n",
       "  array([0.99999995, 0.99999995])],\n",
       " 'J': [0.2,\n",
       "  0.4,\n",
       "  0.6000000000000001,\n",
       "  0.8,\n",
       "  1.0,\n",
       "  1.2,\n",
       "  1.3999999555910838,\n",
       "  1.5999999555910838,\n",
       "  1.7999999111821676,\n",
       "  1.9900000004934348,\n",
       "  1.999022613112433,\n",
       "  1.99990256517331,\n",
       "  1.9999902586534828,\n",
       "  1.9999990258866995,\n",
       "  1.9999999025888835],\n",
       " 'G': [[-0.99],\n",
       "  [-0.96],\n",
       "  [-0.9099999999999999],\n",
       "  [-0.84],\n",
       "  [-0.75],\n",
       "  [-0.64],\n",
       "  [-0.5100000310862408],\n",
       "  [-0.36000003552713244],\n",
       "  [-0.19000007993604717],\n",
       "  [-0.009974999509032356],\n",
       "  [-0.0009771480662851273],\n",
       "  [-9.743245330362527e-05],\n",
       "  [-9.741322793743734e-06],\n",
       "  [-9.741130632123784e-07],\n",
       "  [-9.741111417493897e-08]],\n",
       " 'H': [[], [], [], [], [], [], [], [], [], [], [], [], [], [], []],\n",
       " 'muls': [array([-10.]),\n",
       "  array([-5.]),\n",
       "  array([-3.33333333]),\n",
       "  array([-2.5]),\n",
       "  array([-2.]),\n",
       "  array([-1.66666667]),\n",
       "  array([-1.42857147]),\n",
       "  array([-1.25000003]),\n",
       "  array([-1.11111117]),\n",
       "  array([-1.00502513]),\n",
       "  array([-1.00048893]),\n",
       "  array([-1.00004872]),\n",
       "  array([-1.00000487]),\n",
       "  array([-1.00000049])],\n",
       " 'normxiJ': [0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  1.1102230246251565e-16,\n",
       "  0.0,\n",
       "  1.1102230246251565e-16,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  2.220446049250313e-16,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0],\n",
       " 'normxiJ_save': [0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0,\n",
       "  0.0],\n",
       " 'tolerance': [array([0.02]),\n",
       "  array([0.04]),\n",
       "  array([0.06]),\n",
       "  array([0.08]),\n",
       "  array([0.1]),\n",
       "  array([0.12]),\n",
       "  array([0.14]),\n",
       "  array([0.16]),\n",
       "  array([0.17999999]),\n",
       "  array([0.199]),\n",
       "  array([0.19990226]),\n",
       "  array([0.19999026]),\n",
       "  array([0.19999903]),\n",
       "  array([0.1999999])],\n",
       " 's': [0,\n",
       "  0.1,\n",
       "  0.2,\n",
       "  0.30000000000000004,\n",
       "  0.4,\n",
       "  0.5,\n",
       "  0.599999977795542,\n",
       "  0.699999977795542,\n",
       "  0.7999999555910839,\n",
       "  0.8950000002467176,\n",
       "  0.8995113065562166,\n",
       "  0.8999512825866551,\n",
       "  0.8999951293267415,\n",
       "  0.8999995129433499,\n",
       "  0.8999999512944419],\n",
       " 'eps': [array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64),\n",
       "  array([], dtype=float64)]}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have access to the objective function and constraint histories, and the value of the Lagrange multiplier:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "i=1\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "plt.plot(results[i]['it'],results[i]['J'])\n",
    "plt.title('J')\n",
    "plt.figure()\n",
    "plt.plot(results[i]['it'],[ g[0] for g in results[i]['G']])\n",
    "plt.title('G')\n",
    "plt.figure()\n",
    "plt.plot(results[i]['it'][:-1],[mu[0] for mu in results[i]['muls']])\n",
    "plt.title('$\\lambda$');\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comment on the numerical values of the constraint G. Observe that the objective function keeps decreasing while maintaining the constraint satisfied.\n",
    "\n",
    "Plot the constraint $x_1x_2=1$ and the optimization trajectories:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import nullspace_optimizer.examples.draw as draw\n",
    "t = np.linspace(1/3,5,100)\n",
    "plt.plot(t,1/t,color='red',linewidth=0.5,label=\"y=1/x\")\n",
    "plt.plot(-t,-1/t,color='red',linewidth=0.5)\n",
    "for i, r in enumerate(results):\n",
    "    draw.drawData(r, f'x{i}', f'C{i}', x0=True, xfinal=True, initlabel=None)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Further works\n",
    "\n",
    "1. Adapt the above codes to display other trajectories.\n",
    "2. Try with other initializations, change the parameter alphaC and alphaJ\n",
    "3. Comment on the trajectories\n",
    "4. Try another equality constrained optimization program below"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Another problem\n",
    "\n",
    "Do the same to solve \n",
    "\n",
    "$$\n",
    "        \\begin{aligned}\n",
    "                \\max_{(x_1,x_2)\\in\\mathbb{R}^{2}} & \\qquad x_2\\\\\n",
    "                s.t. & \\left\\{\\begin{aligned}\n",
    "                    (x_1-0.5)^{2}+x_2^{2} &= 2\\\\\n",
    "                    (x_1+0.5)^{2}+x_2^{2} &= 2.\n",
    "                \\end{aligned}\\right.\n",
    "            \\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "class problem2(EuclideanOptimizable):\n",
    "    def __init__(self,x0):\n",
    "        super().__init__(2)\n",
    "        self.xinit = x0\n",
    "        self.nconstraints = 2\n",
    "        self.nineqconstraints = 0\n",
    "\n",
    "    def x0(self):\n",
    "        return self.xinit\n",
    "\n",
    "    def J(self, x):\n",
    "        return -x[1]\n",
    "\n",
    "    def dJ(self, x):\n",
    "        return [0, -1]\n",
    "\n",
    "    def G(self, x):\n",
    "        return [(x[0]-0.5)**2+x[1]**2-2,(x[0]+0.5)**2+x[1]**2-2]\n",
    "\n",
    "    def dG(self, x):\n",
    "        return [[2*(x[0]-0.5),2*x[1]],[2*(x[0]+0.5),2*x[1]]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "\n",
      "Optimization completed.\n",
      "\u001b[38;5;4m31. J=-1.323 G=[-5.529e-08,-5.529e-08] H=[]\u001b[0m\n",
      "\n",
      "\n",
      "Optimization completed.\n",
      "\u001b[38;5;4m37. J=-1.323 G=[-3.685e-08,3.784e-08] H=[]\u001b[0m\n",
      "\n",
      "\n",
      "Optimization completed.\n",
      "\u001b[38;5;4m68. J=1.323 G=[-4.121e-08,-3.668e-08] H=[]\u001b[0m\n",
      "\n",
      "\n",
      "Optimization completed.\n",
      "\u001b[38;5;4m39. J=1.323 G=[-2.874e-08,-6.206e-08] H=[]\u001b[0m\n"
     ]
    }
   ],
   "source": [
    "def run_problems():\n",
    "    xinits = ([0,0], [1.0, 0],  [3, -1], [-1.5,-0.5])\n",
    "    # Write xinits\n",
    "    problems = [problem2(x0=x0) for x0 in xinits]\n",
    "    params = {'dt': 0.05,  'alphaJ':2, 'alphaC':1, 'debug': -1}\n",
    "    return [nlspace_solve(pb, params) for pb in problems]\n",
    "\n",
    "results=run_problems()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t=np.linspace(0,2*np.pi,100)\n",
    "plt.plot(0.5+np.sqrt(2)*np.cos(t),np.sqrt(2)*np.sin(t),color='red',linewidth='0.5')\n",
    "plt.plot(-0.5+np.sqrt(2)*np.cos(t),np.sqrt(2)*np.sin(t),color='red',linewidth='0.5')\n",
    "plt.axis('equal')\n",
    "for i, r in enumerate(results):\n",
    "    draw.drawData(r, f'x{i}', f'C{i}', x0=True, xfinal=True, initlabel=None)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is the behavior of the trajectories `x2` and `x3` surprising ?"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.8.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}