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Got a basic optimality condition function working

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youainti 5 years ago
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commit d1840a9ed7
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6
Code/README.md
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# COMPUTATIONAL TODO # COMPUTATIONAL TODO
## Next steps ## Completed steps
- implement 'launch function as a function' portion - implement 'launch function as a function' portion
- This will probably use a neural network as a function type approach.
- substitute the transition functions into the optimality conditions. - substitute the transition functions into the optimality conditions.
## Next steps
- create the iterated optimality conditions
- use these optimality conditions to create a loss function - use these optimality conditions to create a loss function
- add boundary conditions to loss function - add boundary conditions to loss function
- get a basic gradient descent/optimization of launch function working. - get a basic gradient descent/optimization of launch function working.

175
Code/connect_transition_to_optimality.ipynb
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{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 1, "execution_count": 1,
"id": "behavioral-session", "id": "expired-austria",
"metadata": { "metadata": {
"tags": [] "tags": []
}, },
@ -16,7 +16,7 @@
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "vocational-rover", "id": "damaged-accountability",
"metadata": {}, "metadata": {},
"source": [ "source": [
"# Setup Functions\n", "# Setup Functions\n",
@ -26,7 +26,7 @@
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 2, "execution_count": 2,
"id": "physical-guidance", "id": "modular-memorabilia",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -73,7 +73,7 @@
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "difficult-drinking", "id": "direct-picture",
"metadata": {}, "metadata": {},
"source": [ "source": [
"# functions related to transitions" "# functions related to transitions"
@ -81,12 +81,12 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 3, "execution_count": 37,
"id": "beautiful-northwest", "id": "bridal-ordinary",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
"def single_transition(item_to_iterate, laws_motion, profit, stocks, debris ):\n", "def single_transition(item_to_iterate, laws_motion_fn, profit_fn, stocks, debris, launch_fn):\n",
" \"\"\"\n", " \"\"\"\n",
" This function represents the inverted envelope conditions.\n", " This function represents the inverted envelope conditions.\n",
" It allows us to describe the derivatives of the value function evaluated at time $t+1$ in terms based in time period $t$.\n", " It allows us to describe the derivatives of the value function evaluated at time $t+1$ in terms based in time period $t$.\n",
@ -101,7 +101,7 @@
" #it consists of the derivative of the laws of motion with respect to stocks and debris\n", " #it consists of the derivative of the laws of motion with respect to stocks and debris\n",
" \n", " \n",
" #Get the jacobian\n", " #Get the jacobian\n",
" a = jacobian(laws_motion, (stocks,debris))\n", " a = jacobian(laws_motion_fn, (stocks,debris, launch_fn(stocks,debris)))\n",
" \n", " \n",
" #Reassemble the Jacobian nested tuples into the appropriate tensor\n", " #Reassemble the Jacobian nested tuples into the appropriate tensor\n",
" A = BETA * torch.cat((torch.cat((a[0][0],a[0][1]),dim=1),torch.cat((a[1][0],a[1][1]),dim=1)), dim=0)\n", " A = BETA * torch.cat((torch.cat((a[0][0],a[0][1]),dim=1),torch.cat((a[1][0],a[1][1]),dim=1)), dim=0)\n",
@ -112,9 +112,16 @@
" # - EigVal(A) ~= 0\n", " # - EigVal(A) ~= 0\n",
" # - A.inverse() with a try catch system to record types of returns\n", " # - A.inverse() with a try catch system to record types of returns\n",
" #Alternatively, \n", " #Alternatively, \n",
" #if abs(a.det())\n",
" \n", " \n",
" #Calculate the item to transition\n", " #Calculate the item to transition\n",
" T = item_to_iterate - torch.cat(jacobian(profit,(stocks, debris))) \n", " f_jacobians = jacobian(profit_fn,(stocks, debris, launch_fn(stocks,debris)))\n",
"\n",
" #issue with shape here: my launch function is for all launches, not just a single launch.\n",
" f_theta = torch.cat([f_jacobians[0][0], f_jacobians[1][0]],axis=0) \n",
"\n",
" T = item_to_iterate - f_theta\n",
"\n",
" #Includes rearranging the jacobian of profit.\n", " #Includes rearranging the jacobian of profit.\n",
"\n", "\n",
" #Return the transitioned values\n", " #Return the transitioned values\n",
@ -125,30 +132,31 @@
" \"\"\"\n", " \"\"\"\n",
" \"\"\"\n", " \"\"\"\n",
" #unpack states and functions\n", " #unpack states and functions\n",
" stocks, debris,profit, laws_motion, item_to_transition = data_in\n", " stocks, debris,profit_fn, laws_motion_fn, item_to_transition,launch_fn = data_in\n",
" \n", " \n",
" #Calculate new states\n", " #Calculate new states\n",
" new_stocks, new_debris = laws_motion(stocks,debris)\n", " new_stocks, new_debris = laws_motion_fn(stocks,debris, launch_fn(stocks,debris))\n",
" \n", " \n",
" #WARNING: RECURSION: You may break your head...\n", " #WARNING: RECURSION: You may break your head...\n",
" #This gets the transition of the value function derivatives over time.\n", " #This gets the transition of the value function derivatives over time.\n",
" transitioned = single_transition(\n", " transitioned = single_transition(\n",
" item_to_transition, #item to iterate, i.e. the derivatives of the value function\n", " item_to_transition, #item to iterate, i.e. the derivatives of the value function\n",
" #functions\n", " #functions\n",
" laws_motion, \n", " laws_motion_fn, \n",
" profit, \n", " profit_fn, \n",
" stocks, debris #states\n", " stocks, debris, #states\n",
" launch_fn #launch function\n",
" )\n", " )\n",
" \n", " \n",
" #collects the data back together for return, including the updated state variables\n", " #collects the data back together for return, including the updated state variables\n",
" data_out = new_stocks, new_debris, profit, laws_motion, transitioned\n", " data_out = new_stocks, new_debris, profit_fn, laws_motion_fn, transitioned, launch_fn\n",
" \n", " \n",
" return data_out" " return data_out"
] ]
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "entertaining-theorem", "id": "miniature-karaoke",
"metadata": {}, "metadata": {},
"source": [ "source": [
"## Setup functions related to the problem" "## Setup functions related to the problem"
@ -156,24 +164,13 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 4, "execution_count": 38,
"id": "modern-kentucky", "id": "bright-minimum",
"metadata": {}, "metadata": {
"tags": []
},
"outputs": [], "outputs": [],
"source": [ "source": [
"### Classes\n",
"class States():\n",
" \"\"\"\n",
" This class represents the state variables associated with the problems.\n",
" \n",
" In this problem, the two types of states are constellation stocks and debris.\n",
" \n",
" I'm not sure how useful it will be. We'll see. It is missing a lot\n",
" \"\"\"\n",
" def __init__(self, satellite_stock, debris):\n",
" self.stock = satellite_stock\n",
" self.debris = debris\n",
" \n",
" \n", " \n",
"\n", "\n",
"### functions\n", "### functions\n",
@ -192,34 +189,33 @@
" \"\"\"\n", " \"\"\"\n",
" return torch.ones(5, requires_grad=True)\n", " return torch.ones(5, requires_grad=True)\n",
"\n", "\n",
"def laws_of_motion(stock, debris):\n", "def laws_of_motion(stock, debris, launch):\n",
" \"\"\"\n", " \"\"\"\n",
" This function updates state variables (stock and debris), according \n", " This function updates state variables (stock and debris), according \n",
" to the laws of motion.\n", " to the laws of motion.\n",
" \n", " \n",
" It returns the state variables as \n", " It returns the state variables as \n",
" \"\"\"\n", " \"\"\"\n",
" l = launches(stock,debris)\n", "\n",
" #Notes: Launches is a global function.\n",
" s = survival(stock,debris)\n", " s = survival(stock,debris)\n",
" #Notes: Survival is a global function.\n", " #Notes: Survival is a global function.\n",
" \n", " \n",
" new_stock = stock*s + l\n", " new_stock = stock*s + launch\n",
" \n", " \n",
" \n", " \n",
" #TODO: Currently Ignoring autocatalysis\n", " #TODO: Currently Ignoring autocatalysis\n",
" new_debris = (1-DELTA)*debris + LAUNCH_DEBRIS_RATE * l.sum() + COLLISION_DEBRIS_RATE*(1-s) @ stock\n", " new_debris = (1-DELTA)*debris + LAUNCH_DEBRIS_RATE * launch.sum() + COLLISION_DEBRIS_RATE*(1-s) @ stock\n",
" \n", " \n",
" return (new_stock, new_debris)\n", " return (new_stock, new_debris)\n",
"\n", "\n",
"#This is not a good specification of the profit function, but it will work for now.\n", "#This is not a good specification of the profit function, but it will work for now.\n",
"def profit(stock, debris):\n", "def profit(stock, debris, launches):\n",
" return UTIL_WEIGHTS @ stock" " return UTIL_WEIGHTS @ stock - LAUNCH_COST*launches"
] ]
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "broadband-technique", "id": "conservative-ukraine",
"metadata": {}, "metadata": {},
"source": [ "source": [
"# Actual calculations" "# Actual calculations"
@ -227,8 +223,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 5, "execution_count": 39,
"id": "adjustable-harvey", "id": "initial-mathematics",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -245,13 +241,15 @@
"#CHANGE LATER: Launch is currently a value, should be a function (i.e. neural network)\n", "#CHANGE LATER: Launch is currently a value, should be a function (i.e. neural network)\n",
"launches = test_launch\n", "launches = test_launch\n",
"\n", "\n",
"#compose the functions together.\n", "#Starting point\n",
"base_data = (stocks,debris, profit, laws_of_motion, torch.ones(6, requires_grad=True))\n", "# Stocks, debris, profit fn, laws of motion, \n",
"base_data = (stocks,debris, profit, laws_of_motion, torch.ones(6, requires_grad=True),launches)\n",
"\n", "\n",
"#Parameters\n", "#Parameters\n",
"SCALING = torch.ones(5)\n", "SCALING = torch.ones(5)\n",
"DELTA = 0.9\n", "DELTA = 0.9\n",
"LAUNCH_DEBRIS_RATE = 0.005\n", "LAUNCH_DEBRIS_RATE = 0.005\n",
"LAUNCH_COST = 1.0\n",
"COLLISION_DEBRIS_RATE = 0.0007\n", "COLLISION_DEBRIS_RATE = 0.0007\n",
"UTIL_WEIGHTS = torch.tensor([1,-0.2,0,0,0])\n", "UTIL_WEIGHTS = torch.tensor([1,-0.2,0,0,0])\n",
"BETA = 0.95" "BETA = 0.95"
@ -259,8 +257,8 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 6, "execution_count": 75,
"id": "cordless-wages", "id": "nuclear-definition",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
@ -304,16 +302,97 @@
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "shaped-zambia", "id": "casual-annex",
"metadata": {}, "metadata": {},
"source": [ "source": [
"Also, maybe I can create a `Model` class that upon construction will capture the necesary constants, functions, etc.\n" "Also, maybe I can create a `Model` class that upon construction will capture the necesary constants, functions, etc.\n"
] ]
}, },
{
"cell_type": "markdown",
"id": "stopped-socket",
"metadata": {},
"source": [
"# Optimatility conditions"
]
},
{
"cell_type": "code",
"execution_count": 193,
"id": "excessive-script",
"metadata": {},
"outputs": [],
"source": [
"#Optimality condition\n",
"def optimality(stocks,debris,profit_fn,laws_motion_fn,launch_fn, iterated_item):\n",
" #Derivative of the value function with respect to choice functions\n",
" #this returns derivatives with respect to every launch, so I've removed that\n",
" fx = jacobian(profit_fn, (stocks,debris,launch_fn(stocks,debris)))[-1][:,0]\n",
" \n",
" \n",
" #The following returns a tuple of tuples of tensors.\n",
" #the first tuple contains jacobians related to laws of motion for stocks\n",
" #the second tuple contains jacobians related to laws of motion for debris.\n",
" #we need the derivatives related to both\n",
" b = jacobian(laws_of_motion,(stocks,debris,launches(stocks,debris)))\n",
" B = torch.cat((b[0][2],b[1][2].T),axis=1)\n",
"\n",
"\n",
" return fx + BETA * B @ iterated_item"
]
},
{
"cell_type": "code",
"execution_count": 195,
"id": "unlikely-coverage",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor(49.4968)"
]
},
"execution_count": 195,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sum(optimality(stocks,debris,profit,laws_of_motion,launches,tmp_result)**2)"
]
},
{
"cell_type": "markdown",
"id": "endless-occupation",
"metadata": {},
"source": [
"## Now to set up the recursive set of optimatliy conditions"
]
},
{
"cell_type": "code",
"execution_count": 179,
"id": "valuable-bleeding",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([0.0300, 2.0300, 3.0300, 4.0300, 5.0300])"
]
},
"execution_count": 179,
"metadata": {},
"output_type": "execute_result"
}
],
"source": []
},
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": null,
"id": "canadian-excitement", "id": "subjective-chassis",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [] "source": []

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