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"cells": [
{
"cell_type": "markdown",
"id": "0",
"metadata": {},
"source": [
"# Solving constrained optimization problems\n",
"\n",
"---\n",
"\n",
"In this example, we consider the following constrained optimization problem:\n",
"\n",
"$$\n",
"\\begin{array}{r c}\n",
"\\text{maximize } & a + b + c \\\\\n",
"\\text{subject to } & 2a + 3b \\leq 45 \\\\\n",
" & 5a + 2c \\leq 75 \\\\\n",
" & 3b + c \\leq 50 \\\\\n",
" & -100 \\leq a \\leq 100 \\\\\n",
" & -100 \\leq b \\leq 100 \\\\\n",
" & -100 \\leq c \\leq 100 \\\\\n",
"\\end{array}\n",
"$$\n",
"\n",
"We will now solve this optimization problem using:\n",
"\n",
"- Evolutionary computation (using PGPE with constraint penalties)\n",
"- Penalty method (using gradient-based search with ClipUp, with penalty multipliers iteratively incremented)\n",
"- Interior points method (with the help of the functional API of PyTorch and the log-barrier function of EvoTorch)\n",
"\n",
"This notebook demonstrates:\n",
"\n",
"- How to use functional algorithms of EvoTorch (evolutionary and gradient-based)\n",
"- How to use constraint penalization utilities of EvoTorch and use them as building blocks for constrained optimization, both for evolutionary and for gradient-based optimization"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1",
"metadata": {},
"outputs": [],
"source": [
"from evotorch.algorithms.functional import pgpe, pgpe_ask, pgpe_tell, clipup, clipup_ask, clipup_tell\n",
"from evotorch.decorators import expects_ndim\n",
"from evotorch.tools import penalty, log_barrier\n",
"\n",
"import torch\n",
"import torch.func as tfunc\n",
"from typing import Union\n",
"from functools import partial\n",
"\n",
"from matplotlib import pyplot as plt\n",
"import numpy as np\n",
"\n",
"from datetime import datetime"
]
},
{
"cell_type": "markdown",
"id": "2",
"metadata": {},
"source": [
"## Fitness function implementation\n",
"\n",
"We begin by implementing our fitness function.\n",
"This fitness function has two arguments:\n",
"\n",
"- `penalty_multiplier`: Expected as a scalar, this value represents the multiplier for the negative penalty quantities that will be added onto the fitness. Higher values for `penalty_multiplier` will result in stronger penalizations.\n",
"- `x`: A 1-dimensional tensor, that represents the solution that will be evaluated.\n",
"\n",
"Notice how the fitness function below is decorated by `@expects_ndim(0, 1)`. This decorators declares that the first positional argument (`penalty_multipliers`) expects a 0-dimensional tensor, and the second positional argument (`x`) expects a 1-dimensional tensor. If any of these arguments are given with more dimensions, those extra dimensions will be considered as batch dimensions by the decorated function. This auto-batching feature is very helpful, because it allows the decorated function `f` to work with not just a single solution, but with a population of solutions (or a batch of populations of solutions) when `x` is given as an n-dimensional tensor with `n>1`. We will use this auto-batching feature with PGPE."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3",
"metadata": {},
"outputs": [],
"source": [
"@expects_ndim(0, 1)\n",
"def f(penalty_multiplier: torch.Tensor, x: torch.Tensor) -> torch.Tensor:\n",
" a = x[0]\n",
" b = x[1]\n",
" c = x[2]\n",
" objective = a + b + c\n",
"\n",
" constraints = [\n",
" [(2 * a) + (3 * b), \"<=\", 45],\n",
" [(5 * a) + (2 * c), \"<=\", 75],\n",
" [(3 * b) + c, \"<=\", 50],\n",
" [a, \">=\", -100],\n",
" [a, \"<=\", 100],\n",
" [b, \">=\", -100],\n",
" [b, \"<=\", 100],\n",
" [c, \">=\", -100],\n",
" [c, \"<=\", 100],\n",
" ]\n",
"\n",
" penalty_amount = 0.0\n",
" for lhs, op, rhs in constraints:\n",
" # For each constraint, we add a penalty (if there is violation)\n",
" penalty_amount = penalty_amount + penalty(\n",
" # Left-hand-side, comparison operator, and the right-hand-side:\n",
" lhs,\n",
" op,\n",
" rhs,\n",
" #\n",
" # Because this is a function we wish to maximize, the penalties should be in the opposite direction.\n",
" # Therefore, we declare the sign of the penalties as \"-\", making them negative quantities:\n",
" penalty_sign=\"-\",\n",
" #\n",
" # There will be a penalty in the form: (linear * amount_of_violation)\n",
" linear=1.0,\n",
" #\n",
" # There will also be a penalty in the form: (amount_of_violation ** exp)\n",
" exp=2.0,\n",
" #\n",
" # The exponential penalties are not allowed to exceed this amount:\n",
" exp_inf=5000.0,\n",
" #\n",
" # There will also be a penalty in the form: (step if amount_of_violation > 0 else 0)\n",
" step=10000.0,\n",
" )\n",
"\n",
" # Scale the accumulated penalty by `penalty_multiplier`, add it onto the objective, then return it\n",
" return objective + (penalty_amount * penalty_multiplier)"
]
},
{
"cell_type": "markdown",
"id": "4",
"metadata": {},
"source": [
"---\n",
"\n",
"## Evolutionary computation"
]
},
{
"cell_type": "markdown",
"id": "5",
"metadata": {},
"source": [
"As the evolutionary algorithm, we use the functional `pgpe`."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6",
"metadata": {},
"outputs": [],
"source": [
"pgpe_state = pgpe(\n",
" # We want to maximize the evaluation results:\n",
" objective_sense=\"max\",\n",
"\n",
" # The center point of the initial search distribution is given as 0 vector:\n",
" center_init=torch.zeros(3),\n",
"\n",
" # Learning rates for the center and the standard deviation of the\n",
" # search distribution:\n",
" center_learning_rate=0.1,\n",
" stdev_learning_rate=0.1,\n",
"\n",
" # The ranking method is \"centered\", which ranks the worst solution as\n",
" # -0.5, and the best solution as +0.5:\n",
" ranking_method=\"centered\",\n",
"\n",
" # We use the ClipUp optimizer:\n",
" optimizer=\"clipup\",\n",
"\n",
" # In the case of this example problem, we use the following max_speed:\n",
" optimizer_config={\"max_speed\": 1.0},\n",
"\n",
" # The standard deviation for the initial search distribution:\n",
" stdev_init=100.0,\n",
"\n",
" # Relative maximum allowed change for the standard deviation.\n",
" # 0.2 means that a standard deviation value cannot change more than 20%\n",
" # of its original value.\n",
" stdev_max_change=0.2,\n",
"\n",
" # Minimum standard deviation value. The standard deviation is cannot shrink\n",
" # to values lower than this:\n",
" stdev_min=0.01,\n",
")\n",
"\n",
"pgpe_state"
]
},
{
"cell_type": "markdown",
"id": "7",
"metadata": {},
"source": [
"Main loop of the evolutionary computation:"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8",
"metadata": {},
"outputs": [],
"source": [
"num_generations = 1000 # For how many iterations will PGPE work\n",
"\n",
"time_interval = 1 # With this period, we will print the status\n",
"last_report_time = datetime.now()\n",
"\n",
"# Initialize the variables that will store the best solution and the best evaluation result\n",
"pgpe_best_eval = float(\"-inf\")\n",
"pgpe_best = None\n",
"\n",
"for generation in range(1, 1 + num_generations):\n",
" # Ask for a population from PGPE\n",
" population = pgpe_ask(pgpe_state, popsize=1000)\n",
"\n",
" # Compute the fitnesses\n",
" fitnesses = f(1, population)\n",
"\n",
" # Inform PGPE of the latest fitnesses, and get its next state\n",
" pgpe_state = pgpe_tell(pgpe_state, population, fitnesses)\n",
"\n",
" # From the most recent population and fitnesses, update the best solution and the best evaluation result\n",
" pop_best_index = torch.argmax(fitnesses)\n",
" pop_best = population[pop_best_index]\n",
" pop_best_eval = fitnesses[pop_best_index]\n",
" if pop_best_eval > pgpe_best_eval:\n",
" pgpe_best_eval = pop_best_eval\n",
" pgpe_best = population[pop_best_index, :]\n",
"\n",
" # If it is time to report, print the status\n",
" tnow = datetime.now()\n",
" if ((tnow - last_report_time).total_seconds() > time_interval) or (generation == num_generations):\n",
" print(\"best solution:\", pgpe_best, \"best evaluation:\", pgpe_best_eval)\n",
" last_report_time = tnow"
]
},
{
"cell_type": "markdown",
"id": "9",
"metadata": {},
"source": [
"Best solution:"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "10",
"metadata": {},
"outputs": [],
"source": [
"pgpe_best, pgpe_best_eval"
]
},
{
"cell_type": "markdown",
"id": "11",
"metadata": {},
"source": [
"---\n",
"\n",
"## Penalty method\n",
"\n",
"As an alternative to evolutionary computation, we now use the gradient-based penalty method below.\n",
"The main points of the penalty method are as follows:\n",
"\n",
"- Use a gradient-based search on a fitness function augmented by penalties (i.e. by penalty terms that are computed according to how much the constraints are penalized)\n",
"- Periodically increase the multipliers for the penalty terms (when the penalty multiplier is increased, we use the previous optimization's solution as the new starting point)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "12",
"metadata": {},
"outputs": [],
"source": [
"# Try with these penalty multipliers, ordered from small to large:\n",
"penalty_multipliers = [0.1, 1, 4]\n",
"\n",
"# For each penalty multiplier, we do the search for this number of iterations:\n",
"num_iterations = 1000\n",
"\n",
"# Initialize the variables that will store the best solution and the best evaluation result\n",
"clipup_best_eval = float(\"-inf\")\n",
"clipup_best = None\n",
"\n",
"# Initialize the search point as a 0 vector:\n",
"x = torch.zeros(3)\n",
"\n",
"for penalty_multiplier in penalty_multipliers:\n",
" # Initialize the ClipUp algorithm for the currently considered penalty multiplier\n",
" clipup_state = clipup(center_init=x, center_learning_rate=0.1, max_speed=1.0)\n",
"\n",
" # Optimization loop for the current penalty multiplier\n",
" for iteration in range(1, 1 + num_iterations):\n",
" # Ask ClipUp for the current search point\n",
" x = clipup_ask(clipup_state)\n",
"\n",
" # Compute the gradient and the fitness\n",
" gradient, fitness = tfunc.grad_and_value(partial(f, penalty_multiplier))(x)\n",
"\n",
" # Update the best-known solution so far\n",
" if fitness > clipup_best_eval:\n",
" clipup_best_eval = fitness\n",
" clipup_best = x\n",
"\n",
" # Inform ClipUp of the latest gradient, and get its next state\n",
" clipup_state = clipup_tell(clipup_state, follow_grad=gradient)\n",
"\n",
" # After each optimization loop, print the best known solution\n",
" print(\"best solution:\", clipup_best, \" best evaluation:\", clipup_best_eval)"
]
},
{
"cell_type": "markdown",
"id": "13",
"metadata": {},
"source": [
"---\n",
"\n",
"## Interior points method"
]
},
{
"cell_type": "markdown",
"id": "14",
"metadata": {},
"source": [
"Although, as its name implies, the main focus of EvoTorch is evolutionary computation, we now demonstrate that it is possible to implement an interior points method by combining:\n",
"\n",
"- `torch.func.grad`\n",
"- `torch.func.hessian`\n",
"- `evotorch.tools.log_barrier`\n",
"\n",
"The `grad(...)` and `hessian(...)` functions are basic building blocks for implementing a Newton-Raphson search. We penalize proximities to the infeasible regions with the help of `log_barrier(...)`.\n",
"\n",
"We begin with a modified implementation of our fitness function, where `evotorch.tools.penalty(...)` are replaced by `evotorch.tools.log_barrier(...)`."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "15",
"metadata": {},
"outputs": [],
"source": [
"@expects_ndim(0, 1)\n",
"def f_with_log_barriers(sharpness: torch.Tensor, x: torch.Tensor) -> torch.Tensor:\n",
" a = x[0]\n",
" b = x[1]\n",
" c = x[2]\n",
" objective = a + b + c\n",
"\n",
" constraints = [\n",
" [(2 * a) + (3 * b), \"<=\", 45],\n",
" [(5 * a) + (2 * c), \"<=\", 75],\n",
" [(3 * b) + c, \"<=\", 50],\n",
" [a, \">=\", -100],\n",
" [a, \"<=\", 100],\n",
" [b, \">=\", -100],\n",
" [b, \"<=\", 100],\n",
" [c, \">=\", -100],\n",
" [c, \"<=\", 100],\n",
" ]\n",
"\n",
" penalty_amount = 0.0\n",
" for lhs, op, rhs in constraints:\n",
" penalty_amount = penalty_amount + log_barrier(lhs, op, rhs, penalty_sign=\"-\", sharpness=sharpness)\n",
"\n",
" # Return the objective with the log-barrier penalties added onto it.\n",
" # Notice that in the end, we are inverting the sign of the returned quantity.\n",
" # This will allow us to implement the Newton-Raphson's search method from a minimization perspective.\n",
" return -(objective + penalty_amount)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "16",
"metadata": {},
"outputs": [],
"source": [
"# Start the search from the 0 vector:\n",
"x = torch.zeros(3)\n",
"\n",
"# Try with these sharpness values for the log barrier, ordered from small to large:\n",
"sharpness_values = [1, 10, 100, 1000, 10000]\n",
"\n",
"# Learning rate for when taking a step\n",
"learning_rate = 0.1\n",
"\n",
"# An identity matrix multiplied by the constant below will be added to the Hessian matrix.\n",
"# When a diagonal element of the Hessian matrix is 0 because of numerical issues, this trick will allow the\n",
"# algorithm to still take a step.\n",
"I_multiplier = 0.01"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "17",
"metadata": {},
"outputs": [],
"source": [
"# With this interval (in seconds), we wish to report the status\n",
"reporting_interval = 1\n",
"last_report_time = datetime.now()\n",
"\n",
"# The interior points method will run for this amount of iterations:\n",
"num_iterations = 200\n",
"\n",
"for sharpness in sharpness_values:\n",
" print(\"sharpness:\", sharpness)\n",
" print()\n",
"\n",
" for iteration in range(1, 1 + num_iterations):\n",
" # Compute the gradient and the solution cost\n",
" g, cost = tfunc.grad_and_value(partial(f_with_log_barriers, sharpness))(x)\n",
"\n",
" # Compute the Hessian matrix\n",
" H = tfunc.hessian(partial(f_with_log_barriers, sharpness))(x)\n",
"\n",
" # Add the identity matrix multiplied by a constant to the Hessian\n",
" H = H + (I_multiplier * torch.eye(H.shape[-1]))\n",
"\n",
" # Take the inverse of the Hessian matrix\n",
" invH = torch.linalg.inv(H)\n",
"\n",
" # Move the center point\n",
" x = x - learning_rate * (invH @ g)\n",
"\n",
" # If it is time to report, print the status\n",
" tnow = datetime.now()\n",
" if (tnow - last_report_time).total_seconds() > reporting_interval:\n",
" print(\"Iteration:\", iteration, \" Gradient norm:\", torch.linalg.norm(g), \" Solution cost:\", cost)\n",
" last_report_time = tnow\n",
"\n",
" # Print the current search point\n",
" print()\n",
" print(\"x:\", x)\n",
" print()"
]
}
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