Many engineering problems can be formulated as mathematical optimization problems, and solved by numerical solvers. A generic mathematical optimization problem can be formulated as \begin{align} \begin{array}{rl} \min_x \; & f(x) \\[0.1pc]\text{subject to} \; & x \in\mathcal{S} \end{array} \hspace{1.45em} {\scriptsize \begin{array}{|ll|} \hline \text{The real-valued decision variable is} & x \\[-0.21pc] \text{The real-valued cost function is} & f(x) \\[-0.21pc] \text{The constraint set is} & \mathcal{S} \\[-0.21pc] \text{The optimal $x$ that minimizes the cost function is} & x^* \\[0.03pc]\hline \end{array} } \end{align}
where $x$ is the real-valued decision variable(s), $f(x)$ is the real-valued cost function, $\mathcal{S}$ is the constraint set for $x$. Our goal is to find the optimal $x^*$ within the constraint set $\mathcal{S}$, such that $x^*$ minimizes the cost function $f(x)$.
For example, the following optimization problem determines the value of $x$ that minimizes $x^3 + 2x + 1$ subject to $x \ge 1$. \begin{align} \begin{array}{rl} \min_x & x^3 + 2x + 1 \\[0.1pc] \text{subject to} & x \ge 1 \end{array} \hspace{1.45em} {\scriptsize \begin{array}{|ll|} \hline \text{The real-valued decision variable is} & x \\[-0.21pc] \text{The real-valued cost function $f(x)$ is} & x^3 + 2x + 1 \\[-0.21pc] \text{The set $\mathcal{S}$ of constraints is} & x \ge 1 \\[-0.21pc] \text{The value that minimizes the cost function is} & x^* = 1 \\[0.03pc]\hline \end{array}} \end{align}
In general, how an optimization problem is solved depends on its categorization (categories include Linear Programming, Quadratic Programming, Mixed-integer Programming, etc.). Categorization depends on properties of both the cost function $f(x)$ and the constraint set $\mathcal{S}$. For example, if the cost function $f(x)$ is a linear function of $x$, and the constraint $\mathcal{S}$ is a linear set $\mathcal{S} = \{x | Ax\le b\}$, then we have a linear programming problem, which is efficiently solved with certain solvers.
There are multiple solvers for each category of optimization problems,
but each solver has its own API and data structures.
Frequently, users need to rewrite code when they switch solvers.
To remedy this, Drake provides a common API through the MathematicalProgram class.
In addition to avoiding solver-specific code,
the constraint and cost functions can be written in symbolic form (which makes code more readable).
In these ways, Drake's MathematicalProgram is akin to YALMIP in MATLAB or JuMP in Julia, and we support both Python and C++.
Note: Drake supports many solvers
(some are open-source and some require a license).
Drake can formulate and solve the following categories of optimization problems
This tutorial provides the basics of Drake's MathematicalProgram. Advanced tutorials are available at the bottom of this document.
Drake's MathematicalProgram class contains the mathematical formulation of an optimization problem, namely the decision variables $x$, the cost function $f(x)$, and the constraint set $\mathcal{S}$.
To initialize this class, first create an empty MathematicalProgram as
from pydrake.solvers.mathematicalprogram import MathematicalProgram
import numpy as np
# Create an empty MathematicalProgram named prog (with no decision variables,
# constraints or cost function)
prog = MathematicalProgram()
Shown below, the function NewContinuousVariables
adds two new continuous decision variables to prog
. The newly added variables are returned as x
in a numpy array.
Note the range of the variable is a continuous set, as opposed to binary variables which only take discrete value 0 or 1.
x = prog.NewContinuousVariables(2)
The default names of the variable in x are "x(0)" and "x(1)". The next line prints the default names and types in x
, whereas the second line prints the symbolic expression "1 + 2x[0] + 3x[1] + 4x[1]".
print(x)
print(1 + 2*x[0] + 3*x[1] + 4*x[1])
To create an array y
of two variables named "dog(0)"" and "dog(1)", pass the name "dog" as a second argument to NewContinuousVariables()
. Also shown below is the printout of the two variables in y
and a symbolic expression involving y
.
y = prog.NewContinuousVariables(2, "dog")
print(y)
print(y[0] + y[0] + y[1] * y[1] * y[1])
To create a $3 \times 2$ matrix of variables named "A", type
var_matrix = prog.NewContinuousVariables(3, 2, "A")
print(var_matrix)
The simplest way to add a constraint is with MathematicalProgram.AddConstraint()
.
# Add the constraint x(0) * x(1) = 1 to prog
prog.AddConstraint(x[0] * x[1] == 1)
You can also add inequality constraints to prog
such as
prog.AddConstraint(x[0] >= 0)
prog.AddConstraint(x[0] - x[1] <= 0)
prog
automatically analyzes these symbolic inequality constraint expressions and determines they are all linear constraints on $x$.
In a complicated optimization problem, it is often convenient to write the total cost function $f(x)$ as a sum of individual cost functions \begin{align} f(x) = \sum_i g_i(x) \end{align}
The simplest way to add an individual cost function $g_i(x)$ to the total cost function $f(x)$ is with the MathematicalProgram.AddCost()
method (as shown below).
# Add a cost x(0)**2 + 3 to the total cost. Since prog doesn't have a cost before, now the total cost is x(0)**2 + 3
prog.AddCost(x[0] ** 2 + 3)
To add another individual cost function $x(0) + x(1)$ to the total cost function $f(x)$, simply call AddCost()
again as follows
prog.AddCost(x[0] + x[1])
now the total cost function becomes $x(0)^2 + x(0) + x(1) + 3$.
prog
can analyze each of these individual cost functions and determine that $x(0) ^ 2 + 3$ is a convex quadratic function, and $x(0) + x(1)$ is a linear function of $x$.
Once all the decision variables/constraints/costs are added to prog
, we are ready to solve the optimization problem.
The simplest way to solve the optimization problem is to call Solve()
function. Drake's MathematicalProgram analyzes the type of the constraints/costs, and then calls an appropriate solver for your problem. The result of calling Solve()
is stored inside the return argument. Here is a code snippet
"""
Solves a simple optimization problem
min x(0)^2 + x(1)^2
subject to x(0) + x(1) = 1
x(0) <= x(1)
"""
from pydrake.solvers.mathematicalprogram import Solve
# Set up the optimization problem.
prog = MathematicalProgram()
x = prog.NewContinuousVariables(2)
prog.AddConstraint(x[0] + x[1] == 1)
prog.AddConstraint(x[0] <= x[1])
prog.AddCost(x[0] **2 + x[1] ** 2)
# Now solve the optimization problem.
result = Solve(prog)
# print out the result.
print("Success? ", result.is_success())
# Print the solution to the decision variables.
print('x* = ', result.GetSolution(x))
# Print the optimal cost.
print('optimal cost = ', result.get_optimal_cost())
# Print the name of the solver that was called.
print('solver is: ', result.get_solver_id().name())
Notice that we can then retrieve optimization result from the return argument of Solve
. For example, the solution $x^*$ is retrieved from result.GetSolution()
, and the optimal cost from result.get_optimal_cost()
.
Some optimization solution is infeasible (doesn't have a solution). For example in the following code example, result.get_solution_result()
will not report kSolutionFound
.
"""
An infeasible optimization problem.
"""
prog = MathematicalProgram()
x = prog.NewContinuousVariables(1)[0]
y = prog.NewContinuousVariables(1)[0]
prog.AddConstraint(x + y >= 1)
prog.AddConstraint(x + y <= 0)
prog.AddCost(x)
result = Solve(prog)
print("Success? ", result.is_success())
print(result.get_solution_result())
"""
Demo on manually choosing a solver
Solves the problem
min x(0)
s.t x(0) + x(1) = 1
0 <= x(1) <= 1
"""
from pydrake.solvers.ipopt import IpoptSolver
prog = MathematicalProgram()
x = prog.NewContinuousVariables(2)
prog.AddConstraint(x[0] + x[1] == 1)
prog.AddConstraint(0 <= x[1])
prog.AddConstraint(x[1] <= 1)
prog.AddCost(x[0])
# Choose IPOPT as the solver.
# First instantiate an IPOPT solver.
solver = IpoptSolver()
# The initial guess is [1, 1]. The third argument is the options for Ipopt solver,
# and we set no solver options.
result = solver.Solve(prog, np.array([1, 1]), None)
print(result.get_solution_result())
print("x* = ", result.GetSolution(x))
print("Solver is ", result.get_solver_id().name())
print("Ipopt solver status: ", result.get_solver_details().status,
", meaning ", result.get_solver_details().ConvertStatusToString())
Note that solver.Solve()
expects three input arguments, the optimization program prog
, the initial guess of the decision variable values ([1, 1]
in this case), and an optional setting for the solver (None
in this case, we use the default IPOPT setting). If you don't have an initial guess, you can substitute np.array([1, 1])
with None
. The solver will choose an initial guess automatically, as shown in the code below
result = solver.Solve(prog, None, None)
print(result.get_solution_result())
print("x* = ", result.GetSolution(x))
Also note that if we know which solver is called, then we can access some solver-specific result, by calling result.get_solver_details()
. For example, IpoptSolverDetails
contains a field status
, namely the status code of the IPOPT solver, we could access this info by
print("Ipopt solver status: ", result.get_solver_details().status,
", meaning ", result.get_solver_details().ConvertStatusToString())
Each solver has its own details. You should refer to XXXSolverDetails
class on what is stored inside the return argument of result.get_solver_details()
. For example, if you know that IPOPT is called, then refer to IpoptSolverDetails
class; for OSQP solver, refer to OsqpSolverDetails
, etc.