# Linear Program (LP) Tutorial¶

For instructions on how to run these tutorial notebooks, please see the README.

## Important Note¶

Please refer to mathematical program tutorial for constructing and solving a general optimization program in Drake.

## Linear Program¶

A linear program (LP) is a special type of optimization problem. The cost and constraints in an LP is a linear (affine) function of decision variables. The mathematical formulation of a general LP is \begin{align} \min_x \;c^Tx + d\\ \text{subject to } Ax\leq b \end{align}

A linear program can be solved by many open source or commercial solvers. Drake supports some solvers including SCS, Gurobi, Mosek, etc. Please see our Doxygen page for a complete list of supported solvers. Note that some commercial solvers (such as Gurobi and Mosek) are not included in the pre-compiled Drake binaries, and therefore not on Binder/Colab.

Drake's API supports multiple functions to add linear cost and constraints. We briefly go through some of the functions in this tutorial. For a complete list of functions, please check our Doxygen.

The easiest way to add linear cost is to call AddLinearCost function. We first demonstrate how to construct an optimization program with 2 decision variables, then we will call AddLinearCost to add the cost.

In [ ]:
from pydrake.solvers.mathematicalprogram import MathematicalProgram, Solve
import numpy as np

# Create an empty MathematicalProgram named prog (with no decision variables,
# constraints or costs)
prog = MathematicalProgram()
# Add two decision variables x[0], x[1].
x = prog.NewContinuousVariables(2, "x")


We can call AddLinearCost(expression) to add a new linear cost. expression is a symbolic linear expression of the decision variables.

In [ ]:
# Add a symbolic linear expression as the cost.
cost1 = prog.AddLinearCost(x[0] + 3 * x[1] + 2)
# Print the newly added cost
print(cost1)
# The newly added cost is stored in prog.linear_costs().
print(prog.linear_costs()[0])


If we call AddLinearCost again, the total cost stored in prog is the summation of all the costs. You can see that prog.linear_costs() will have two entries.

In [ ]:
cost2 = prog.AddLinearCost(2 * x[1] + 3)
print(f"number of linear cost objects: {len(prog.linear_costs())}")


If you know the coefficient of the linear cost as a vector, you could also add the cost by calling AddLinearCost(e, f, x) which will add a linear cost $e^Tx + f$ to the optimization program

In [ ]:
# We add a linear cost 3 * x[0] + 4 * x[1] + 5 to prog by specifying the coefficients
# [3., 4] and the constant 5 in AddLinearCost
cost3 = prog.AddLinearCost([3., 4.], 5., x)
print(cost3)


Lastly, the user can call AddCost to add a linear expression to the linear cost. Drake will analyze the structure of the expression, if Drake determines the expression is linear, then the added cost is linear.

In [ ]:
print(f"number of linear cost objects before calling AddCost: {len(prog.linear_costs())}")
# Call AddCost to add a linear expression as linear cost. After calling this function,
# len(prog.linear_costs()) will increase by 1.
cost4 = prog.AddCost(x[0] + 3 * x[1] + 5)
print(f"number of linear cost objects after calling AddCost: {len(prog.linear_costs())}")


We have three types of linear constraints

• Bounding box constraint. A lower/upper bound on the decision variable: $lower \le x \le upper$.
• Linear equality constraint: $Ax = b$.
• Linear inequality constraint: $lower <= Ax <= upper$.

The easiest way to add linear constraints is to call AddConstraint or AddLinearConstraint function, which can handle all three types of linear constraint. Compared to the generic AddConstraint function, AddLinearConstraint does more sanity will refuse to add the constraint if it is not linear.

In [ ]:
prog = MathematicalProgram()
x = prog.NewContinuousVariables(2, "x")
y = prog.NewContinuousVariables(3, "y")

# Call AddConstraint to add a bounding box constraint x[0] >= 1
print(f"number of bounding box constraint objects: {len(prog.bounding_box_constraints())}")

# Call AddLinearConstraint to add a bounding box constraint x[1] <= 2
print(f"number of bounding box constraint objects: {len(prog.bounding_box_constraints())}")

# Call AddConstraint to add a linear equality constraint x[0] + y[1] == 3
linear_eq1 = prog.AddConstraint(x[0] + y[1] == 3.)
print(f"number of linear equality constraint objects: {len(prog.linear_equality_constraints())}")

# Call AddLinearConstraint to add a linear equality constraint x[1] + 2 * y[2] == 1
linear_eq2 = prog.AddLinearConstraint(x[1] + 2 * y[2] == 1)
print(f"number of linear equality constraint objects: {len(prog.linear_equality_constraints())}")

# Call AddConstraint to add a linear inequality constraint x[0] + 3*x[1] + 2*y[2] <= 4
linear_ineq1 = prog.AddConstraint(x[0] + 3*x[1] + 2*y[2] <= 4)
print(f"number of linear inequality constraint objects: {len(prog.linear_constraints())}")

# Call AddLinearConstraint to add a linear inequality constraint x[1] + 4 * y[1] >= 2
linear_ineq2 = prog.AddLinearConstraint(x[1] + 4 * y[1] >= 2)
print(f"number of linear inequality constraint objects: {len(prog.linear_constraints())}")


AddLinearConstraint will check if the constraint is actually linear, and throw an exception if the constraint is not linear.

In [ ]:
# Add a nonlinear constraint square(x[0]) == 2 by calling AddLinearConstraint. This should
# throw an exception
try:
except RuntimeError as err:
print(err.args)


If the users know the coefficients of the constraint as a matrix, they could also call AddLinearConstraint(A, lower, upper, x) to add a constraint $lower \le Ax \le upper$. This version of the method does not construct any symbolic representations, and will be more efficient especially when A is very large.

In [ ]:
# Add a linear constraint 2x[0] + 3x[1] <= 2, 1 <= 4x[1] + 5y[2] <= 3.
# This is equivalent to lower <= A * [x;y[2]] <= upper with
# lower = [-inf, 1], upper = [2, 3], A = [[2, 3, 0], [0, 4, 5]].
A=[[2., 3., 0], [0., 4., 5.]],
lb=[-np.inf, 1],
ub=[2., 3.],
vars=np.hstack((x, y[2])))
print(linear_constraint)


If your constraint is a bounding box constraint (i.e. $lower \le x \le upper$), apart from calling AddConstraint or AddLinearConstraint, you could also call AddBoundingBoxConstraint(lower, upper, x), which will be slightly faster than AddConstraint and AddLinearConstraint.

In [ ]:
# Add a bounding box constraint -1 <= x[0] <= 2, 3 <= x[1] <= 5
bounding_box3 = prog.AddBoundingBoxConstraint([-1, 3], [2, 5], x)
print(bounding_box3)


If the variables share the same lower or upper bound, you could use a scalar lower or upper value in AddBoundingBoxConstraint. For example

In [ ]:
# Add a bounding box constraint 3 <= y[i] <= 5 for all i.
print(bounding_box4)


If your constraint is a linear equality constraint (i.e. $Ax = b$), apart from calling AddConstraint or AddLinearConstraint, you could also call AddLinearEqualityConstraint to be more specific (and slightly faster than AddConstraint and AddLinearConstraint).

In [ ]:
# Add a linear equality constraint 4 * x[0] + 5 * x[1] == 1
linear_eq3 = prog.AddLinearEqualityConstraint(np.array([[4, 5]]), np.array([1]), x)
print(linear_eq3)


### Solving Linear Program.¶

Once all the constraints and costs are added to the program, we can call Solve function to solve the program and call GetSolution to obtain the results.

In [ ]:
# Solve an optimization program
# min -3x[0] - x[1] - 5x[2] -x[3] + 2
# s.t 3x[0] + x[1] + 2x[2] = 30
#     2x[0] + x[1] + 3x[2] + x[3] >= 15
#     2x[1] + 3x[3] <= 25
#     -100 <= x[0] + 2x[2] <= 40
#   x[0], x[1], x[2], x[3] >= 0, x[1] <= 10
prog = MathematicalProgram()
# Declare x as decision variables.
x = prog.NewContinuousVariables(4)
# Add linear costs. To show that calling AddLinearCosts results in the sum of each individual
# cost, we add two costs -3x[0] - x[1] and -5x[2]-x[3]+2
# Add linear equality constraint 3x[0] + x[1] + 2x[2] == 30
prog.AddLinearConstraint(3*x[0] + x[1] + 2*x[2] == 30)