Updating costs and constraints in MathematicalProgram

Often cases after we solve an optimization problem, we want to tweak its constraints and costs slightly, and then resolve the updated problem. One use case is in model predictive control, where in each time instance we solve a new optimization problem, whose constraints/costs are just slightly different from the one in the previous time instance.

Instead of constructing a new MathematicalProgram object, we could update the constraints/costs in the old MathematicalProgram object, and then solve the updated problem. To do so, many constraints/costs object provide an "update" function. In this tutorial we show how to update certain types of constraints/costs

Updating LinearCost

For a linear cost $a^Tx + b$, we could call LinearCost.UpdateCoefficients() function to update the linear coefficient a vector or the constant term b

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

prog = MathematicalProgram()
x = prog.NewContinuousVariables(2)
cost1 = prog.AddLinearCost(2*x[0] + 3 * x[1] + 2)
print(f"original cost: {cost1}")
prog.AddBoundingBoxConstraint(-1, 1, x)
result = Solve(prog)
print(f"optimal solution: {result.GetSolution(x)}")
print(f"original optimal cost: {result.get_optimal_cost()}")

# Now update the cost to 3x[0] - 4x[1] + 5
cost1.evaluator().UpdateCoefficients(new_a=[3, -4], new_b=5)
print(f"updated cost: {cost1}")
# Solve the optimization problem again with the updated cost.
result = Solve(prog)
print(f"updated optimal solution: {result.GetSolution(x)}")
print(f"updated optimal cost: {result.get_optimal_cost()}")

Updating QuadraticCost

For a quadratic cost in the form $0.5x^TQx + b'x + c$, we can also call QuadraticCost.UpdateCoefficients to update its $Q, b, c$ terms

In [ ]:
prog = MathematicalProgram()
x = prog.NewContinuousVariables(2)
cost1 = prog.AddQuadraticCost(x[0]**2 + 2 * x[1]**2 + x[0]*x[1] + 3*x[0] + 5)
print(f" original cost: {cost1}")
cost1.evaluator().UpdateCoefficients(new_Q=[[1., 2], [2., 4]], new_b=[1, 2.], new_c= 4)
print(f" updated cost: {cost1}")

Updating the bounds for any constraint

For any constraint $lower \le f(x) \le upper$, we can update its bounds by

In [ ]:
prog = MathematicalProgram()
x = prog.NewContinuousVariables(2)
constraint1 = prog.AddLinearConstraint(x[0] + 3 * x[1] <= 2)
print(f"original constraint: {constraint1}")
constraint1.evaluator().UpdateLowerBound([-1])
print(f"updated constraint: {constraint1}")
constraint1.evaluator().UpdateUpperBound([3])
print(f"updated constraint: {constraint1}")
constraint1.evaluator().set_bounds(new_lb=[-5], new_ub=[10])
print(f"updated constraint: {constraint1}")

Update linear constraint coefficients and bounds

For a linear constraint $lower \le Ax \le upper$, we can call LinearConstraint.UpdateCoefficients(new_A, new_lb, new_ub) to update the constraint as $\text{new_lb} \le \text{new_A}* x\le \text{new_ub}$.

For a linear equality constraint $Ax = b$, we can call LinearEqualityConstraint.UpdateCoefficients(Aeq, beq) to update the constraint to $\text{Aeq}*x=\text{beq}$.

In [ ]:
prog = MathematicalProgram()
x = prog.NewContinuousVariables(2)
linear_constraint = prog.AddLinearConstraint(3 * x[0] + 4 * x[1] <= 5)
linear_eq_constraint = prog.AddLinearConstraint(5 * x[0] + 2 * x[1] == 3)
print(f"original linear constraint: {linear_constraint}")
linear_constraint.evaluator().UpdateCoefficients(new_A = [[1, 3]], new_lb=[-2], new_ub=[3])
print(f"updated linear constraint: {linear_constraint}")

print(f"original linear equality constraint: {linear_eq_constraint}")
linear_eq_constraint.evaluator().UpdateCoefficients(Aeq=[[3, 4]], beq=[2])
print(f"updated linear equality constraint: {linear_eq_constraint}")