Sensitivity Toolbox

The sensitivity toolbox provides a Pyomo interface to sIPOPT to very quickly compute approximate solutions to nonlinear programs with a small perturbation in model parameters. See the sIPOPT documentation or the following paper for additional details:

Pirnay, R. Lopez-Negrete, and L.T. Biegler, Optimal Sensitivity based on IPOPT, Math. Prog. Comp., 4(4):307--331, 2012.

Using the Sensitivity Toolbox

We will start with a motivating example:

$$\begin{aligned} \begin{align*} \min_{x_1,x_2,x_3} \quad & x_1^2 + x_2^2 + x_3^2 \\\ \mathrm{s.t.} \qquad & 6 x_1 + 3 x_2 + 2 x_3 - p_1 = 0 \\\ & p_2 x_1 + x_2 - x_3 - 1 = 0 \\\ & x_1, x_2, x_3 \geq 0 \end{align*} \end{aligned}$$

Here x₁, x₂, and x₃ are the decision variables while p₁ and p₂ are parameters. At first, let's consider p₁ = 4.5 and p₂ = 1.0. Below is the model implemented in Pyomo.

Import Pyomo and sipopt from the sensitivity toolbox

>>> from pyomo.environ import * >>> from pyomo.contrib.sensitivity_toolbox.sens import sipopt

Create a concrete model

>>> m = ConcreteModel()

Define the variables with bounds and initial values

>>> m.x1 = Var(initialize = 0.15, within=NonNegativeReals) >>> m.x2 = Var(initialize = 0.15, within=NonNegativeReals) >>> m.x3 = Var(initialize = 0.0, within=NonNegativeReals)

Define the parameters

>>> m.eta1 = Param(initialize=4.5,mutable=True) >>> m.eta2 = Param(initialize=1.0,mutable=True)

Define the constraints and objective

>>> m.const1 = Constraint(expr=6*m.x1+3*m.x2+2*m.x3-m.eta1 ==0) >>> m.const2 = Constraint(expr=m.eta2*m.x1+m.x2-m.x3-1 ==0) >>> m.cost = Objective(expr=m.x1*2+m.x22+m.x3*2)

The solution of this optimization problem is x₁^* = 0.5, x₂^* = 0.5, and x₃^* = 0.0. But what if we change the parameter values to p̂₁ = 4.0 and p̂₂ = 1.0? Is there a quick way to approximate the new solution x̂₁^*, x̂₂^*, and x̂₃^*? Yes! This is the main functionality of sIPOPT.

Next we define the perturbed parameter values p̂₁ and p̂₂:

>>> m.perturbed_eta1 = Param(initialize = 4.0) >>> m.perturbed_eta2 = Param(initialize = 1.0)

And finally we call sIPOPT:

>>> m_sipopt = sipopt(m,[m.eta1,m.eta2], [m.perturbed_eta1,m.perturbed_eta2], streamSoln=True) Ipopt ...: run_sens=yes ... **************************************************************************** This program contains Ipopt, a library for large-scale nonlinear optimization. Ipopt is released as open source code under the Eclipse Public License (EPL). For more information visit http://projects.coin-or.org/Ipopt**************************************************************************** ... EXIT: Optimal Solution Found.

The first argument is the Pyomo model. The second argument is a list of the original parameters. The third argument is a list of the perturbed parameters. sIPOPT requires these two lists are the same length. The `...` represents extra lines of output that were cut from this page for brevity.

First, we can inspect the initial point:

>>> print("eta1 = %0.3f" % m.eta1()) eta1 = 4.500

>>> print("eta2 = %0.3f" % m.eta2()) eta2 = 1.000

Initial point (not feasible): >>> print("Objective = %0.3f" % m.cost()) Objective = 0.045

>>> print("x1 = %0.3f" % m.x1()) x1 = 0.150

>>> print("x2 = %0.3f" % m.x2()) x2 = 0.150

>>> print("x3 = %0.3f" % m.x3()) x3 = 0.000

Next, we inspect the solution x₁^*, x₂^*, and x₃^*:

Solution with the original parameter values: >>> print("Objective = %0.3f" % m_sipopt.cost()) Objective = 0.500

>>> print("x1 = %0.3f" % m_sipopt.x1()) x1 = 0.500

>>> print("x2 = %0.3f" % m_sipopt.x2()) x2 = 0.500

>>> print("x3 = %0.3f" % m_sipopt.x3()) x3 = 0.000

Finally, we inspect the approximate solution x̂₁^*, x̂₂^*, and x̂₃^*:

New parameter values: >>> print("eta1 = %0.3f" %m_sipopt.perturbed_eta1()) eta1 = 4.000

>>> print("eta2 = %0.3f" % m_sipopt.perturbed_eta2()) eta2 = 1.000

(Approximate) solution with the new parameter values: >>> x1 = m_sipopt.sens_sol_state_1[m_sipopt.x1] >>> x2 = m_sipopt.sens_sol_state_1[m_sipopt.x2] >>> x3 = m_sipopt.sens_sol_state_1[m_sipopt.x3] >>> print("Objective = %0.3f" % (x1*2 + x22 + x3*2)) Objective = 0.556

>>> print("x1 = %0.3f" % x1) x1 = 0.333

>>> print("x2 = %0.3f" % x2) x2 = 0.667

>>> print("x3 = %0.3f" % x3) x3 = -0.000

Installing sIPOPT

The sensitivity toolbox requires sIPOPT is installed and available in your system PATH. See the IPOPT documentation for detailed instructions:

https://coin-or.github.io/Ipopt/INSTALL.html
https://projects.coin-or.org/Ipopt/wiki/sIpopt
https://coin-or.github.io/coinbrew/

Note

If you get an error that ipopt_sens cannot be found, you need to make sure sIPOPT was installed and that it is in the system path.