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PyROS Solver

PyROS (Pyomo Robust Optimization Solver) is a metasolver capability within Pyomo for solving non-convex, two-stage optimization models using adjustable robust optimization.

It was developed by Natalie M. Isenberg and Chrysanthos E. Gounaris of Carnegie Mellon University, in collaboration with John D. Siirola of Sandia National Labs. The developers gratefully acknowledge support from the U.S. Department of Energy's Institute for the Design of Advanced Energy Systems (IDAES).

Methodology Overview

Below is an overview of the type of optimization models PyROS can accomodate.

  • PyROS is suitable for optimization models of continuous variables that may feature non-linearities (including non-convexities) in both the variables and uncertain parameters.
  • PyROS can handle equality constraints defining state variables, including implicit state variables that cannot be eliminated via reformulation.
  • PyROS allows for two-stage optimization problems that may feature both first-stage and second-stage degrees of freedom.

The general form of a deterministic optimization problem that can be passed into PyROS is shown below:

$$\begin{aligned} \begin{align*} \displaystyle \min_{\substack{x \in \mathcal{X}, \\ z \in \mathbb{R}^n, y\in\mathbb{R}^a}} & ~~ f_1\left(x\right) + f_2\left(x,z,y; q^0\right) & \\\ \displaystyle \text{s.t.} \quad \: & ~~ g_i\left(x, z, y; q^0\right) \leq 0 & \forall i \in \mathcal{I} \\\ & ~~ h_j\left(x,z,y; q^0\right) = 0 & \forall j \in \mathcal{J} \\\ \end{align*} \end{aligned}$$

where:

  • x ∈ 𝒳 are the "design" variables (i.e., first-stage degrees of freedom), where 𝒳 ⊆ ℝm is the feasible space defined by the model constraints that only reference these variables
  • z ∈ ℝn are the "control" variables (i.e., second-stage degrees of freedom)
  • y ∈ ℝa are the "state" variables
  • q ∈ ℝw is the vector of parameters that we shall later consider to be uncertain, and q0 is the vector of nominal values associated with those.
  • f1(x) are the terms of the objective function that depend only on design variables
  • f2(x,z,y;q) are the terms of the objective function that depend on control and/or state variables
  • gi(x,z,y;q) is the ith inequality constraint in set ℐ (see Note)
  • hj(x,z,y;q) is the jth equality constraint in set 𝒥 (see Note)

Note

* Applicable bounds on variables z and/or y are assumed to have been incorporated in the set of inequality constraints ℐ. * A key requirement of PyROS is that each value of (x,z,q) maps to a unique value of y, a property that is assumed to be properly enforced by the system of equality constraints 𝒥. If such unique mapping does not hold, then the selection of 'state' (i.e., not degree of freedom) variables y is incorrect, and one or more of the y variables should be appropriately redesignated to be part of either x or z.

In order to cast the robust optimization counterpart formulation of the above model, we shall now assume that the uncertain parameters may attain any realization from within an uncertainty set 𝒬 ⊆ ℝw, such that q0 ∈ 𝒬. The set 𝒬 is assumed to be closed and bounded, while it can be either continuous or discrete.

Based on the above notation, the form of the robust counterpart addressed in PyROS is shown below:

$$\begin{aligned} \begin{align*} \displaystyle \min_{x \in \mathcal{X}} & \displaystyle \max_{q \in \mathcal{Q}} & \displaystyle \min_{z \in \mathbb{R}^n, y \in \mathbb{R}^a} \ \ & \displaystyle ~~ f_1\left(x\right) + f_2\left(x, z, y, q\right) & & \\\ & & \text{s.t.} \quad \:& \displaystyle ~~ g_i\left(x, z, y, q\right) \leq 0 & & \forall i \in \mathcal{I}\\\ & & & \displaystyle ~~ h_j\left(x, z, y, q\right) = 0 & & \forall j \in \mathcal{J} \end{align*} \end{aligned}$$

In order to solve problems of the above type, PyROS implements the Generalized Robust Cutting-Set algorithm developed in [GRCSPaper].

When using PyROS, please consider citing the above paper.

PyROS Required Inputs

The required inputs to the PyROS solver are the following:

  • The determinisitic optimization model
  • List of first-stage ("design") variables
  • List of second-stage ("control") variables
  • List of parameters to be considered uncertain
  • The uncertainty set
  • Subordinate local and global NLP optimization solvers

Below is a list of arguments that PyROS expects the user to provide when calling the solve command. Note how all but the model argument must be specified as kwargs.

model : ConcreteModel

A ConcreteModel object representing the deterministic model.

first_stage_variables : list(Var)

A list of Pyomo Var objects representing the first-stage degrees of freedom (design variables) in model.

second_stage_variables : list(Var)

A list of Pyomo Var objects representing second-stage degrees of freedom (control variables) in model.

uncertain_params : list(Param)

A list of Pyomo Param objects in deterministic_model to be considered uncertain. These specified Param objects must have the property mutable=True.

uncertainty_set : UncertaintySet

A PyROS UncertaintySet object representing uncertainty in the space of those parameters listed in the uncertain_params object.

local_solver : Solver

A Pyomo Solver instance for a local NLP optimization solver.

global_solver : Solver

A Pyomo Solver instance for a global NLP optimization solver.

Note

Any variables in the model not specified to be first- or second-stage variables are automatically considered to be state variables.

PyROS Solver Interface

pyomo.contrib.pyros.PyROS

Note

Solving the master problems globally (via option solve_masters_globally=True) is one of the requirements to guarantee robust optimality; solving the master problems locally can only lead to a robust feasible solution.

Note

Selecting worst-case objective (via option objective_focus=ObjectiveType.worst_case) is one of the requirements to guarantee robust optimality; selecting nominal objective can only lead to a robust feasible solution, albeit one that has optimized the sum of first- and (nominal) second-stage objectives.

Note

To utilize option p_robustness, a dictionary of the following form must be supplied via the kwarg: There must be a key (str) called 'rho', which maps to a non-negative value, where '1+rho' defines a bound for the ratio of the objective that any scenario may exhibit compared to the nominal objective.

PyROS Uncertainty Sets

PyROS contains pre-implemented UncertaintySet specializations for many types of commonly used uncertainty sets. Additional capabilities for intersecting multiple PyROS UncertaintySet objects so as to create custom sets are also provided via the IntersectionSet class. Custom user-specified sets can also be defined via the base UncertaintySet class.

Mathematical representations of the sets are shown below, followed by the class descriptions.

PyROS Uncertainty Sets
Uncertainty Set Type Set Representation
BoxSet $Q_X = \left\{q \in \mathbb{R}^n : q^\ell \leq q \leq q^u\right\} \\ q^\ell \in \mathbb{R}^n \\ q^u \in \mathbb{R}^n : \left\{q^\ell \leq q^u\right\}$
CardinalitySet $Q_C = \left\{q \in \mathbb{R}^n : q = q^0 + (\hat{q} \circ \xi) \text{ for some } \xi \in \Xi_C\right\}\\ \Xi_C = \left\{\xi \in [0, 1]^n : \displaystyle\sum_{i=1}^{n} \xi_i \leq \Gamma\right\} \\ \Gamma \in [0, n] \\ \hat{q} \in \mathbb{R}^{n}_{+} \\ q^0 \in \mathbb{R}^n$
BudgetSet $Q_B = \left\{q \in \mathbb{R}^n_+: \displaystyle\sum_{i \in B_\ell} q_i \leq b_\ell \ \forall \ell \in \left\{1,\ldots,L\right\} \right\} \\ b_\ell \in \mathbb{R}^{L}_+$
FactorModelSet $Q_F = \left\{q \in \mathbb{R}^n: \displaystyle q = q^0 + \Psi \xi \text{ for some }\xi \in \Xi_F\right\} \\ \Xi_F = \left\{ \xi \in \left[-1, 1\right]^F, \left\lvert \displaystyle \sum_{f=1}^{F} \xi_f\right\rvert \leq \beta F \right\} \\ \beta \in [0,1] \\ \Psi \in \mathbb{R}^{n \times F}_+ \\ q^0 \in \mathbb{R}^n$
PolyhedralSet $Q_P = \left\{q \in \mathbb{R}^n: \displaystyle A q \leq b \right\} \\ A \in \mathbb{R}^{m \times n} \\ b \in \mathbb{R}^{m} \\ q^0 \in \mathbb{R}^n: {Aq^0 \leq b}$
AxisAlignedEllipsoidalSet $Q_A = \left\{q \in \mathbb{R}^n: \displaystyle \sum\limits_{i=1 : \atop \left\{ \alpha_i > 0 \right\} } \left(\frac{q_i - q_i^0}{\alpha_i} \right)^2 \leq 1 , \quad q_i = q^0_i \quad \forall i : \left\{\alpha_i=0\right\}\right\} \\ \alpha \in \mathbb{R}^n_+, \\ q^0 \in \mathbb{R}^n$
EllipsoidalSet $Q_E = \left\{q \in \mathbb{R}^n: \displaystyle q = q^0 + P^{1/2} \xi \text{ for some } \xi \in \Xi_E \right\} \\ \Xi_E = \left\{\xi \in \mathbb{R} : \xi^T\xi \leq s \right\} \\ P \in \mathbb{S}^{n\times n}_+ \\ s \in \mathbb{R}_+ \\ q^0 \in \mathbb{R}^n$
UncertaintySet $Q_U = \left\{q \in \mathbb{R}^n: \displaystyle g_i(q) \leq 0 \quad \forall i \in \left\{1,\ldots,m \right\}\right\} \\ m \in \mathbb{N}_+ \\ g_i : \mathbb{R}^n \mapsto \mathbb{R} \, \forall i \in \left\{1,\ldots,m\right\}, \\ q^0 \in \mathbb{R}^n : \left\{g_i(q^0) \leq 0 \ \forall i \in \left\{1,\ldots,m\right\}\right\}$
DiscreteScenariosSet $Q_D = \left\{q^s : s = 0,\ldots,D \right\} \\ D \in \mathbb{N} \\ q^s \in \mathbb{R}^n \forall s \in \left\{ 0,\ldots,D\right\}$
IntersectionSet $Q_I = \left\{q \in \mathbb{R}^n: \displaystyle q \in \bigcap_{i \in \left\{1,\ldots,m\right\}} Q_i\right\} \\ Q_i \subset \mathbb{R}^n \quad \forall i \in \left\{1,\ldots,m\right\}$

Note

Each of the PyROS uncertainty set classes inherits from the UncertaintySet base class.

PyROS Uncertainty Set Classes

pyomo.contrib.pyros.uncertainty_sets.BoxSet

pyomo.contrib.pyros.uncertainty_sets.CardinalitySet

pyomo.contrib.pyros.uncertainty_sets.BudgetSet

pyomo.contrib.pyros.uncertainty_sets.FactorModelSet

pyomo.contrib.pyros.uncertainty_sets.PolyhedralSet

pyomo.contrib.pyros.uncertainty_sets.AxisAlignedEllipsoidalSet

pyomo.contrib.pyros.uncertainty_sets.EllipsoidalSet

pyomo.contrib.pyros.uncertainty_sets.UncertaintySet

pyomo.contrib.pyros.uncertainty_sets.DiscreteScenarioSet

pyomo.contrib.pyros.uncertainty_sets.IntersectionSet

PyROS Usage Example

We will use an example to illustrate the usage of PyROS. The problem we will use is called hydro and comes from the GAMS example problem database in The GAMS Model Library. The model was converted to Pyomo format via the GAMS Convert tool.

This model is a QCQP with 31 variables. Of these variables, 13 represent degrees of freedom, with the additional 18 being state variables. The model features 6 linear inequality constraints, 6 linear equality constraints, 6 non-linear (quadratic) equalities, and a quadratic objective. We have augmented this model by converting one objective coefficient, two constraint coefficients, and one constraint right-hand side into Param objects so that they can be considered uncertain later on.

Note

Per our analysis, the hydro problem satisfies the requirement that each value of (x,z,q) maps to a unique value of y, which indicates a proper partition of variables between (first- or second-stage) degrees of freedom and state variables.

Step 0: Import Pyomo and the PyROS Module

In anticipation of using the PyROS solver and building the deterministic Pyomo model:

>>> # === Required import === >>> import pyomo.environ as pyo >>> import pyomo.contrib.pyros as pyros

>>> # === Instantiate the PyROS solver object === >>> pyros_solver = pyo.SolverFactory("pyros")

Step 1: Define the Deterministic Problem

The deterministic Pyomo model for hydro is shown below.

Note

Primitive data (Python literals) that have been hard-coded within a deterministic model cannot be later considered uncertain, unless they are first converted to Param objects within the ConcreteModel object. Furthermore, any Param object that is to be later considered uncertain must have the property mutable=True.

Note

In case modifying the mutable property inside the deterministic model object itself is not straight-forward in your context, you may consider adding the following statement after import pyomo.environ as pyo but before defining the model object: pyo.Param.DefaultMutable = True. Note how this sets the default mutable property in all Param objects in the ensuing model instance to True; consequently, this solution will not work with Param objects for which the mutable=False property was explicitly enabled inside the model object.

>>> # === Construct the Pyomo model object === >>> m = pyo.ConcreteModel() >>> m.name = "hydro"

>>> # === Define variables === >>> m.x1 = pyo.Var(within=pyo.Reals,bounds=(150,1500),initialize=150) >>> m.x2 = pyo.Var(within=pyo.Reals,bounds=(150,1500),initialize=150) >>> m.x3 = pyo.Var(within=pyo.Reals,bounds=(150,1500),initialize=150) >>> m.x4 = pyo.Var(within=pyo.Reals,bounds=(150,1500),initialize=150) >>> m.x5 = pyo.Var(within=pyo.Reals,bounds=(150,1500),initialize=150) >>> m.x6 = pyo.Var(within=pyo.Reals,bounds=(150,1500),initialize=150) >>> m.x7 = pyo.Var(within=pyo.Reals,bounds=(0,1000),initialize=0) >>> m.x8 = pyo.Var(within=pyo.Reals,bounds=(0,1000),initialize=0) >>> m.x9 = pyo.Var(within=pyo.Reals,bounds=(0,1000),initialize=0) >>> m.x10 = pyo.Var(within=pyo.Reals,bounds=(0,1000),initialize=0) >>> m.x11 = pyo.Var(within=pyo.Reals,bounds=(0,1000),initialize=0) >>> m.x12 = pyo.Var(within=pyo.Reals,bounds=(0,1000),initialize=0) >>> m.x13 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x14 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x15 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x16 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x17 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x18 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x19 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x20 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x21 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x22 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x23 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x24 = pyo.Var(within=pyo.Reals,bounds=(0,None),initialize=0) >>> m.x25 = pyo.Var(within=pyo.Reals,bounds=(100000,100000),initialize=100000) >>> m.x26 = pyo.Var(within=pyo.Reals,bounds=(60000,120000),initialize=60000) >>> m.x27 = pyo.Var(within=pyo.Reals,bounds=(60000,120000),initialize=60000) >>> m.x28 = pyo.Var(within=pyo.Reals,bounds=(60000,120000),initialize=60000) >>> m.x29 = pyo.Var(within=pyo.Reals,bounds=(60000,120000),initialize=60000) >>> m.x30 = pyo.Var(within=pyo.Reals,bounds=(60000,120000),initialize=60000) >>> m.x31 = pyo.Var(within=pyo.Reals,bounds=(60000,120000),initialize=60000)

>>> # === Define parameters === >>> m.set_of_params = pyo.Set(initialize=[0, 1, 2, 3]) >>> nominal_values = {0:82.8*0.0016, 1:4.97, 2:4.97, 3:1800} >>> m.p = pyo.Param(m.set_of_params, initialize=nominal_values, mutable=True)

>>> # === Specify the objective function === >>> m.obj = pyo.Objective(expr=m.p[0]m.x12 + 82.88*m.x1 + 82.8*0.0016*m.x2*2 + ... 82.882.8*8*m.x2 + 82.8*0.0016*m.x3*2 + 82.88*m.x3 + ... 82.8*0.0016*m.x4*2 + 82.88*m.x4 + 82.8*0.0016*m.x5*2 + ... 82.88*m.x5 + 82.8*0.0016*m.x6*2 + 82.88*m.x6 + 248400, ... sense=pyo.minimize)

>>> # === Specify the constraints === >>> m.c2 = pyo.Constraint(expr=-m.x1 - m.x7 + m.x13 + 1200<= 0) >>> m.c3 = pyo.Constraint(expr=-m.x2 - m.x8 + m.x14 + 1500 <= 0) >>> m.c4 = pyo.Constraint(expr=-m.x3 - m.x9 + m.x15 + 1100 <= 0) >>> m.c5 = pyo.Constraint(expr=-m.x4 - m.x10 + m.x16 + m.p[3] <= 0) >>> m.c6 = pyo.Constraint(expr=-m.x5 - m.x11 + m.x17 + 950 <= 0) >>> m.c7 = pyo.Constraint(expr=-m.x6 - m.x12 + m.x18 + 1300 <= 0) >>> m.c8 = pyo.Constraint(expr=12*m.x19 - m.x25 + m.x26 == 24000) >>> m.c9 = pyo.Constraint(expr=12*m.x20 - m.x26 + m.x27 == 24000) >>> m.c10 = pyo.Constraint(expr=12*m.x21 - m.x27 + m.x28 == 24000) >>> m.c11 = pyo.Constraint(expr=12*m.x22 - m.x28 + m.x29 == 24000) >>> m.c12 = pyo.Constraint(expr=12*m.x23 - m.x29 + m.x30 == 24000) >>> m.c13 = pyo.Constraint(expr=12*m.x24 - m.x30 + m.x31 == 24000) >>> m.c14 = pyo.Constraint(expr=-8e-5*m.x7*2 + m.x13 == 0) >>> m.c15 = pyo.Constraint(expr=-8e-5m.x8*2 + m.x14 == 0) >>> m.c16 = pyo.Constraint(expr=-8e-5m.x9*2 + m.x15 == 0) >>> m.c17 = pyo.Constraint(expr=-8e-5m.x10*2 + m.x16 == 0) >>> m.c18 = pyo.Constraint(expr=-8e-5m.x11*2 + m.x17 == 0) >>> m.c19 = pyo.Constraint(expr=-8e-5m.x12*2 + m.x18 == 0) >>> m.c20 = pyo.Constraint(expr=-4.97m.x7 + m.x19 == 330) >>> m.c21 = pyo.Constraint(expr=-m.p[1]m.x8 + m.x20 == 330) >>> m.c22 = pyo.Constraint(expr=-4.97m.x9 + m.x21 == 330) >>> m.c23 = pyo.Constraint(expr=-4.97*m.x10 + m.x22 == 330) >>> m.c24 = pyo.Constraint(expr=-m.p[2]m.x11 + m.x23 == 330) >>> m.c25 = pyo.Constraint(expr=-4.97m.x12 + m.x24 == 330)

Step 2: Define the Uncertainty

First, we need to collect into a list those Param objects of our model that represent potentially uncertain parameters. For purposes of our example, we shall assume uncertainty in the model parameters (m.p[0], m.p[1], m.p[2], m.p[3]), for which we can conveniently utilize the m.p object (itself an indexed Param object).

>>> # === Specify which parameters are uncertain === >>> uncertain_parameters = [m.p] # We can pass IndexedParams this way to PyROS, or as an expanded list per index

Note

Any Param object that is to be considered uncertain by PyROS must have the property mutable=True.

PyROS will seek to identify solutions that remain feasible for any realization of these parameters included in an uncertainty set. To that end, we need to construct an UncertaintySet object. In our example, let us utilize the BoxSet constructor to specify an uncertainty set of simple hyper-rectangular geometry. For this, we will assume each parameter value is uncertain within a percentage of its nominal value. Constructing this specific UncertaintySet object can be done as follows.

>>> # === Define the pertinent data === >>> relative_deviation = 0.15 >>> bounds = [(nominal_values[i] - relative_deviation*nominal_values[i], ... nominal_values[i] + relative_deviation*nominal_values[i]) ... for i in range(4)]

>>> # === Construct the desirable uncertainty set === >>> box_uncertainty_set = pyros.BoxSet(bounds=bounds)

Step 3: Solve with PyROS

PyROS requires the user to supply one local and one global NLP solver to be used for solving sub-problems. For convenience, we shall have PyROS invoke BARON as both the local and the global NLP solver.

>>> # === Designate local and global NLP solvers === >>> local_solver = pyo.SolverFactory('baron') >>> global_solver = pyo.SolverFactory('baron')

Note

Additional solvers to be used as backup can be designated during the solve statement via the config options backup_local_solvers and backup_global_solvers presented above.

The final step in solving a model with PyROS is to designate the remaining required inputs, namely first_stage_variables and second_stage_variables. Below, we present two separate cases.

PyROS Termination Conditions

PyROS will return one of six termination conditions upon completion. These termination conditions are tabulated below.

c

Termination Condition

 Description

pyrosTerminationCondition.robust_optimal

The final solution is robust optimal

pyrosTerminationCondition.robust_feasible

The final solution is robust feasible

pyrosTerminationCondition.robust_infeasible

The posed problem is robust infeasible

pyrosTerminationCondition.max_iter

Maximum number of GRCS iteration reached

pyrosTerminationCondition.time_out

Maximum number of time reached

pyrosTerminationCondition.subsolver_error

Unacceptable return status(es) from a user-supplied sub-solver

A Single-Stage Problem

If we choose to designate all variables as either design or state variables, without any control variables (i.e., all degrees of freedom are first-stage), we can use PyROS to solve the single-stage problem as shown below. In particular, let us instruct PyROS that variables m.x1 through m.x6, m.x19 through m.x24, and m.x31 correspond to first-stage degrees of freedom.

>>> # === Designate which variables correspond to first- and second-stage degrees of freedom === >>> first_stage_variables =[m.x1, m.x2, m.x3, m.x4, m.x5, m.x6, ... m.x19, m.x20, m.x21, m.x22, m.x23, m.x24, m.x31] >>> second_stage_variables = [] >>> # The remaining variables are implicitly designated to be state variables

>>> # === Call PyROS to solve the robust optimization problem === >>> results_1 = pyros_solver.solve(model = m, ... first_stage_variables = first_stage_variables, ... second_stage_variables = second_stage_variables, ... uncertain_params = uncertain_parameters, ... uncertainty_set = box_uncertainty_set, ... local_solver = local_solver, ... global_solver= global_solver, ... options = { ... "objective_focus": pyros.ObjectiveType.worst_case, ... "solve_master_globally": True, ... "load_solution":False ... }) =========================================================================================== PyROS: Pyomo Robust Optimization Solver ... =========================================================================================== ... INFO: Robust optimal solution identified. Exiting PyROS.

>>> # === Query results === >>> time = results_1.time >>> iterations = results_1.iterations >>> termination_condition = results_1.pyros_termination_condition >>> objective = results_1.final_objective_value >>> # === Print some results === >>> single_stage_final_objective = round(objective,-1) >>> print("Final objective value: %s" % single_stage_final_objective) Final objective value: 48367380.0 >>> print("PyROS termination condition: %s" % termination_condition) PyROS termination condition: pyrosTerminationCondition.robust_optimal

PyROS Results Object

The results object returned by PyROS allows you to query the following information from the solve call: total iterations of the algorithm iterations, CPU time time, the GRCS algorithm termination condition pyros_termination_condition, and the final objective function value final_objective_value. If the option load_solution = True (default), the variables in the model will be loaded to the solution determined by PyROS and can be obtained by querying the model variables. Note that in the results obtained above, we set load_solution = False. This is to ensure that the next set of runs shown here can utilize the original deterministic model, as the initial point can affect the performance of sub-solvers.

Note

The reported final_objective_value and final model variable values depend on the selection of the option objective_focus. The final_objective_value is the sum of first-stage and second-stage objective functions. If objective_focus = ObjectiveType.nominal, second-stage objective and variables are evaluated at the nominal realization of the uncertain parameters, q0. If objective_focus = ObjectiveType.worst_case, second-stage objective and variables are evaluated at the worst-case realization of the uncertain parameters, qk* where k* = argmaxk ∈ 𝒦f2(x, zk, yk, qk) .

An example of how to query these values on the previously obtained results is shown in the code above.

A Two-Stage Problem

For this next set of runs, we will assume that some of the previously designated first-stage degrees of freedom are in fact second-stage ones. PyROS handles second-stage degrees of freedom via the use of decision rules, which is controlled with the config option decision_rule_order presented above. Here, we shall select affine decision rules by setting decision_rule_order to the value of 1.

>>> # === Define the variable partitioning >>> first_stage_variables =[m.x5, m.x6, m.x19, m.x22, m.x23, m.x24, m.x31] >>> second_stage_variables = [m.x1, m.x2, m.x3, m.x4, m.x20, m.x21] >>> # The remaining variables are implicitly designated to be state variables

>>> # === Call PyROS to solve the robust optimization problem === >>> results_2 = pyros_solver.solve(model = m, ... first_stage_variables = first_stage_variables, ... second_stage_variables = second_stage_variables, ... uncertain_params = uncertain_parameters, ... uncertainty_set = box_uncertainty_set, ... local_solver = local_solver, ... global_solver = global_solver, ... options = { ... "objective_focus": pyros.ObjectiveType.worst_case, ... "solve_master_globally": True, ... "decision_rule_order": 1 ... }) =========================================================================================== PyROS: Pyomo Robust Optimization Solver ... ... INFO: Robust optimal solution identified. Exiting PyROS.

>>> # === Compare final objective to the singe-stage solution >>> two_stage_final_objective = round(pyo.value(results_2.final_objective_value),-1) >>> percent_difference = 100 * (two_stage_final_objective - single_stage_final_objective)/(single_stage_final_objective) >>> print("Percent objective change relative to constant decision rules objective: %.2f %%" % percent_difference) Percent objective change relative to constant decision rules objective: -24...

In this example, when we compare the final objective value in the case of constant decision rules (no second-stage recourse) and affine decision rules, we see there is a ~25% decrease in total objective value.

The Price of Robustness

Using appropriately constructed hierarchies, PyROS allows for the facile comparison of robust optimal objectives across sets to determine the "price of robustness." For the set we considered here, the BoxSet, we can create such a hierarchy via an array of relative_deviation parameters to define the size of these uncertainty sets. We can then loop through this array and call PyROS within a loop to identify robust solutions in light of each of the specified BoxSet objects.

>>> # This takes a long time to run and therefore is not a doctest
>>> # === An array of maximum relative deviations from the nominal uncertain parameter values to utilize in constructing box sets
>>> relative_deviation_list = [0.00, 0.10, 0.20, 0.30, 0.40]
>>> # === Final robust optimal objectives
>>> robust_optimal_objectives = []
>>> for relative_deviation in relative_deviation_list: # doctest: +SKIP
...   bounds = [(nominal_values[i] - relative_deviation*nominal_values[i],
...                   nominal_values[i] + relative_deviation*nominal_values[i])
...                   for i in range(4)]
...   box_uncertainty_set = pyros.BoxSet(bounds = bounds)
...   results = pyros_solver.solve(model = m,
...                                     first_stage_variables = first_stage_variables,
...                                     second_stage_variables = second_stage_variables,
...                                     uncertain_params = uncertain_parameters,
...                                     uncertainty_set = box_uncertainty_set,
...                                     local_solver = local_solver,
...                                     global_solver = global_solver,
...                                     options = {
...                                        "objective_focus": pyros.ObjectiveType.worst_case,
...                                        "solve_master_globally": True,
...                                        "decision_rule_order": 1
...                                     })
...   if results.pyros_termination_condition != pyros.pyrosTerminationCondition.robust_optimal:
...       print("This instance didn't solve to robust optimality.")
...       robust_optimal_objective.append("-----")
...   else:
...       robust_optimal_objectives.append(str(results.final_objective_value))

For this example, we obtain the following price of robustness results:

c

Uncertainty Set Size (+/-) o

 Robust Optimal Objective % Increase x

0.00

 35,837,659.18 0.00 %

0.10

 36,135,191.59 0.82 %

0.20

 36,437,979.81 1.64 %

0.30

 43,478,190.92 17.57 %

0.40

 robust_infeasible —–

Note how, in the case of the last uncertainty set, we were able to utilize PyROS to show the robust infeasibility of the problem.

o Relative Deviation from Nominal Realization

x Relative to Deterministic Optimal Objective

This clearly illustrates the impact that the uncertainty set size can have on the robust optimal objective values. Price of robustness studies like this are easily implemented using PyROS.

Warning

PyROS is still under a beta release. Please provide feedback and/or report any problems by opening an issue on the Pyomo GitHub page.