[Documentation work in progress...]
robust has a variety of tools beyond the basics to aid engineering design under uncertainty.
All of these methods have been implemented in the robustSPpaper code repository, which defines the models used in [Ozturk, 2019].
There are many possible **options to RobustModel, but the primary function of the options is to be able to choose between the different RGP approximation methods, which have differing levels of conservativeness. We prefer to define the three methods in a dict for ease access, and call RobustModel, for example for Best Pairs, as follows:
methods = [{'name': 'Best Pairs', 'twoTerm': True, 'boyd': False, 'simpleModel': False},
{'name': 'Linearized Perturbations', 'twoTerm': False, 'boyd': False, 'simpleModel': False},
{'name': 'Simple Conservative', 'twoTerm': False, 'boyd': False, 'simpleModel': True}]
method = methods[0] # Best Pairs
robust_model = RobustModel(m, uncertainty_set, gamma=Gamma, twoTerm=method['twoTerm'],
boyd=method['boyd'], simpleModel=method['simpleModel'])For uncertainty_set, 'box' and 'elliptical' are currently supported, and define the ∞- and 2-norms respectively.
Robust optimization provides probabilistic guarantees of constraint satisfaction under the defined uncertainty sets, but it is possible that the real distributions of uncertainty are better defined by probability distributions. Within robust, we have a framework to generate random Monte Carlo (MC) simulations of the uncertain parameters to be able to determine the probability of constraint violation (i.e. probability of failure) of models, as well as the expectation and standard deviation of the objective cost. To generate samples of uncertain parameters and simulate a robust model's performance over these samples, we use the simulate module of robust. Good implementations of robust.simulate are in robustSPpaper.
There is one additional step before GP models can be simulated. Every free variable that is fixed during MC simulation must have a `fix = True` attribute, as following:
A = Variable("A", "-", "aspect ratio", fix = True)This allows for the optimizer to know that these variables (eg. aspect ratio in aircraft design), once the optimization is complete, cannot change. Other variables, eg. state variables such as speed and altitude of an aircraft, can change during MC simulation to fulfill constraints.
There are a daunting number of parameters for the functions in this module. This I would recommend overcoming by creating a parameter function such as the following, which I have filled with defaults:
def pof_parameters():
model = SomeGPModelWithUncertainParams() # A GP model with uncertain parameters
number_of_time_average_solves = 3 # number of solves for computing time cost, set to 1 if not important
## PARAMETERS THAT AFFECT NUMBER OF GP MODELS SAMPLED
# Each combination of methods, uncertainty_sets and gammas generates a
# new RGP model designed with these inputs, to be tested against MC samples.
methods = [{'name': 'Best Pairs', 'twoTerm': True, 'boyd': False, 'simpleModel': False},
{'name': 'Linearized Perturbations', 'twoTerm': False, 'boyd': False, 'simpleModel': False},
{'name': 'Simple Conservative', 'twoTerm': False, 'boyd': False, 'simpleModel': True}
]
uncertainty_sets = ['box', 'elliptical']
nGammas = 11
gammas = np.linspace(0, 1.0, nGammas)
# LINEARIZATION PARAMETERS
# These exist because RGPs/RSPs implement robust linear programs in their backend.
# Defaults are great for engineering purposes.
min_num_of_linear_sections = 3
max_num_of_linear_sections = 99
linearization_tolerance = 1e-4
# PARALLELIZATION (will be available in future, please keep as False for now)
parallel = False
# SAMPLING PARAMETERS
number_of_iterations = 100 # number of MC samples
distribution = 'normal' # or 'uniform'
# GENERATING SAMPLES (contained in directly_uncertain_vars_subs)
nominal_solution, nominal_solve_time, nominal_number_of_constraints, directly_uncertain_vars_subs = \
simulate.generate_model_properties(model, number_of_time_average_solves, number_of_iterations, distribution)
verbosity = 1 # controls printouts for simulations
return [model, methods, gammas, number_of_iterations,
min_num_of_linear_sections, max_num_of_linear_sections, verbosity, linearization_tolerance,
number_of_time_average_solves, uncertainty_sets, nominal_solution, directly_uncertain_vars_subs, parallel,
nominal_number_of_constraints, nominal_solve_time]For Monte Carlo simulations, it is important to note that the solution time is proportional to the product of number of methods, uncertainty sets, gammas and samples. Simulating a model is equivalent to optimizing the model over its remaining free variables (without the 'fix' attribute). As such, one MC simulation usually takes similar time to one solution to the un-robustified model. Furthermore, MC simulations take longer for robustified models vs. unrobust ones, since infeasibility can be detected faster than a feasible solution. If in a time crunch, it is recommended that one method, set and gamma is chosen for simulation purposes.
Once the parameters are generated, the robust.simulate module can be used to generate MC data.
solutions, solve_times, simulation_results, number_of_constraints = simulate.variable_gamma_results(
model, methods, gammas, number_of_iterations,
min_num_of_linear_sections,
max_num_of_linear_sections, verbosity, linearization_tolerance,
number_of_time_average_solves,
uncertainty_sets, nominal_solution, directly_uncertain_vars_subs, parallel=parallel)It is highly recommended that you save/pickle the results, since MC simulations can have a large time cost. robustSPpaper/SimPleAC_save and robustSPpaper/SimPleAC_pof_simulate have simple demonstrations of saving/pickling.