Constrained Multi-Output Min-Max

In [1]:

import obsidian
print(f'obsidian version: ' + obsidian.__version__)

import pandas as pd
import plotly.express as px
import plotly.io as pio
pio.renderers.default = "plotly_mimetype+notebook"

obsidian version: 0.8.0

Introduction¶

In this tutorial, we will see how to use obsidian for multi-output optimization. To demonstrate the versatility of the approach, we will seek to maximize one response while minimizing the other.

$$\underset{X}{argmax} HV\left(+f\left(y_1\right) -f\left(y_2\right)\right)$$

Furthermore, we will apply a linear constraint on the input variables; requiring that the $X_1 + X_2 \leq 6 $.

Set up parameter space and initialize a design¶

In [2]:

from obsidian import Campaign, Target, ParamSpace, BayesianOptimizer
from obsidian.parameters import Param_Continuous

In [3]:

params = [
    Param_Continuous('X1', 0, 10),
    Param_Continuous('X2', 0, 10),
    ]

X_space = ParamSpace(params)
target = [
    Target('Response 1', aim='max'),
    Target('Response 2', aim='min')
]
campaign = Campaign(X_space, target, seed=0)
X0 = campaign.designer.initialize(4, 'LHS')

X0

Out[3]:

	X1	X2
0	6.25	8.75
1	1.25	6.25
2	3.75	1.25
3	8.75	3.75

Collect results (e.g. from a simulation)¶

In [4]:

from obsidian.experiment import Simulator
from obsidian.experiment.benchmark import branin_currin

simulator = Simulator(X_space, branin_currin, name='Response', eps=0.05)
y0 = simulator.simulate(X0)
Z0 = pd.concat([X0, y0], axis=1)

campaign.add_data(Z0)
campaign.data

Out[4]:

	X1	X2	Response 1	Response 2	Iteration
Observation ID
0	6.25	8.75	-154.239634	-5.248404	0
1	1.25	6.25	-8.751383	-6.534766	0
2	3.75	1.25	-29.269633	-13.059490	0
3	8.75	3.75	-26.305781	-7.544486	0

Fit an optimizer and visualize results¶

In [5]:

campaign.fit()

GP model has been fit                       to data with a train-score of: 1 for response: Response 1
GP model has been fit                       to data with a train-score of: 1 for response: Response 2

In [6]:

from obsidian.plotting import surface_plot, optim_progress

In [7]:

surface_plot(campaign.optimizer)

Optimize new experiment suggestions¶

In [8]:

from obsidian.constraints import InConstraint_Generic

Note: It is a good idea to balance a set of acquisition functions with those that prefer design-space exploration. This helps to ensure that the optimizer is not severely misled by deficiencies in the dataset, particularly for small data. It also helps to ascertain a global optimum.

A simple choice is Space Filling (SF) although Negative Integrated Posterior Variance (NIPV) is available for single-output optimizations; and there are various other acquisiiton functions whose hyperparameters can be tuned to manage the "explore-exploit" balance.

In [9]:

X_suggest, eval_suggest = campaign.optimizer.suggest(acquisition = [{'NEHVI':{'ref_point':[-350, -20]}}, 'SF'],
                                                     # X1 + X2 <= 6, written as -X1 - X2 >= -6
                                                     ineq_constraints = [InConstraint_Generic(X_space, indices=[0,1], coeff=[-1,-1], rhs=-6)])

In [10]:

pd.concat([X_suggest, eval_suggest], axis=1)

Out[10]:

	X1	X2	Response 1 (pred)	Response 1 lb	Response 1 ub	Response 2 (pred)	Response 2 lb	Response 2 ub	f(Response 1)	f(Response 2)	aq Value	aq Value (joint)	aq Method	Expected Hypervolume (joint)	Expected Pareto
0	1.855368e+00	3.467079e-15	-38.805388	-101.625151	24.014386	-11.443693	-8.463541	-14.423845	0.236315	0.973132	4.233924	4.233924	NEHVI	3.389429	False
1	5.782685e-15	2.974608e+00	-29.479032	-100.736606	41.778553	-9.269232	-5.810519	-12.727946	0.375486	0.340895	0.350581	0.350581	SF	3.406860	False

Collect data at new suggestions¶

In [11]:

y_iter1 = pd.DataFrame(simulator.simulate(X_suggest))
Z_iter1 = pd.concat([X_suggest, y_iter1, eval_suggest], axis=1)
campaign.add_data(Z_iter1)
campaign.data.tail()

Out[11]:

	X1	X2	Response 1	Response 2	Iteration	Response 1 (pred)	Response 1 lb	Response 1 ub	Response 2 (pred)	Response 2 lb	...	f(Response 2)	aq Value	aq Value (joint)	aq Method	Expected Hypervolume (joint)	Expected Pareto	Response 1 (max) (iter)	Response 2 (max) (iter)	Hypervolume (iter)	Pareto Front
Observation ID
1	1.250000e+00	6.250000e+00	-8.751383	-6.534766	0	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	-8.751383	-5.248404	949.270665	True
2	3.750000e+00	1.250000e+00	-29.269633	-13.059490	0	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	-8.751383	-5.248404	949.270665	False
3	8.750000e+00	3.750000e+00	-26.305781	-7.544486	0	NaN	NaN	NaN	NaN	NaN	...	NaN	NaN	NaN	NaN	NaN	NaN	-8.751383	-5.248404	949.270665	False
4	1.855368e+00	3.467079e-15	-116.993962	-14.357309	1	-38.805388	-101.625151	24.014386	-11.443693	-8.463541	...	0.973132	4.233924	4.233924	NEHVI	3.389429	False	-8.751383	-2.714921	1355.934274	False
5	5.782685e-15	2.974608e+00	-178.155376	-2.714921	1	-29.479032	-100.736606	41.778553	-9.269232	-5.810519	...	0.340895	0.350581	0.350581	SF	3.406860	False	-8.751383	-2.714921	1355.934274	True

5 rows × 22 columns

Repeat as desired¶

In [12]:

for iter in range(5):
    campaign.fit()
    X_suggest, eval_suggest = campaign.optimizer.suggest(acquisition = [{'NEHVI':{'ref_point':[-350, -20]}}, 'SF'],
                                                         ineq_constraints = [InConstraint_Generic(X_space, indices=[0,1], coeff=[-1,-1], rhs=-6)])
    y_iter = pd.DataFrame(simulator.simulate(X_suggest))
    Z_iter = pd.concat([X_suggest, y_iter, eval_suggest], axis=1)
    campaign.add_data(Z_iter)

GP model has been fit                       to data with a train-score of: 0.998 for response: Response 1
GP model has been fit                       to data with a train-score of: 0.999 for response: Response 2
GP model has been fit                       to data with a train-score of: 1 for response: Response 1
GP model has been fit                       to data with a train-score of: 1 for response: Response 2
GP model has been fit                       to data with a train-score of: 1 for response: Response 1
GP model has been fit                       to data with a train-score of: 1 for response: Response 2
GP model has been fit                       to data with a train-score of: 0.999 for response: Response 1
GP model has been fit                       to data with a train-score of: 1 for response: Response 2
GP model has been fit                       to data with a train-score of: 1 for response: Response 1
GP model has been fit                       to data with a train-score of: 1 for response: Response 2

In [13]:

optim_progress(campaign)

In [14]:

surface_plot(campaign.optimizer, response_id = 0)

In [15]:

surface_plot(campaign.optimizer, response_id = 1)