Extremum seeking

Extremum Seeking Optimization¶

In this example we demonstrate extremum seeking optimization. The optimum of the test evaluate function would drift around a center point and we would be trying to follow the trend by applying extremum seeking technique.

In [1]:

# If you encounter the "Initializing libomp.dylib, but found libomp.dylib already initialized." error
# Please run this cell

import os

# If you encounter the "Initializing libomp.dylib, but found libomp.dylib already initialized." error # Please run this cell import os

In [2]:

# set values if testing
SMOKE_TEST = os.environ.get("SMOKE_TEST")
NUM_STEPS = 10 if SMOKE_TEST else 1000

# set values if testing SMOKE_TEST = os.environ.get("SMOKE_TEST") NUM_STEPS = 10 if SMOKE_TEST else 1000

In [3]:

import numpy as np
from xopt.generators.es.extremumseeking import ExtremumSeekingGenerator
from xopt.vocs import VOCS
from xopt.evaluator import Evaluator
from xopt import Xopt
from tqdm.auto import tqdm
import warnings
warnings.filterwarnings("ignore")

import numpy as np from xopt.generators.es.extremumseeking import ExtremumSeekingGenerator from xopt.vocs import VOCS from xopt.evaluator import Evaluator from xopt import Xopt from tqdm.auto import tqdm import warnings warnings.filterwarnings("ignore")

Extremum seeking test problem¶

This test problem is a 10-D quadratic function, with its optimum drifting around the initial position. We also add some noise to make the problem more realistic.

In [4]:

np.random.seed(42)  # set deterministic run
import pandas as pd
nES = 10

# This global dict is used as a counter to emulate drifting
states = {
    'count': 0
}

noise = 0.1 * np.random.randn(NUM_STEPS)

# This is the unknown optimal point
p_opt = 1.5 * (2 * np.random.rand(nES) - 1)

# Various frequencies for unknown points
w_opt = 0.25 + 2 * np.random.rand(nES)

def f_ES_minimize(input_dict):
    p = []
    for i in range(10):
        p.append(input_dict[f'p{i}'])
    p = np.array(p)
    
    # Vary the optimal point with time
    p_opt_i = np.zeros(nES)
    i = states['count']
    
    outcome_dict = {}
    for n in np.arange(nES):
        p_opt_i[n] = p_opt[n] * (1 + np.sin(2 * np.pi * w_opt[n] * i / 2000))
    # This simple cost will be distance from the optimal point
    f_val = np.sum((p - p_opt_i) ** 2) + noise[i]
    
    states['count'] += 1
    outcome_dict = {'f': f_val, 'p_opt': pd.Series(p_opt_i)}
    
    return outcome_dict

np.random.seed(42) # set deterministic run import pandas as pd nES = 10 # This global dict is used as a counter to emulate drifting states = { 'count': 0 } noise = 0.1 * np.random.randn(NUM_STEPS) # This is the unknown optimal point p_opt = 1.5 * (2 * np.random.rand(nES) - 1) # Various frequencies for unknown points w_opt = 0.25 + 2 * np.random.rand(nES) def f_ES_minimize(input_dict): p = [] for i in range(10): p.append(input_dict[f'p{i}']) p = np.array(p) # Vary the optimal point with time p_opt_i = np.zeros(nES) i = states['count'] outcome_dict = {} for n in np.arange(nES): p_opt_i[n] = p_opt[n] * (1 + np.sin(2 * np.pi * w_opt[n] * i / 2000)) # This simple cost will be distance from the optimal point f_val = np.sum((p - p_opt_i) ** 2) + noise[i] states['count'] += 1 outcome_dict = {'f': f_val, 'p_opt': pd.Series(p_opt_i)} return outcome_dict

Run ES on the test problem (YAML method)¶

In [5]:

YAML = """
max_evaluations: 5000
generator:
    name: extremum_seeking
    k: 2.0
    oscillation_size: 0.1
    decay_rate: 1.0
evaluator:
    function: __main__.f_ES_minimize
vocs:
    variables:
        p0: [-2, 2]
        p1: [-2, 2]
        p2: [-2, 2]
        p3: [-2, 2]
        p4: [-2, 2]
        p5: [-2, 2]
        p6: [-2, 2]
        p7: [-2, 2]
        p8: [-2, 2]
        p9: [-2, 2]
    objectives:
        f: MINIMIZE
"""

X = Xopt.from_yaml(YAML)
X.max_evaluations = NUM_STEPS

X

YAML = """ max_evaluations: 5000 generator: name: extremum_seeking k: 2.0 oscillation_size: 0.1 decay_rate: 1.0 evaluator: function: __main__.f_ES_minimize vocs: variables: p0: [-2, 2] p1: [-2, 2] p2: [-2, 2] p3: [-2, 2] p4: [-2, 2] p5: [-2, 2] p6: [-2, 2] p7: [-2, 2] p8: [-2, 2] p9: [-2, 2] objectives: f: MINIMIZE """ X = Xopt.from_yaml(YAML) X.max_evaluations = NUM_STEPS X

Out[5]:

            Xopt
________________________________
Version: 0+untagged.1.ge872ea5
Data size: 0
Config as YAML:
dump_file: null
evaluator:
  function: __main__.f_ES_minimize
  function_kwargs: {}
  max_workers: 1
  vectorized: false
generator:
  decay_rate: 1.0
  k: 2.0
  name: extremum_seeking
  oscillation_size: 0.1
max_evaluations: 1000
serialize_inline: false
serialize_torch: false
strict: true
vocs:
  constants: {}
  constraints: {}
  objectives:
    f: MINIMIZE
  observables: []
  variables:
    p0:
    - -2.0
    - 2.0
    p1:
    - -2.0
    - 2.0
    p2:
    - -2.0
    - 2.0
    p3:
    - -2.0
    - 2.0
    p4:
    - -2.0
    - 2.0
    p5:
    - -2.0
    - 2.0
    p6:
    - -2.0
    - 2.0
    p7:
    - -2.0
    - 2.0
    p8:
    - -2.0
    - 2.0
    p9:
    - -2.0
    - 2.0

In [6]:

# Reset global counter to guarantee deterministic optimization
states['count'] = 0

X.random_evaluate(1)
X.step()

# Reset global counter to guarantee deterministic optimization states['count'] = 0 X.random_evaluate(1) X.step()

Now you can go directly to the Visualization section and check out the results.

Run ES on the test problem (API method)¶

VOCS¶

We'll set the bounds for all the variables pi to [-2, 2].

In [7]:

variables = {}
for i in range(nES):
    variables[f'p{i}'] = [-2, 2]

vocs = VOCS(
    variables=variables,
    objectives={'f': 'MINIMIZE'},
)

variables = {} for i in range(nES): variables[f'p{i}'] = [-2, 2] vocs = VOCS( variables=variables, objectives={'f': 'MINIMIZE'}, )

In [8]:

vocs

vocs

Out[8]:

VOCS(variables={'p0': [-2.0, 2.0], 'p1': [-2.0, 2.0], 'p2': [-2.0, 2.0], 'p3': [-2.0, 2.0], 'p4': [-2.0, 2.0], 'p5': [-2.0, 2.0], 'p6': [-2.0, 2.0], 'p7': [-2.0, 2.0], 'p8': [-2.0, 2.0], 'p9': [-2.0, 2.0]}, constraints={}, objectives={'f': 'MINIMIZE'}, constants={}, observables=[])

Evaluator¶

In [9]:

evaluator = Evaluator(function=f_ES_minimize)

evaluator = Evaluator(function=f_ES_minimize)

Generator¶

In [10]:

generator = ExtremumSeekingGenerator(vocs=vocs)

generator = ExtremumSeekingGenerator(vocs=vocs)

In [11]:

generator.dict()

generator.dict()

Out[11]:

{'k': 2.0, 'oscillation_size': 0.1, 'decay_rate': 1.0}

Note that ES has 3 hyper-parameters: k, oscillation_size, and decay_rate.

• k: ES feedback gain (set k < 0 for maximization instead of minimization)
• oscillation_size: ES dithering size
• decay_rate: This value is optional, it causes the oscillation sizes to naturally decay. If you want the parameters to persistently oscillate without decay, set decay_rate = 1.0

Run the optimization¶

In [12]:

X = Xopt(vocs=vocs, evaluator=evaluator, generator=generator)

X = Xopt(vocs=vocs, evaluator=evaluator, generator=generator)

In [13]:

X.max_evaluations = NUM_STEPS

X.max_evaluations = NUM_STEPS

In [14]:

# Reset global counter to guarantee deterministic optimization
states['count'] = 0

for i in tqdm(range(NUM_STEPS)):
    X.step()

# Reset global counter to guarantee deterministic optimization states['count'] = 0 for i in tqdm(range(NUM_STEPS)): X.step()

Visualization¶

In [15]:

import matplotlib.pyplot as plt

import matplotlib.pyplot as plt

In [16]:

# Plot all results
plt.figure(1,figsize=(8,10))

plt.subplot(2,1,1)
plt.plot(X.data['f'])
plt.ylabel('ES cost')
plt.xticks([])


plt.subplot(2,1,2)
plt.plot(X.data[[f'p{i}' for i in range(10)]],alpha=0.25)
_p_opt = np.vstack(X.data['p_opt'].values).astype(float)  # do not use p_opt as var name!
plt.plot(_p_opt, 'k--')
plt.plot(2+np.zeros(NUM_STEPS),'r')
plt.plot(-2+np.zeros(NUM_STEPS),'r')
plt.legend(frameon=False)
plt.ylabel('ES parameter')
plt.xlabel('ES step')

plt.tight_layout()

# Plot all results plt.figure(1,figsize=(8,10)) plt.subplot(2,1,1) plt.plot(X.data['f']) plt.ylabel('ES cost') plt.xticks([]) plt.subplot(2,1,2) plt.plot(X.data[[f'p{i}' for i in range(10)]],alpha=0.25) _p_opt = np.vstack(X.data['p_opt'].values).astype(float) # do not use p_opt as var name! plt.plot(_p_opt, 'k--') plt.plot(2+np.zeros(NUM_STEPS),'r') plt.plot(-2+np.zeros(NUM_STEPS),'r') plt.legend(frameon=False) plt.ylabel('ES parameter') plt.xlabel('ES step') plt.tight_layout()

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In [17]:

# Plot Individual Parameter Trajectories
plt.figure(2,figsize=(15,8))

for n in np.arange(nES):
    plt.subplot(2,5,n+1)
    plt.plot(X.data[f'p{n}'],label=f'$p^{{ES}}_{n+1}$')
    plt.plot(_p_opt[:,n],'k--',label=f'$p^*_{n+1}$')
    plt.plot(2+np.zeros(NUM_STEPS),'r--')
    plt.plot(-2+np.zeros(NUM_STEPS),'r--')
    plt.ylim([-3,5])
    plt.legend(frameon=False,loc=1)
    if n == 0:
        plt.ylabel('parameters')
    elif n == 5:
        plt.ylabel('parameters')
    else:
        plt.yticks([])
    if n > 4:
        plt.xlabel('ES step')
    else:
        plt.xticks([])

plt.tight_layout()

# Plot Individual Parameter Trajectories plt.figure(2,figsize=(15,8)) for n in np.arange(nES): plt.subplot(2,5,n+1) plt.plot(X.data[f'p{n}'],label=f'$p^{{ES}}_{n+1}$') plt.plot(_p_opt[:,n],'k--',label=f'$p^*_{n+1}$') plt.plot(2+np.zeros(NUM_STEPS),'r--') plt.plot(-2+np.zeros(NUM_STEPS),'r--') plt.ylim([-3,5]) plt.legend(frameon=False,loc=1) if n == 0: plt.ylabel('parameters') elif n == 5: plt.ylabel('parameters') else: plt.yticks([]) if n > 4: plt.xlabel('ES step') else: plt.xticks([]) plt.tight_layout()

In [17]: