The implementation of the method for solving multi-objective optimization probelem. The main algorithm can be found on the article [1]. Unlike methods based on evolutionary computation such as a genetic algorithm, descent methods can only provide a single solution, so the algorithm must be run iteratively to obtain a Pareto optimal set.
- Python 3.8 or above.
Define the objective function to be minimized in the obj_func.py. We also define a list of objective functions in the same file. An example is as follows:
class Obj:
def f(self, x: NDArray[(1, ...), np.float64]) -> Any:
return x[0]**2 + 3 * (x[1] - 1)**2
def g(self, x: NDArray[(1, ...), np.float64]) -> Any:
return 2 * (x[0] - 1)**2 + x[1]**2
def Fs(self, x: NDArray[(1, ...), np.float64]) -> NDArray[(1, ...), np.float64]:
return np.array([self.f(x), self.g(x)])
def Fss(self):
return np.array([self.f, self.g])The main algorithm lies in steepest_descent.py, and we can run a test code by calling steepest() method. An example is as follows:
import numpy as np
from steepest_descent import SteepestDescent
from obj_func import Obj
sd = SteepestDescent(
ndim=2,
nu=0.8,
sigma=0.8,
eps=1e-5,
)
obj = Obj()
x_init = np.array([1, 2])
f_opt = sd.steepest(x_init)
print(obj.Fs(f_opt)) # Pareto optimal| Parameter | Type | Default | Description |
|---|---|---|---|
| ndim | int (x >= 1) | required | The dimension of decision variable |
| nu | float (0.0 < x < 1.0) | required | The parameter for an algorithm |
| sigma | float (0.0 < x < 1.0) | required | The parameter for an algorithm |
| eps | float (x > 0) | required | Absolute error in xopt between iterations that is acceptable for convergence. |