無拘束最佳化

Python 最佳化入門

Jasmin Ludolf

Content Developer

什麼是無拘束最佳化？

對輸入變數不設任何限制下，求函式的極大或極小值

SciPy

呈正向趨勢並向上箭頭的圖

單變數無拘束最佳化

from scipy.optimize import minimize_scalar


def objective_function(x): 
  return x**2 - 12*x + 4


result = minimize_scalar(objective_function) 
print(result)

     fun: -32.0
 message: 'Optimization terminated successfully; The
 returned value satisfies the termination criteria 
 (using xtol = 1.48e-08 )'
    nfev: 10
     nit: 4
 success: True
       x: 6.000000000000001

2 * x**2 - 3 * x + 1 的圖形

單變數無拘束最佳化結果

     fun: -32.0
 message: 'Optimization terminated successfully; The
 returned value satisfies the termination criteria 
 (using xtol = 1.48e-08 )'
    nfev: 10
     nit: 4
 success: True
       x: 6.000000000000001

result.x

6.000000000000001

fun：在極小值處的函式值
message：狀態訊息
nfev：演算法評估函式的次數
nit：達到解所需的迭代次數
success：是否找到最適解的布林值
x：最佳解

尋找極大值

極大化問題的曲線圖。

def objective_function(x): 
  return 40 * q - 0.5 * q**2

-1*(40 * q - 0.5 * q**2)
-40 * q + 0.5 * q**2

def negated_function(x): 
  return -40 * q + 0.5 * q**2
result = minimize_scalar(negated_function) 
print(f"The maximum is {result.x:.2f} in two 
      decimals")

The maximum is 40.00 in two decimals

多變數無拘束最佳化

from scipy.optimize import minimize


def objective_function(a): 
  return (a[0] - 2)**2 + (a[1] - 3)**2 + 1.5


x0 = [1, 2]

result = minimize(objective_function, x0)

a[0] 代表 x，a[1] 代表 y

多變數無拘束最佳化結果

print(result)
print(f"minimum is (x, y) = ({result.x[0]:.2f}, {result.x[1]:.2f}) in two decimals.")

      fun: 1.5000000000000002

 hess_inv: array([[ 0.75, -0.25],
       [-0.25,  0.75]])

      jac: array([0., 0.])

  message: 'Optimization terminated successfully.'
     nfev: 9
      nit: 2
     njev: 3

   status: 0

  success: True
        x: array([1.99999999, 2.99999999])
minimum is (x, y) = (2.00, 3.00) in two decimals.

一起來練習吧！

Python 最佳化入門