带不等式的凸约束优化

Python 优化入门

Jasmin Ludolf

Content Developer

角点解 vs. 内点解

  • Alexia 只有 5 小时可工作

$$ w\leq 5$$

  • 两个约束：
    • 一天 24 小时
    • 工作 5 小时
  • 约束形成角点
  • 内点解不是交点

无差异曲线与约束。最优效用的无差异曲线经过约束的交点。

Python 优化入门

带产能约束的制造商

  • 汽车厂在两座工厂 $A$、$B$ 生产同款车型
    • 产量：$q_A$, $q_B$
    • 产能：$q_A\leq 90$, $q_B\leq 90$
    • 成本：$C_A(q)=3q$, $C_B(q)=3.5q$
  • 需求：
    • $P=120-Q$
  • 合同：
    • $Q\geq 92$
  • 目标：最大化利润
    • $\displaystyle\max \Pi(q_A,q_B)$

汽车装配线

Python 优化入门

利润最大化

  • 目标：
    • $\Pi = R-C$
      • $R = PQ$
Python 优化入门

利润最大化

  • 目标：
    • $\Pi = R-C$
      • $R = PQ = (120-Q)Q $
Python 优化入门

利润最大化

  • 目标：

    • $\Pi = R-C$
      • $R = PQ = (120-Q)Q=\left[120-(q_A+q_B)\right](q_A+q_B)$
      • $C=C_A+C_B=3q_A+3.5q_B$

  • 边界

    • $0\leq q_A, q_B\leq 90$
  • 约束
    • $Q\geq92\Leftrightarrow 92\leq q_A+q_B$
Python 优化入门

模型表述

$$\max_{q_A,q_B}R(q_A,q_B)-C(q_A,q_B)$$

$$s.t.$$

$$R(q_A,q_B)=\left[120-(q_A+q_B)\right](q_A+q_B)$$

$$\ \ \ \ \ C(q_A,q_B)=3q_A+3.5q_B$$

$$\ \ \ 0\leq q_A, q_B\leq 90$$

$$ \ \ \ \ 92\leq q_A+q_B$$

from scipy.optimize import minimize,\
        Bounds, LinearConstraint


def R(q):
    return (120 - (q[0] + q[1]
       )) * (q[0] + q[1])

def C(q): return 3*q[0] + 3.5*q[1] 

def profit(q): return R(q) - C(q)


bounds = Bounds([0, 0], [90, 90])


constraints = LinearConstraint([1, 1], 
                               lb=92)
Python 优化入门

用 SciPy 最大化利润

result = minimize(lambda q: -profit(q),

                  [50, 50],

                  bounds=bounds,
                  constraints=constraints)

print(result.message)
print(f'The optimal number of cars produced in plant A is: {result.x[0]:.2f}')
print(f'The optimal number of cars produced in plant B is: {result.x[1]:.2f}')
print(f'The firm made: ${-result.fun:.2f}')

Optimization terminated successfully
The optimal number of cars produced in plant A is: 90.00
The optimal number of cars produced in plant B is: 2.00
The firm made: $2299.00
Python 优化入门

SciPy 中的非线性约束

from scipy.optimize import NonlinearConstraint 
import numpy as np


constraints = NonlinearConstraint(lambda q: q[0] + q[1], lb=92, ub=np.inf)


result = minimize(lambda q: -profit(q), 
                  [50, 50], 
                  bounds=Bounds([0, 0], [90, 90]),
                  constraints=constraints)
Python 优化入门

使用 NonlinearConstraint 的解

print(result.message)
print(f'The optimal number of cars produced in plant A is: {result.x[0]:.2f}')
print(f'The optimal number of cars produced in plant B is: {result.x[1]:.2f}')
print(f'The firm made: ${-result.fun:.2f}') 

Optimization terminated successfully
The optimal number of cars produced in plant A is: 90.00
The optimal number of cars produced in plant B is: 2.00
The firm made: $2299.00
  • 线性问题用 LinearConstraint()
  • 其他情况用 NonlinearConstraint()
Python 优化入门

Passons à la pratique !

Python 优化入门

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