Linera Optimization

优化问题的要素:

objective，欲优化的量。比如某函数的最大值或最小值。
constraints, 约束变量。基于问题的特定需求，对可行解进行约束。

线性优化用于计算一组线性关系建模问题的最优解。

谷歌提供的开源库Glop可用来求解该问题。

解决问题步骤:

创建变量
定义约束
定义目标函数
声明求解器
调用求解器
展示结果

eg1:

objective: max(x+y)
constraints:
- 0 ≤ x ≤ 1
- 0 ≤ y ≤ 2

from ortools.linear_solver import pywraplp

def main():
  solver = pywraplp.Solver('SolveSimpleSystem',
                           pywraplp.Solver.GLOP_LINEAR_PROGRAMMING)
  # Create the variables x and y.
  x = solver.NumVar(0, 1, 'x')
  y = solver.NumVar(0, 2, 'y')
  # Create the objective function, x + y.
  objective = solver.Objective()
  objective.SetCoefficient(x, 1)
  objective.SetCoefficient(y, 1)
  objective.SetMaximization()
  # Call the solver and display the results.
  solver.Solve()
  print('Solution:')
  print('x = ', x.solution_value())
  print('y = ', y.solution_value())

if __name__ == '__main__':
  main()

运行得

Solution:
x =  1.0
y =  2.0

eg2:

objective: max(3x + 4y)
constraints:
- x + 2y ≤ 14
- 3x – y ≥ 0
- x – y ≤ 2

"""Linear optimization example"""
from ortools.linear_solver import pywraplp

def main():
  # Instantiate a Glop solver, naming it LinearExample.
  solver = pywraplp.Solver('LinearExample',
                           pywraplp.Solver.GLOP_LINEAR_PROGRAMMING)

# Create the two variables and let them take on any value.
  x = solver.NumVar(-solver.infinity(), solver.infinity(), 'x')
  y = solver.NumVar(-solver.infinity(), solver.infinity(), 'y')

  # Constraint 1: x + 2y <= 14.
  constraint1 = solver.Constraint(-solver.infinity(), 14)
  constraint1.SetCoefficient(x, 1)
  constraint1.SetCoefficient(y, 2)

  # Constraint 2: 3x - y >= 0.
  constraint2 = solver.Constraint(0, solver.infinity())
  constraint2.SetCoefficient(x, 3)
  constraint2.SetCoefficient(y, -1)

  # Constraint 3: x - y <= 2.
  constraint3 = solver.Constraint(-solver.infinity(), 2)
  constraint3.SetCoefficient(x, 1)
  constraint3.SetCoefficient(y, -1)

  # Objective function: 3x + 4y.
  objective = solver.Objective()
  objective.SetCoefficient(x, 3)
  objective.SetCoefficient(y, 4)
  objective.SetMaximization()

  # Solve the system.
  solver.Solve()
  opt_solution = 3 * x.solution_value() + 4 * y.solution_value()
  print('Number of variables =', solver.NumVariables())
  print('Number of constraints =', solver.NumConstraints())
  # The value of each variable in the solution.
  print('Solution:')
  print('x = ', x.solution_value())
  print('y = ', y.solution_value())
  # The objective value of the solution.
  print('Optimal objective value =', opt_solution)
if __name__ == '__main__':
  main()

运行得

Number of variables = 2
Number of constraints = 3
Solution:
x =  5.999999999999998
y =  3.9999999999999996
Optimal objective value = 33.99999999999999

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