注：中文非直接翻译英文，而是理解加工后的笔记，记录英文仅为学其专业表述。
概述
Constraint optimization, or constraint programming(CP)，约束优化(规划)，用于在一个非常大的候选集合中找到可行解，其中的问题可以用任意约束来建模。
CP基于可行性(寻找可行解)而不是优化(寻找最优解)，它更关注约束和变量，而不是目标函数。
SP-SAT Solver
CP-SAT求解器技术上优于传统CP求解器。
The CP-SAT solver is technologically superior to the original CP solver and should be preferred in almost all situations.
为了增加运算速度，CP求解器处理的都是整数。
如果有非整数项约束，可以先将其乘一个整数，使其变成整数项。
If you begin with a problem that has constraints with non-integer terms, you need to first multiply those constraints by a sufficiently large integer so that all terms are integers
来看一个简单例子：寻找可行解。
- 变量x，y，z，每个只能取值0，1，2。
- eg:
x = model.NewIntVar(0, num_vals - 1, 'x')
- eg:
- 约束条件：x ≠ y
- eg:
model.Add(x != y)
- eg:
核心步骤：
- 声明模型
- 创建变量
- 创建约束条件
- 调用求解器
- 展示结果
from ortools.sat.python import cp_model
def SimpleSatProgram():
"""Minimal CP-SAT example to showcase calling the solver."""
# Creates the model.
model = cp_model.CpModel()
# Creates the variables.
num_vals = 3
x = model.NewIntVar(0, num_vals - 1, 'x')
y = model.NewIntVar(0, num_vals - 1, 'y')
z = model.NewIntVar(0, num_vals - 1, 'z')
# Creates the constraints.
model.Add(x != y)
# Creates a solver and solves the model.
solver = cp_model.CpSolver()
status = solver.Solve(model)
if status == cp_model.FEASIBLE:
print('x = %i' % solver.Value(x))
print('y = %i' % solver.Value(y))
print('z = %i' % solver.Value(z))
SimpleSatProgram()运行得
x = 1
y = 0
z = 0status = solver.Solve(model)返回值状态含义:
- OPTIMAL 找到了最优解。
- FEASIBLE 找到了一个可行解，不过不一定是最优解。
- INFEASIBLE 无可行解。
- MODEL_INVALID 给定的CpModelProto没有通过验证步骤。可以通过调用
ValidateCpModel(model_proto)获得详细的错误。 - UNKNOWN 由于达到了搜索限制，模型的状态未知。
加目标函数，寻找最优解
假设目标函数是求x + 2y + 3z的最大值，在求解器前加
model.Maximize(x + 2 * y + 3 * z)并替换展示条件
if status == cp_model.OPTIMAL:运行得
x = 1
y = 2
z = 2加回调函数，展示所有可行解
去掉目标函数，加上打印回调函数
from ortools.sat.python import cp_model
class VarArraySolutionPrinter(cp_model.CpSolverSolutionCallback):
"""Print intermediate solutions."""
def __init__(self, variables):
cp_model.CpSolverSolutionCallback.__init__(self)
self.__variables = variables
self.__solution_count = 0
def on_solution_callback(self):
self.__solution_count += 1
for v in self.__variables:
print('%s=%i' % (v, self.Value(v)), end=' ')
print()
def solution_count(self):
return self.__solution_count
def SearchForAllSolutionsSampleSat():
"""Showcases calling the solver to search for all solutions."""
# Creates the model.
model = cp_model.CpModel()
# Creates the variables.
num_vals = 3
x = model.NewIntVar(0, num_vals - 1, 'x')
y = model.NewIntVar(0, num_vals - 1, 'y')
z = model.NewIntVar(0, num_vals - 1, 'z')
# Create the constraints.
model.Add(x != y)
# Create a solver and solve.
solver = cp_model.CpSolver()
solution_printer = VarArraySolutionPrinter([x, y, z])
status = solver.SearchForAllSolutions(model, solution_printer)
print('Status = %s' % solver.StatusName(status))
print('Number of solutions found: %i' % solution_printer.solution_count())
SearchForAllSolutionsSampleSat()运行得
x=0 y=1 z=0
x=1 y=2 z=0
x=1 y=2 z=1
x=1 y=2 z=2
x=1 y=0 z=2
x=1 y=0 z=1
x=2 y=0 z=1
x=2 y=1 z=1
x=2 y=1 z=2
x=2 y=0 z=2
x=0 y=1 z=2
x=0 y=1 z=1
x=0 y=2 z=1
x=0 y=2 z=2
x=1 y=0 z=0
x=2 y=0 z=0
x=2 y=1 z=0
x=0 y=2 z=0
Status = OPTIMAL
Number of solutions found: 18展示 intermediate solutions
与展示所有可行解的异同主要在:
- 加
model.Maximize(x + 2 * y + 3 * z) - 换
status = solver.SolveWithSolutionCallback(model, solution_printer)
输出结果为:
x=0 y=1 z=0
x=0 y=2 z=0
x=0 y=2 z=1
x=0 y=2 z=2
x=1 y=2 z=2
Status = OPTIMAL
Number of solutions found: 5与官网结果不一致。https://developers.google.cn/optimization/cp/cp_solver?hl=es-419#run_intermediate_sol