【数学建模】Python+Gurobi求解非线性规划模型
目录
1 概述
2 算例
2.1 算例
2.2 参数设置
2.3 Python代码实现
2.4 求解结果
1 概述
如果目标函数或约束条件中包含非线性函数，就称这种规划问题为非线性规划问题。
参考：（非线性规划Python）计及动态约束及节能减排环保要求的经济调度
2 算例
2.1 算例
2.2 参数设置
求解NLP/非凸问题时，Python+Gurobi 的参数设置
1）需要设置参数'NonConvex=2';
2）Gurobi接受的幂函数y =中，α的最大值为2.c.虽然有解，但不一定是最优值。
3）求解时间看模型规模。
观察模型可知，要想求此模型，需要在编程中把多个变量乘积变成两两相乘的形式。令、y23=x2x3即可实现此功能。建模问题解决了，调用Gurobi求解器即可求解了。可是运行模型的时候，出问题了: GurobiError: Q matrix is notpositive semi-definite (PSD). Set NonConvex parameter to 2 to solvemodel.错误很明显，告诉你需要设定参数NonConvex=2，Python+Gurobi设定参数的方法，即:
M_NLP.Params.NonConvex=2
2.3 Python代码实现
from gurobipy import *
# 创建模型
NLP=Model("NLP")
# 变量声明
x1 =NLP.addVar(lb=1,ub=3, name="x1")
x2 =NLP.addVar(lb=1,ub=3, name="x2")
x3 =NLP.addVar(lb=1,ub=3, name="x3")
x4 =NLP.addVar(lb=1,ub=3, name="x4")
y14 =NLP.addVar(lb=0,ub=300, name="y14")
y23 =NLP.addVar(lb=0,ub=300, name="y23")
# 设置目标函数
NLP.setObjective(y14*(x1+x2+x3)+x2,GRB.MAXIMIZE)
# 添加约束
NLP.addConstr(y14*y23>=20,"Con1")
NLP.addConstr(x1*x1+x2*x2+x3*x3+x4*x4==30,"Con2")
# 表示乘积项
NLP.addConstr(y14==x1*x4,"Con_y14")
NLP.addConstr(y23==x2*x3,"Con_y23")
NLP.Params.NonConvex=2
# Optimize model
NLP.optimize()
NLP.write("NLP.lp")
print('**************')
print(' The optimal solution ')
print('**************')
print('Obj is :',NLP.ObjVal) # 输出目标值
print('x1 is :',x1.x) # 输出 x1 的值
print('x2 is :',x2.x) # 输出 x2 的值
print('x3 is :',x3.x) # 输出 x3 的值
print('x4 is :',x4.x) # 输出 x4 的值from gurobipy import *
# 创建模型
NLP=Model("NLP")
# 变量声明
x1 =NLP.addVar(lb=1,ub=3, name="x1")
x2 =NLP.addVar(lb=1,ub=3, name="x2")
x3 =NLP.addVar(lb=1,ub=3, name="x3")
x4 =NLP.addVar(lb=1,ub=3, name="x4")
y14 =NLP.addVar(lb=0,ub=300, name="y14")
y23 =NLP.addVar(lb=0,ub=300, name="y23")
# 设置目标函数
NLP.setObjective(y14*(x1+x2+x3)+x2,GRB.MAXIMIZE)
# 添加约束
NLP.addConstr(y14*y23>=20,"Con1")
NLP.addConstr(x1*x1+x2*x2+x3*x3+x4*x4==30,"Con2")
# 表示乘积项
NLP.addConstr(y14==x1*x4,"Con_y14")
NLP.addConstr(y23==x2*x3,"Con_y23")
NLP.Params.NonConvex=2
# Optimize model
NLP.optimize()
NLP.write("NLP.lp")
print('**************')
print(' The optimal solution ')
print('**************')
print('Obj is :',NLP.ObjVal) # 输出目标值
print('x1 is :',x1.x) # 输出 x1 的值
print('x2 is :',x2.x) # 输出 x2 的值
print('x3 is :',x3.x) # 输出 x3 的值
print('x4 is :',x4.x) # 输出 x4 的值
2.4 求解结果
Set parameter NonConvex to value 2
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64)
Thread count: 8 physical cores, 16 logical processors, using up to 16 threads
Optimize a model with 0 rows, 6 columns and 0 nonzeros
Model fingerprint: 0x11cd71a3
Model has 3 quadratic objective terms
Model has 4 quadratic constraints
Coefficient statistics:
Matrix range [0e+00, 0e+00]
QMatrix range [1e+00, 1e+00]
QLMatrix range [1e+00, 1e+00]
Objective range [1e+00, 1e+00]
QObjective range [2e+00, 2e+00]
Bounds range [1e+00, 3e+02]
RHS range [0e+00, 0e+00]
QRHS range [2e+01, 3e+01]
Continuous model is non-convex -- solving as a MIP
Presolve time: 0.00s
Presolved: 28 rows, 14 columns, 74 nonzeros
Presolved model has 1 quadratic constraint(s)
Presolved model has 9 bilinear constraint(s)
Variable types: 14 continuous, 0 integer (0 binary)
Root relaxation: objective 7.463397e+01, 16 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 74.63397 0 3 - 74.63397 - - 0s
H 0 0 73.6047208 74.63397 1.40% - 0s
0 0 73.62217 0 3 73.60472 73.62217 0.02% - 0s
0 0 73.61845 0 3 73.60472 73.61845 0.02% - 0s
0 0 73.61845 0 3 73.60472 73.61845 0.02% - 0s
0 2 73.61845 0 3 73.60472 73.61845 0.02% - 0s
Cutting planes:
PSD: 1
Explored 5 nodes (37 simplex iterations) in 0.02 seconds (0.00 work units)
Thread count was 16 (of 16 available processors)
Solution count 1: 73.6047
Optimal solution found (tolerance 1.00e-04)
Best objective 7.360472078597e+01, best bound 7.361117907001e+01, gap 0.0088%
**************
The optimal solution
**************
Obj is : 73.60472078597236
x1 is : 2.9999999998974762
x2 is : 2.5748464524640973
x3 is : 2.317361807798582
x4 is : 2.9999999997674647
Process finished with exit code 0