第三章 非线性规划

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 3.1 非线性规划的Matlab解法

Matlab中非线性规划的数学模型：

[egin{gathered}
egin{array}{*{20}{c}}
{}
end{array}egin{array}{*{20}{c}}
{}
end{array}min egin{array}{*{20}{c}}
{}
end{array}f(x) hfill \
s.t.left{ egin{gathered}
A cdot x leqslant b, hfill \
Aeq cdot x = beq, hfill \
c(x) leqslant 0, hfill \
ceq(x) = 0, hfill \
lb leqslant x leqslant ub. hfill \
end{gathered} ight. hfill \
end{gathered} ]

Matlab中的命令是：

[x,fval]=fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

例 求下列非线性规划：

[egin{gathered}
egin{array}{*{20}{c}}
{}
end{array}egin{array}{*{20}{c}}
{}
end{array}min egin{array}{*{20}{c}}
{}
end{array}f(x) = x_1^2 + x_2^2 + x_3^2 + 8, hfill \
s.t.left{ egin{gathered}
x_1^2 - {x_2} + x_3^2 geqslant 0, hfill \
{x_1} + x_2^2 + x_3^2 leqslant 20, hfill \
- {x_1} - x_2^2 + 2 = 0, hfill \
{x_2} + 2x_3^2 = 3, hfill \
{x_1},{x_2},{x_3} geqslant 0. hfill \
end{gathered} ight. hfill \
end{gathered} ]

Matlab 编程如下：

共分为三部分：

子函数1：

function f=fun1(x);
f=sum(x.^2)+8;

子函数2：

function [g,h]=fun2(x);
g=[-x(1)^2+x(2)-x(3)^2
x(1)+x(2)^2+x(3)^3-20];  %非线性不等式约束
h=[-x(1)-x(2)^2+2
x(2)+2*x(3)^2-3]; %非线性等式约束

主函数：

[x,y]=fmincon('fun1',rand(3,1),[],[],[],[],zeros(3,1),[],'fun2')

　

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3.2约束极值问题

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3.2.1二次规划

[egin{gathered}
min egin{array}{*{20}{c}}
{}
end{array}frac{1}{2}{{mathbf{x}}^T}{mathbf{Hx}} + {f^T}{mathbf{x}} hfill \
s.t.left{ egin{gathered}
{mathbf{Ax}} leqslant b, hfill \
Aeq cdot {mathbf{x}} = beq, hfill \
lb leqslant {mathbf{x}} leqslant ub. hfill \
end{gathered} ight. hfill \
end{gathered} ]

Matlab中求解二次规划的命令是：

[x,fval]=quadprog(H,f,A,b,Aeq,beq,lb,ub,x0,options)

　　

例子：

[egin{gathered}
egin{array}{*{20}{c}}
{}
end{array}egin{array}{*{20}{c}}
{}
end{array}min f(x) = 2x_1^2 - 4{x_1}{x_2} + 4x_2^2 - 6{x_1} - 3{x_2} hfill \
s.t.left{ egin{gathered}
{x_1} + {x_2} leqslant 3, hfill \
4{x_1} + {x_2} leqslant 9, hfill \
{x_1},{x_2} geqslant 0. hfill \
end{gathered} ight. hfill \
end{gathered} ]

解：程序如下

h=[4,-4;-4,8];
f=[-6;-3];
a=[1,1;4,1];
b=[3;9];
[x,value]=quadprog(h,f,a,b,[],[],zeros(2,1))