当 objective 函数处于约束中时使用 fmincon MATLAB 求解优化

Question

我要解决：

我使用以下 MATLAB 代码，但它不起作用。

有人可以指导我吗？

function f=objfun

f=-f;

function [c1,c2,c3]=constraint(x)
a1=1.1; a2=1.1; a3=1.1;
c1=f-log(a1)-log(x(1)/(x(1)+1)); 
c2=f-log(a2)-log(x(2)/(x(2)+1))-log(1-x(1)); 
c3=f-log(a3)-log(1-x(1))-log(1-x(2));


x0=[0.01;0.01]; 
[x,fval]=fmincon('objfun',x0,[],[],[],[],[0;0],[1;1],'constraint')

Answer 1

你需要稍微翻转一下问题。您正在尝试找到使 3 个 LHS 函数中的最小值最大的点 x（即 (l_1,l_2)）。所以，你可以用伪代码重写你的问题，

maximise, by varying x in [0,1] X [0,1]
       min([log(a1)+log(x(1)/(x(1)+1)) ...
            log(a2)+log(x(2)/(x(2)+1))+log(1-x(1)) ...
            log(a3)+log(1-x(1))+log(1-x(2))])

由于Matlab有fmincon，将其重写为最小化问题，

minimise, by varying x in [0,1] X [0,1]
       max(-[log(a1)+log(x(1)/(x(1)+1)) ...
             log(a2)+log(x(2)/(x(2)+1))+log(1-x(1)) ...
             log(a3)+log(1-x(1))+log(1-x(2))])

所以实际代码是

F=@(x) max(-[log(a1)+log(x(1)/(x(1)+1)) ...
             log(a2)+log(x(2)/(x(2)+1))+log(1-x(1)) ...
             log(a3)+log(1-x(1))+log(1-x(2))])
[L,fval]=fmincon(F,[0.5 0.5])

哪个returns

L =
    0.3383    0.6180
fval =
    1.2800

Answer 2

也可以在convex optimization package CVX中用下面的MATLAB代码解决这个问题：

cvx_begin
 variables T(1);
 variables x1(1);
 variables x2(1);

 maximize(T)
 subject to:
  log(a1) + x1 - log_sum_exp([0, x1]) >= T;
  log(a2) + x2 - log_sum_exp([0, x2]) + log(1 - exp(x1)) >= T;
  log(a3) +  log(1 - exp(x1)) +  log(1 - exp(x2)) >= T;
  x1 <= 0;
  x2 <= 0;
cvx_end
l1 = exp(x1); l2 = exp(x2);

要使用 CVX，每个约束和 objective 函数都必须以使用 CVX 规则集证明是凸的方式编写。进行替换 x1 = log(l1) 和 x2 = log(l2) 可以做到这一点。注意：log_sum_exp([0,x1]) = log(exp(0) + exp(x1)) = log(1 + l1)

这也是 returns 答案：l1 = .3383，l2 = .6180，T = -1.2800

当 objective 函数处于约束中时使用 fmincon MATLAB 求解优化

Solve optimization using fmincon MATLAB when objective function is in constraints

matlab

optimization

minimization