How can I minimize a function of functions?

Hello,

I have to minimize a function of functions and I am not sure how can I do it. Below I show the functions and the independent variables. The objective function is g and I have linear constraints ( 0<x1<5 and 0<x2<0.2). I will appreciate it if someone can help me!

syms x1 x2

y1 = x1^2 + x2;

y2 = x1+1+ x2*x1;

y10 = (5*x1)^2 + x2 + x1;

y20 = 3*x1 + 1 + x2*x1 + 2*x2;

g = abs(y1-y10)+ abs(y2-y20);

Answer 1

[Rik]

You can probably do it with the symbolic toolbox as well, but below is a numerical solution.

y1 =@(x1,x2) x1^2 + x2;
y2 =@(x1,x2) x1+1+ x2*x1;
y10 =@(x1,x2) (5*x1)^2 + x2 + x1; 
y20 =@(x1,x2) 3*x1 + 1 + x2*x1 + 2*x2;
g =@(x1,x2) abs(y1(x1,x2)-y10(x1,x2))+ abs(y2(x1,x2)-y20(x1,x2));

fun=@(x)g(x(1),x(2));
x0=[2.5;0.1];%initial guess
[A,b,Aeq,beq]=deal([]);
lb=[0;0];
ub=[5;0.2];

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x = 2×1

1.0e+-7 * 0.3351 0.5012

So the solution is essentially at (0,0). This is not unexpected if you plot the value of g over the ranges of x1 and x2:

[X1,X2]=ndgrid(linspace(0,5),linspace(0,0.2));
surf(X1,X2,arrayfun(g,X1,X2))

[Carolina Bernal Botero]

Thank you very much Rik, your answer was very useful!

[Rik]

You're welcome. If my answer helped you, please consider marking my answer as accepted answer.