MATLAB: Minimize function under inequality conditions

inequality conditionminimizeOptimization Toolbox

I am currently trying to find a minimum to the following function:
rent=@(x)8000-1000*norm(x);
where x is the vector [x,y]
There are furthermore 4 inequality constraint that need to be satisfied. They all state that the distance between x=[x;y] and a point in space does not exceed 10 units. So they are all of the following form:
sqrt((x-6)^2+(y-2)^2)<=10
My code so far looks the following:
P1=[6;2];
P2=[-1;5];
P3=[-5;-1];
P4=[0;-8];
d=10; %maximal distance
x=sym('x',[2,1]);
%Inequality conditions
S1=@(x)norm([P1(1,1)-x(1,1);P1(2,1)-x(2,1)])-d;%<=0



S2=@(x)norm([P2(1,1)-x(1,1);P2(2,1)-x(2,1)])-d;%<=0
S3=@(x)norm([P3(1,1)-x(1,1);P3(2,1)-x(2,1)])-d;%<=0
S4=@(x)norm([P4(1,1)-x(1,1);P4(2,1)-x(2,1)])-d;%<=0
%Function to minimize
rent=@(x)8000-1000*norm(x);
Is it possible to solve the problem using fmincon or what function should i use?

Best Answer

FMINCON is appropriate, but you need to express your objective and constraints in terms of differentiable functions, e.g.,
(x-6)^2+(y-2)^2-100 <= 0
Similarly, minimizing the non-differentiable function 8000-1000*norm(x) is the same as minimizing the differentiable function -norm(x)^2.
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