Check if function handle is nonlinear with Matlab

Hello,

is it possible to check with matlab if a Function Handle is nonlinear?

For example:

f1=@(x) x(1)+x(2)-5;
f2=@(x)10*x(1)-x(1)^2+4*x(2)-x(2)^2;

So here i would need something which tells me that f1 is linear and f2 is nonlinear. Thanks

[Paul]

The first function is linear in x, I suppose, but is not linear from a system theoretric perspective, as it doesn't satisfy superposition or scaling. Just want to make sure we're clear on the definition of linear in the question.

Answer 1

[Torsten]

Call your function handle f with symbolic vectors x and y and a symbolic scalar a and check whether

f(x+y) - (f(x) + f(y)) - f(0) == 0
f(a*x) - a*f(x) == 0

[Paul]

Is the second equation missing a term in f(0)?

[Torsten]

For the second case, I think it must read

f(a*x) - f(0) - a*(f(x) - f(0)) == 0

thus

f(a*x) - a*f(x) - (1-a)*f(0) == 0

and for the first case, it must read

f(x+y) - f(0) - (f(x) - f(0) + f(y) - f(0)) == 0

thus

f(x+y) - f(x) - f(y) + f(0) == 0

as I notice now.

[Daniela Würmseer]

Thank you for your answers.

Perhaps i could solve my Problem differently: My Problem is that i have an Algorithm which takes an Optimization Problem with two functions f1, f2 which can be linear or nonlinear. And the Algorithm solves this Problem by a kind of direction Method. So it solves first a "new" Optimization Problem with one function (max alpha) and the constraints are the constraints of the origin Problem and we add as constraints for example: (those are only for illustration and not the real algorithm):

f1(x)-alpha <= 0 and f2(x)-alpha <=0.

I could just define those functions in nonlincon of fmincon but nonlincon is normally for nonlinear functions. Would it be a problem to use nonlincon with linear functions (normally not or)?

For example if f1 and f2 are linear to still use nonlincon like this:

%Definitions of f, x0, Aneq.....
nonlincon = @constr;
    function [cneq,ceq] = constr(x)
            ceq = [];
            cneq(1) = f1(x)-alpha;
            cneq(2) = f2(x)-alpha;
    end
options = optimoptions('fmincon','Algorithm','sqp');
[x,alpha]=fmincon(f,x0,Aneq,bneq,Aeq,beq,lb,[],nonlincon,options);

[Torsten]

I could just define those functions in nonlincon of fmincon but nonlincon is normally for nonlinear functions. Would it be a problem to use nonlincon with linear functions (normally not or)?

No problem. Implementing linear constraints either in lb,ub or A, Aeq, b, beq should only make the code run faster.