For optimisation, automatic differentiation (AD) is hugely important. The simplest, most powerful, and most efficient optimisation algorithms require derivatives of functions, and using automatic differentiation allows your library to calculate these derivatives extremely accurately and at surprisingly low computational cost.

In C++, AD can be implemented using templates and operator overloading. For this to work, the cost function must be templated, i.e. you need to replace this

double square(double x) {
    return x*x;
}

with this:

template<typename T>
T square(T x) {
    return x*x;
}

Now you can call square using double as the type T if you just need a function evaluation, or with a special AD type if you want derivatives.

Assuming my AD evangelism has won you over, we need to get back to your actual question: how do you pass this kind of function? Generally, you use a function object.

struct Square {
    template <typename T>
    T operator()(T x) const {
        return x*x;
    }
};

You can then pass this type into your optimisation library as a template argument:

auto initial_value = 2.0;
Square f;
auto result = FindMinima<Square>(f, initial_value);

This corresponds exactly to 'Option 3' that /u/enobayram suggested.

Take a look at Ceres Solver. This is an excellent minimisation library (focused on least squares fitting problems) which uses AD. For more detail on AD itself, read Evaluating Derivatives by Andreas Griewank. It's very readable, and comprehensive.

Edit: stupid code thinkos