Bounds on objective function and dependent state with FMINCON

I'm minimizing the function

fun = f(x)

with FMINCON. As the documentation shows, it's easy to apply bounds on x. So let's say:

min f(x)
0 <= x <= 1

But fun as well as a variable y which is dependent of x are bounded. So what I'd like to implement is:

min f(x)
0 <= x <= 1
fun_min <= fun <= fun_max
y_min <= y <= y_max

The dependency of x and y prohibits me from using y as a second optimization variable. What I'm looking for is "minimize fun = f(x) under consideration of the bounds b=[fun_min, fun_max, y_min, y_max]" Any ideas? Thanks for your help

[Walter Roberson]

Do we understand correctly that you want to find the minimum of f(x), but that the same time you want to impose a bound on what the minimum can be? If you know it can be the minimum, then why bother searching for the minimum?

Answer 1

[Jeff Miller]

Would it work to include a penalty for too-low y at the end of your function f? Something like this:

function y = f(x)
...
if y<y_min
   y = y_min + 100*(y_min-y);
end
end % function f

I don't quite see why you need to enforce y_max, since fmincon will try to minimize your function anyway.

[Jacob Mannhardt]

Thank you guys! You are right, the objective function has bounds to it, which should be enforced implicitly by the function's definition. So let's focus on the other problem: Bounds and constraints on variables, which are not optimization variables. These are not independent variables, but have to meet the constraints. To put that into context. My problem is following: I have to types of energy storage, defined by their outlet power and want to minimize the dissipated energy due to conversion. The one power is dependent of the other which is my optimization variable, but I can only allow a minimum and maximum power on both of these storage types. Any ideas how to enforce these bounds? Cheers

[Walter Roberson]

If you can, work it out as a linear constraint. But if you cannot do that, then implement a nonlinear constraint. Nonlinear constraints can do any amount of computation they need to in order to emit their results.

[Jacob Mannhardt]

Thanks Walter,

so my approach to constrain

y=g(x), y_min <= y <= y_max

should be

g(x) >= y_min, g(x) <= y_max

to express y as a function of x and implicitly constrain x?

[Walter Roberson]

ceq = [y_min - y, y - y_max]

Answer 2

[Jacob Mannhardt]

It works perfectly with the nonlinear constraints when you express the second state as a function of the optimization variable. Thanks guys! Quick and perfect help!