How to apply constraints for polynomial to polynomial parameters?

Question

I have a polynomial that I use for least squares regression on experimental data to find coefficients $a_0,a_1,a_2$:

$$f(x)=a_0+a_1 x+a_2 x^2$$

I have a constraint for $f(x)$, e.g. :

$$Y<f(x)<Z$$

and I know the range of the variable $x$. How do I apply this knowledge to constrain parameters $a_0$,$a_1$,$a_2$?

My solution was to create a linearly spaced vector of values in the allowed range of $x$ and supplying them to the polynomial. The result of the polynomial was then checked against the constraint. Is there a better way to do this? I used Matlab and fmincon function.

Answer 1

If you're using a computer, then all you really need to check are the two end points of your range, and if $x=-\frac{a_1}{2a_2}$ is within your range then you need to check that too.

In fact, that's probably the easiest thing to do by hand as well.

The reason that this works is that $x=-\frac{a_1}{2a_2}$ is the extremum point of your parabola, and on either side of that point the function is monotonic.

Answer 2

That's the simple way to do it, although you should probably not use fmincon, as the problem is a simple quadratic program (quadratic objective from your least-squares objective, and your constraints will be linear in the parameters)

Since you are in MATLAB, here is a quick-start via the Toolbox YALMIP (disclaimer, developed by me)

% Data
t = 0:0.1:1;
y = sin(pi*t)+0.2*randn(1,length(t));
clf;plot(t,y);

% Quadratic polynomial
sdpvar a0 a1 a2
yhat = a0 + a1*t + a2*t.^2;

% Grid points to evaluate constraints
xi = 0:0.01:1;
yi = a0 + a1*xi + a2*xi.^2;

% Solve the problem. Best possible solver will automatically be selected
Constraints = [0 <= yi <= 1];
Objective = sum((y-yhat).^2);
optimize(Constraints,Objective)

% Plot the computed polynomial in the evaluation points
hold on
plot(xi,value(yi))

% The parameters
value([a0 a1 a2])

More advanced polynomial regression and design examples https://yalmip.github.io/tutorial/quadraticprogramming/

https://yalmip.github.io/polynomialdesign