I'm optimizing a family of objectives over $x_1,\ldots,x_n$ with a single constraint that $x$'s add up to $1$, and when using method of Lagrange multipliers, the single multiplier always ends up $0$ for members of this family… does this tell us anything interesting about these objectives?
Edit: the answer was — it means the constraint it inactive, ie, removing the constraint doesn't change the answer of optimization problem
Best Answer
One of two things that I can think of are happening here. The global minimum (or max, whatever your goal is here) of your function may actually lie within the simplex defined by $\sum_{i=1}^{4} q_{i} = 1$. In this case your algorithm is finding the correct solution and the constraint is trivially satisfied. (Consider minimizing $x^2+y^2$ constrained to the line $x = 0$, so the Lagrange formulation is $L(x,y,\nu) = x^2 + y^2 + \nu x$. Solving $\nabla L=0$ demands that $\nu=0$. )
Alternatively, your algorithm could be sampling only points that already satisfy the constraint - so you are only checking those points (intentionally or otherwise). Then the algorithm is hopefully finding a local minimum (or max) along this constraint and your multiplier will always be zero.
I hope this helps you identify where a problem may be in your model - or clears up a conceptual issue and allows you to move forward with the solution you have. As far as interpreting the meaning of Lagrange multipliers, look here.