How to find maxima and minima of a function with multiple variables each independently bounded in some range?

Question

Function, $$F(q_1,q_2)=aq_1^2+bq_2^2+cq_1+dq_2+gq_1q_2+l$$ Constraints, $$0<=q_1<=1$$$$0<=q_2<=1$$ How to find maxima and minima of $F$?

Answer 1

You have an optimization problem on a bounded, closed set of the plane $(q_1,q_2)$, na ely a square. Therefore the max and the min are achieved. You should start by finding interior critical points, that is points inside the region where the gradient is equal to zero. Then, you restrict your function to each part of the boundary, for ex. you take $q_1=0$ and $q_2\in[0,1]$ and find the min/max of the corresponding function of one variable $q_2$. You do the same for the four sides of your square. Finally, you just compare the values of the function at all the points you have found (points of min/max on the boundary and interior critical points) and choose the point with the lowest/highest value.