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How to find the the minimum non-negative value of a function:

$$f(x,y)=ax^2+by^2+cx+dy+e$$

s.t. $x$ lies in $[0, A]$ and $y$ lies in $[A, \infty),$

where $A$ is a known constant.

or simply $0\leq x\leq A\leq y$

Example: $f(x,y)=-x^2 + 2y^2 + 3y +8$ has a minimum positive value of $12$ for $A=1$. I have found this graphically but I would like to find the solution analytically.

Any help would be beneficial.

  • 0
    You can check for critical points inside the domain, and also check for minimum values along the boundary.2012-03-27
  • 0
    yes, but its possible that critical points do not fall inside the constraint2012-03-27

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