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I am interested in the minimum of a general multivariate quadratic equation for non-negative real numbers:

$$ \begin{aligned} & \underset{x_i}{\text{minimize}} & & \sum^{n}_{i=1}\sum^{n}_{j=i}a_{ij}x_{i}x_{j} + \sum^{n}_{i=1}b_{i}x_{i} + c \\ & \text{subject to} & & x_i \geq 0 \end{aligned} $$

If we restrict ourselves to the simplest form, that is a two non-negative variables equation: $$ \begin{aligned} & \underset{x,y}{\text{minimize}} & &ax^{2} + bxy + cy^{2} + dx + ey + f \end{aligned} $$

1) What is the condition under which the uniqueness of the solution can be guaranteed? Is there an analytic solution to it?

2) What are the solution concepts for this problem, say when $x=0$, or $y=0$?

I am very welcoming to any suggestion, pointers or an actual solution.

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  • $\begingroup$ Did you mean for the objective function to be $\sum^{n}_{i=1}\sum^{n}_{j=i}a_{ij}x_{i}x_{j} + \sum^{n}_{i=1}b_{i}x_{i} + c$? As currently written, the objective function depends on $j$. $\endgroup$
    – JimmyK4542
    Commented Aug 8, 2015 at 20:18
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    $\begingroup$ In that case, this is just a quadratic program with linear constraints. You might want to read the Wikipedia article on Quadratic programming. For a unique solution to exist, it is sufficient (but not necessary) for the matrix $A$ formed by the $a_{ij}$'s to be positive definite. $\endgroup$
    – JimmyK4542
    Commented Aug 8, 2015 at 20:25
  • $\begingroup$ @JimmyK4542 Thanks for the response. It is indeed a QP with linear constraints. $\endgroup$
    – Amir
    Commented Aug 8, 2015 at 20:31

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