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3 Fixed LaTeX

Ms. Podolskaya,

Your 2nd question is related to matrix copositivity, I believe. Take a look at the 5th chapter of Parrilo's doctoral dissertation.

A quadratic form in $\mathbb{R}[x_1,x_2,x_3]$ is of the form

$P$P (x_1,x_2,x_3) = \left[\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right]^T \left[\begin{array}{ccc} q_{11} & q_{12} & q_{13}\\ q_{12} & q_{22} & q_{23}\\ q_{13} & q_{23} & q_{33}\end{array}\right] \left[\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right]$x_3\end{array}\right]$$or, more compactly, P (x) = x^T Q x. You ask: when is P positive on the positive octant? If P > 0 when x > 0, then (\forall (\forall x \in \mathbb{R^3}) (x > 0 \implies x^T Q x > 0)0)$$ and, in theory, one could use quantifier elimination to obtain conditions on the$q_{ij}$coefficients so that$P > 0$on the positive octant. The following REDLOG script % positivity on the positive octant load_package redlog; rlset ofsf; % define quadratic form P := 1 * q11 * x1 * x1 + + 1 * q22 * x2 * x2 + + 1 * q33 * x3 * x3 + + 2 * q12 * x1 * x2 + + 2 * q13 * x1 * x3 + + 2 * q23 * x2 * x3; % define universally quantified formula phi := all({x1,x2,x3}, (x1 > 0 and x2 > 0 and x3 > 0) impl P>0); % perform quantifier elimination rlqe phi; end;  produces results in a few seconds, but the conditions on the$q_{ij}$coefficients are enormously long quantifier-free formulas. So enormous that REDUCE crashed! If$P$is nonnegative on the nonnegative octant, then$(\forall $(\forall x \in \mathbb{R^3}) (x \geq 0 \implies x^T Q x \geq 0)$0)

which is equivalent to saying that matrix $Q = Q^T$ is copositive.

2 added 39 characters in body

Ms. Podolskaya,

Your 2nd question is related to matrix copositivity, I believe. Take a look at the 5th chapter of Parrilo's doctoral dissertation.

A quadratic form in $\mathbb{R}[x_1,x_2,x_3]$ is of the form

$P (x_1,x_2,x_3) = \left[\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right]^T \left[\begin{array}{ccc} q_{11} & q_{12} & q_{13}\\ q_{12} & q_{22} & q_{23}\\ q_{13} & q_{23} & q_{33}\end{array}\right] \left[\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right]$

or, more compactly, $P (x) = x^T Q x$. You ask: when is $P$ positive on the positive octant? If $P > 0$ when $x > 0$, then

$(\forall x \in \mathbb{R^3}) (x > 0 \implies x^T Q x > 0)$

and, in theory, one could use quantifier elimination to obtain conditions on the $q_{ij}$ coefficients so that $P > 0$ on the positive octant. The following REDLOG script

% positivity on the positive octant

rlset ofsf;

P := 1 * q11 * x1 * x1 +
+ 1 * q22 * x2 * x2 +
+ 1 * q33 * x3 * x3 +
+ 2 * q12 * x1 * x2 +
+ 2 * q13 * x1 * x3 +
+ 2 * q23 * x2 * x3;

% define universally quantified formula
phi := all({x1,x2,x3}, (x1 > 0 and x2 > 0 and x3 > 0) impl P>0);

% perform quantifier elimination
rlqe phi;

end;


produces results in a few seconds, but the conditions on the $q_{ij}$ coefficients are enormously long quantifier-free formulas. So enormous that REDLOGREDUCE crashed!

If $P$ is nonnegative on the nonnegative octant, then

$(\forall x \in \mathbb{R^3}) (x \geq 0 \implies x^T Q x \geq 0)$

which is equivalent to saying that matrix $Q = Q^T$ is copositive.

1

Ms. Podolskaya,

Your 2nd question is related to matrix copositivity, I believe. Take a look at the 5th chapter of Parrilo's doctoral dissertation.

A quadratic form in $\mathbb{R}[x_1,x_2,x_3]$ is of the form

$P (x_1,x_2,x_3) = \left[\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right]^T \left[\begin{array}{ccc} q_{11} & q_{12} & q_{13}\\ q_{12} & q_{22} & q_{23}\\ q_{13} & q_{23} & q_{33}\end{array}\right] \left[\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right]$

or, more compactly, $P (x) = x^T Q x$. You ask: when is $P$ positive on the positive octant? If $P > 0$ when $x > 0$, then

$(\forall x \in \mathbb{R^3}) (x > 0 \implies x^T Q x > 0)$

and, in theory, one could use quantifier elimination to obtain conditions on the $q_{ij}$ coefficients so that $P > 0$ on the positive octant. The following REDLOG script

% positivity on the positive octant

rlset ofsf;

P := 1 * q11 * x1 * x1 +
+ 1 * q22 * x2 * x2 +
+ 1 * q33 * x3 * x3 +
+ 2 * q12 * x1 * x2 +
+ 2 * q13 * x1 * x3 +
+ 2 * q23 * x2 * x3;

% define universally quantified formula
phi := all({x1,x2,x3}, (x1 > 0 and x2 > 0 and x3 > 0) impl P>0);

% perform quantifier elimination
rlqe phi;

end;


produces results in a few seconds, but the conditions on the $q_{ij}$ coefficients are enormously long quantifier-free formulas. So enormous that REDLOG crashed!

If $P$ is nonnegative on the nonnegative octant, then

$(\forall x \in \mathbb{R^3}) (x \geq 0 \implies x^T Q x \geq 0)$

which is equivalent to saying that matrix $Q = Q^T$ is copositive.