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Bounty Ended with 50 reputation awarded by mathworker21
Tried to clarify where we were using the quadratic form.
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Bill Bradley
Bill Bradley
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You might consider the "sum-of-squares" approach. The idea is to find a set of polynomials so that your quadratic formexpression is the sum of the squaresquares of the elements in the region of interest. For your case, you could replace each $x_i$ with a new variable $z_i^2$, and ask; you are now asking if the quadratic form is a SOS as an (unconstrained)corresponding 4-th degree unconstrained polynomial over $z$is non-negative.

This restatement may not sound like an improvement, but it turns out that SOS problems can be attacked using semidefinite programming techniques (e.g., see this page). You can use a freely available SDP solvers.

This is a sufficient approach, i.e., it may prove that your original quadratic form is positive, but it can't disprove it. Since you're trying to solve a specific problem, though, it may be worth the gamble.

You might consider the "sum-of-squares" approach. The idea is to find a set of polynomials so that your quadratic form is the sum of the square of the elements in the region of interest. For your case, you could replace each $x_i$ with a new variable $z_i^2$, and ask if the quadratic form is a SOS as an (unconstrained) polynomial over $z$.

This restatement may not sound like an improvement, but it turns out that SOS problems can be attacked using semidefinite programming techniques (e.g., see this page). You can use a freely available SDP solvers.

This is a sufficient approach, i.e., it may prove that your quadratic form is positive, but it can't disprove it. Since you're trying to solve a specific problem, though, it may be worth the gamble.

You might consider the "sum-of-squares" approach. The idea is to find a set of polynomials so that your expression is the sum of squares of the elements in the region of interest. For your case, you could replace each $x_i$ with a new variable $z_i^2$; you are now asking if the corresponding 4-th degree unconstrained polynomial is non-negative.

This restatement may not sound like an improvement, but it turns out that SOS problems can be attacked using semidefinite programming techniques (e.g., see this page). You can use a freely available SDP solvers.

This is a sufficient approach, i.e., it may prove that your original quadratic form is positive, but it can't disprove it. Since you're trying to solve a specific problem, though, it may be worth the gamble.

Source Link
Bill Bradley
Bill Bradley
  • 4k
  • 20
  • 25

You might consider the "sum-of-squares" approach. The idea is to find a set of polynomials so that your quadratic form is the sum of the square of the elements in the region of interest. For your case, you could replace each $x_i$ with a new variable $z_i^2$, and ask if the quadratic form is a SOS as an (unconstrained) polynomial over $z$.

This restatement may not sound like an improvement, but it turns out that SOS problems can be attacked using semidefinite programming techniques (e.g., see this page). You can use a freely available SDP solvers.

This is a sufficient approach, i.e., it may prove that your quadratic form is positive, but it can't disprove it. Since you're trying to solve a specific problem, though, it may be worth the gamble.