I want to optimize over sum of squares, but with constraints on the coefficients of the polynomials. I know that this can be done by adding constraints; however, this often seems wasteful. For example I want to restrict the polynomial to be even. Is there a way, without defining a new type of cone, or adding constraints to define this type of problem in Hypatia. I know from the Papp and Yildiz paper that this basically invlolves using only a subset of the bases polynomials in defining the problem. Is there an easy way to do this?
My guess for Hypatia is “without defining a new type of cone, no.” But @lkapelevich or @chriscoey may be able to correct me.
Have you seen https://github.com/jump-dev/SumOfSquares.jl?
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Hey @geo.
Chris and I have discussed using only subsets of the basis polynomials, but we never tried it so I’m not sure how easy it will be/what it will involve. The WSOS dual cone doesn’t know anything about degree/number of variables, but the code that we use to get the P matrices assumes we will want to work with dense polynomials up to a fixed degree/number of variables. You might be able to work out how to get the right P matrices manually to do what you want.
Since @odow mentioned SumOfSquares, just bear in mind that currently using SumOfSquares with Hypatia will still result in a PSD reformulation: https://github.com/chriscoey/Hypatia.jl/issues/283
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Thank you for your replies. I understand that the use of polynomial coefficients and the semidefinite programming formulation would make it fairly easy put constraints on the polynomial constraints, but then I would not be able to take advantage of the better numerical properties of the WSOS formulation in Hypatia. It seems that it is either easy formulation or better numerical properties, but not both!
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