@article{davis_papp_2022, title={DUAL CERTIFICATES AND EFFICIENT RATIONAL SUM-OF-SQUARES DECOMPOSITIONS FOR POLYNOMIAL OPTIMIZATION OVER COMPACT SETS}, volume={32}, ISSN={["1095-7189"]}, url={https://doi.org/10.1137/21M1422574}, DOI={10.1137/21M1422574}, abstractNote={We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual cone certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate certifies a full-dimensional convex cone of WSOS polynomials. As a result, rational WSOS certificates can be constructed from numerically computed dual certificates at little additional cost, without any rounding or projection steps applied to the numerical certificates. As an additional algorithmic application, we present an almost entirely numerical hybrid algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence.}, number={4}, journal={SIAM JOURNAL ON OPTIMIZATION}, author={Davis, Maria M. and Papp, David}, year={2022}, pages={2461–2492} }
 @article{hatam_johnson_liu_macaulay_2021, title={Determinantal Formulas for SEM Expansions of Schubert Polynomials}, ISSN={["0219-3094"]}, DOI={10.1007/s00026-021-00558-z}, abstractNote={We show that for any permutation w that avoids a certain set of 13 patterns of length 5 and 6, the Schubert polynomial $${\mathfrak {S}}_w$$ can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner similar to the Jacobi–Trudi identity. For such w, this determinantal formula is equivalent to a (signed) subtraction-free expansion of $$\mathfrak S_w$$ in the basis of standard elementary monomials.}, journal={ANNALS OF COMBINATORICS}, author={Hatam, Hassan and Johnson, Joseph and Liu, Ricky Ini and Macaulay, Maria}, year={2021}, month={Oct} }