Sum-of-squares exactness for quartically regularized cubic polynomials
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Does adding arbitrarily strong positive quartic regularization to a fixed cubic objective make its minimum-shifted polynomial a sum of polynomial squares? Nonnegativity at the exact minimum need not imply such a representation.
Why it matters and what counts as progress
Why it matters: The answer tests whether quartic regularization can eliminate a gap in sum-of-squares optimization certificates.
What counts as progress: A fixed cubic retaining a strict SOS gap for every positive regularization parameter, or a theorem identifying sufficient conditions; a unique nondegenerate minimizer is a stronger counterexample variant.
How the problem developed
- 28 Jan 2026 · Sum of squares threshold [1]
Wenqi Zhu and Coralia Cartis Proved that sufficiently strong Euclidean quartic regularization makes the specified nonnegative quartic polynomials sums of squares, with an explicit threshold; some structured subclasses are exact for every regularization level.
AI contribution
Persistent failure of Euclidean quartic SOS exactness
An explicit fixed rational cubic in 14 variables gives a quartically regularized polynomial whose exact minimum shift is nonnegative but not a sum of polynomial squares for every positive regularization parameter. A second explicit cubic has a unique global minimizer with positive-definite Hessian there and retains a strict SOS gap for every positive parameter.
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