Convergence and finite convergence of the multiplier hierarchy

Determine conditions under which the multiplier hierarchy for products of AHI polynomials converges to the constrained optimum, establish the degree at which finite convergence occurs, and characterize its relationship with the constrained SONC and geometric-programming framework.

Background

For a product of AHI polynomials whose factors share variables and lack a common minimizer, the paper introduces shifted nonnegative factors and a hierarchy of SOS-polynomial multiplier certificates. The hierarchy produces successively stronger lower bounds for the product optimization problem, beginning with the factorized lower bound at degree zero.

The paper does not establish whether the sequence of bounds converges to the true constrained optimum, when finite convergence occurs, or how the construction relates to constrained SONC and geometric-programming methods. It also notes that an Archimedean-type condition on the module generated by the shifted factors would be required for convergence arguments.

References

A detailed analysis of eq:multiplier-hierarchy for AHI factors --- in particular conditions under which $\gamma_d \to f^{*}_K$, the degree at which finite convergence occurs, and the relation to the constrained SONC and geometric programming framework of \citet{dressler2019approach} --- is left for future work.

eq:multiplier-hierarchy:

$f - \gamma \;=\; \sum_{S \subseteq \{1,\dots,k\}} \sigma_{S}\prod_{l\in S} h_l, \qquad \sigma_{S}\ \text{SOS with } \deg \sigma_S \le d,$

— The AHI family of sum of squares polynomials (2608.28015 - Naskar et al., 28 Aug 2026) in Section 4.4, “Level C: multiplier hierarchy for overlapping factors”

Future work may address the convergence of the multiplier hierarchy eq:multiplier-hierarchy, a characterization of $\capC_{n,2d}$, and extensions of the construction to other classical inequalities.