Self-concordance and allocation for spatial power sums

Abstract

We study self-concordance of the negative logarithm of one-sided spatial power-sum utilities. For binary generating polynomials whose logarithms are concave on the nonnegative orthant and whose coefficients do not increase when an element is added, one absolute scaling factor C_* makes the original potential a standard self-concordant barrier at every real power p>3, independently of dimension and support rank. Independent coordinate powers p_i>3 are also allowed, with barrier parameter at most 3C_*Σᵢ pᵢ. The strict threshold is sharp for a uniform theorem with global C³ regularity. Matroid independent-set polynomials satisfy the structural premise by established log-concavity results. For arbitrary decreasing continuous or lattice inputs, a power linear in dimension is sufficient; sparse lattice support replaces dimension by the number of positive coordinates in each state. Coefficient deletion ratios up to M admit powers of order (1+log M)^2, and a rank-one obstruction shows this asymptotic order is necessary for uniform convexity. For an explicitly listed m-state input, a balanced start has objective gap at most log m, enabling classical damped Newton minimization. For the quartic matroid utility, a balanced allocation box and finite-variance median-of-means estimates give an independence-oracle allocation guarantee with sample dependence linear in ε⁻¹, up to logarithmic factors. The optimization and sampling methods are classical; the verified thresholds and uniform resource bounds concern these particular utilities.