Self-concordance and allocation for spatial power sums

Overview

Abstract

We study the negative logarithm of one-sided product-power smoothing of arbitrary coordinatewise decreasing weights. For continuous locally integrable inputs and decreasing lattice arrays in dimension d, a common real power p≥16d makes this particular potential a standard self-concordant barrier, with parameter at most 2pd. If the lattice support has at most r positive coordinates at each state, the sufficient power improves to p≥32r, independently of the magnitudes of the occupancies and of the ambient dimension. The proof uses residual Pareto tails, an exact third-cumulant identity and a sparse L^3 operator bound; it does not assume independence or log-concavity of the input. A decreasing binary mixture shows that powers must grow linearly in r for a uniform convexity theorem over this input class. For log-concave binary generating polynomials with deletion bounds, an absolute scaling retains every fixed power p≥4, independently of rank; matroid independent-set polynomials supply this premise classically. For coefficient deletion ratios up to M≥1, powers of order (1+log M)^2 give an absolute scale; a rank-one obstruction shows squared-logarithmic order necessity for uniform convexity in the stated asymptotic regime. We also give an exact quartic scalar example and an explicit budget-allocation consequence: for a listed m-state input, a balanced starting point has objective gap at most log m, permitting classical damped Newton minimization. For quartic matroid utilities, a balanced KKT box and clipped sampling also give a polynomial expected-cost projected-gradient guarantee using only an independence oracle. Both optimization methods are classical.

Keywords Optimization and Control · Classical and Real Analysis · mathematical research