Self-concordance and allocation for spatial power sums
- Primary subject
- Mathematics
- Subject subfields
- Optimization and Control · Classical and Real Analysis
- Paper type
- Technical Report
- Manuscript stage
- Preprint
This mathematical supplement studies the negative logarithm of one-sided product-power smoothing of arbitrary coordinatewise decreasing weights. A common real power p >= 16d gives a standard self-concordant barrier with parameter at most 2pd. For decreasing lattice inputs with at most r positive coordinates per state, p >= 32r suffices independently of ambient dimension and occupancy magnitudes. Residual Pareto tails, an exact third-cumulant identity and a sparse L3 bound supply the proof without input independence or log-concavity. Decreasing binary mixtures require power growing linearly with r for uniform convexity. Log-concave binary generating polynomials with deletion bounds admit dimension-free fixed-power statements and squared-logarithmic activity dependence. Classical damped Newton and projected-gradient methods give explicit allocation consequences. Constants are not sharp and no bit-complexity bound is asserted.