For the hypercube [0,1]ⁿ with generators gᵢ = xᵢ - xᵢ², let C_(n,2d) be the smallest constant such that x₁… xₙ + C_(n,2d) lies in the degree-2d truncated quadratic module M_(n,2d)(g), and let Cₙ = C_(n,n). De Klerk and Laurent conjectured Cₙ = 1/(n(n+2)) fo…
Optimization and ControlAI reviewed · Human review open
Let W_N be the worst-case value of min_(0 ≤ k ≤ N)norm(nabla f(xₖ))² over L-smooth convex functions with norm(x₀-xₛtar) ≤ R, where x₀,…,x_N are the points at which Nesterov's fast gradient method queries the gradient. Du (arXiv:2608.26719) proves W_N=L²R²/S…
Optimization and ControlAI reviewed · Human review open
For a nonnegative nondecreasing submodular set function f on n elements and a marginal vector x ∈ [0,1]ⁿ, let f⁺(x) be the largest expectation of f over all distributions on subsets with marginals x, and f⁺⁺(x) the same maximum over pairwise independent dis…
Optimization and ControlAI reviewed · Human review open
For z₁,…,zₙ ∈ ℂ summing to zero put pv(zᵢ,zⱼ)=|zᵢ|+|zⱼ|+|zᵢ+zⱼ| and δ(z)=max_(i<j)pv(zᵢ,zⱼ)/Σₖ|zₖ|; the quantity pim(n)=inf_zδ(z), the infimum taken over the z with every zₖ ≠ 0, measures where the eigenvalue set of the locally positive semidefinite matrice…
Optimization and ControlAI reviewed · Human review open
