Effective curvature dimension in smooth DC optimization

Abstract

We introduce the effective curvature dimension of a smooth DC decomposition, defined as the dimension of the span of the Hessian ranges of its first convex component. We prove that the smallest attainable effective dimension over all smooth DC decompositions equals the minimum rank of a positive semidefinite matrix that uniformly majorizes the Hessian of the DC objective. Hence, an optimal dimension is always attained by a quadratic convexifier. We further characterize feasible curvature-free subspaces through generalized Schur-complement criteria, revealing an obstruction caused by cross-curvature.As an algorithmic consequence of the effective dimension framework, we show that the number of globally solved lower model subproblems depends on the effective dimension rather than the ambient dimension. Finally, for quadratic first components, we identify the minimum number of tangent supports required for uniform approximation with an internal covering number and obtain a matching dimensional exponent under a nondegeneracy condition.