On the existence of convex decompositions of partially separable functions

Abstract

The concept of a partially separable functionf developed in [4] is generalized to include all functionsf that can be expressed as a finite sum of element functionsfi whose Hessians have nontrivial nullspacesNi, Such functions can be efficiently minimized by the partitioned variable metric methods described in [5], provided that each element functionfi is convex. If this condition is not satisfied, we attempt toconvexify the given decomposition by shifting quadratic terms among the originalfi such that the resulting modified element functions are at least locally convex. To avoid tests on the numerical value of the Hessian, we study the totally convex case where all locally convexf with the separability structureNi1 have a convex decomposition. It is shown that total convexity only depends on the associated linear conditions on the Hessian matrix. In the sparse case, when eachNi is spanned by Cartesian basis vectors, it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a (permuted) LDLT factorization without fill-in.