Large-Scale Molecular Optimization from Distance Matrices by a D.C. Optimization Approach

A so-called DCA method based on a d.c.\ (difference of convex functions) optimization approach (algorithm) for solving large-scale distance geometry problems is developed. Different formulations of equivalent d.c.\ programs in the $l_{1}$-approach are stated via the Lagrangian duality without gap relative to d.c.\ programming, and new nonstandard nonsmooth reformulations in the $l_{\infty }$-approach (resp., the $l_{1}-l_{\infty }$-approach) are introduced. Substantial subdifferential calculations permit us to compute sequences of iterations in the DCA quite simply. The computations actually require matrix-vector products and only one Cholesky factorization (resp., with an additional solution of a convex program) in the $l_{1}$-approach (resp., the $l_{1}-l_{\infty }$-approach) and allow the exploitation of sparsity in the large-scale setting. Two techniques---respectively, using shortest paths between all pairs of atoms to generate the complete dissimilarity matrix and the spanning trees procedure---are investigated in order to compute a good starting point for the DCA. Finally, many numerical simulations of the molecular optimization problems with up to 12567 variables are reported, which prove the practical usefulness of the nonstandard nonsmooth reformulations, the globality of found solutions, and the robustness and efficiency of our algorithms.