DC Programming Approaches for Distance Geometry Problems

In this chapter, a so-called DCA method based on a DC (difference of convex functions) optimization approach for solving large-scale distance geometry problems is developed. Two main problems are considered: the exact and the general distance geometry problems. Different formulations of equivalent DC programs are introduced. Substantial subdifferential calculations permit to compute sequences of iterations in the DCA quite simply and allow exploiting sparsity in the large-scale setting. For improving the computational efficiency of the DCA schemes we investigate several techniques. A two-phase algorithm using shortest paths between all pairs of atoms to generate the complete dissimilarity matrix, a spanning trees procedure, and a smoothing technique are investigated in order to compute a good starting point (SP) for the DCAs. An important issue in the DC optimization approach is well exploited, say the nice effect of DC decompositions of the objective functions. For this purpose we propose several equivalent DC formulations based on the stability of Lagrangian duality and the regularization techniques. Finally, many numerical simulations of the molecular optimization problems with up to 12,567 variables are reported which prove the practical usefulness of the nonstandard nonsmooth reformulations, the globality of found solutions, the robustness, and the efficiency of our algorithms.