Global Continuation for Distance Geometry Problems

Abstract

Distance geometry problems arise in the interpretation of NMR data and in the determination of protein structure. We formulate the distance geometry problem as a global minimization problem with special structure, and show that global smoothing techniques and a continuation approach for global optimization can be used to determine solutions of distance geometry problems with a nearly 100% probability of success.

Positive semidefinite relaxations for distance geometry problems

“…Also, when the problem is specified by the upper and lower bounds, we have numerically demonstrated that quite accurate configurations are generated when n is up to 300. It should be emphasized that the existing algorithms such as [7,10,18,25,19] need to iterate the search procedures from a large number of starting points, while our algorithm generates accurate configurations without those iterations, which would be the advantage of our algorithm.…”

“…Moreover, these results indicate that the number of specified pairs ISI does not affect the quality of the configurations. Compared to the results in [10,18,25], our algorithm generates acceptable configurations even when the number of the specified pairwise distances, ISI, are small, which enables us to solve the larger size problems with up to 729 points. Figure 2 and Figure 3 show the typical behavior of the algorithm.…”

“…Recently, Alfakih et al [1] have also considered to minimize the same error function by solving a positive semidefinite programming problem with a quadratic objective function. Also, smoothing techniques by Moré and Wu [18] and Stochastic/perturbation approaches by Zou et al [25] have been applied to the S-stress function. We note that the differentiability of the S-stress function brings rich mathematical properties such as those developed in the literature [9,11,1].…”

“…Bij First, we have tested our algorithm over artificially generated distance geometry problems which are the same problems studied in [18,25]. In these problem, points are located in the three-dimensional (i.e., r = 3) lattice defined by…”

The flexible alignment of molecular structures using simulated annealing with weighted Lagrangian multipliers

“…Molecular structure generating algorithms incorporating distance geometry techniques have continued to be published since originally championed by Crippen and Havel; [7][8][9][10][11][12][13] Of particular interest is Crippen's linearized embedding approach. 7 In Crippen's algorithm, 7 a molecule to be embedded in Euclidean 3-space is represented by a spanning tree rooted at the most central atom.…”