Nuclear Magnetic Resonance (NMR) Spectroscopy is a widely used technique to predict the native structure of proteins. However, NMR machines are only able to report approximate and partial distances between pair of atoms. To build the protein structure one has to solve the Euclidean distance geometry problem given the incomplete interval distance data produced by NMR machines. In this paper, we propose a new genetic algorithm for solving the Euclidean distance geometry problem for protein structure prediction given sparse NMR data. Our genetic algorithm uses a greedy mutation operator to intensify the search, a twin removal technique for diversification in the population and a random restart method to recover stagnation. On a standard set of benchmark dataset, our algorithm significantly outperforms standard genetic algorithms.
The molecule problem is that of determining the relative locations of a set of objects in Euclidean space relying only upon a sparse set of pairwise distance measurements. This NP-hard problem has applications in the determination of molecular conformation. The molecule problem can be naturally expressed as a continuous, global optimization problem, but it also has a rich combinatorial structure. This paper investigates how that structure can be exploited to simplify the optimization problem. In particular, we present a novel divide-and-conquer algorithm in which a large global optimization problem is replaced by a sequence of smaller ones. Since the cost of the optimization can grow exponentially with problem size, this approach holds the promise of a substantial improvement in performance. Our algorithmic development relies upon some recently published results in graph theory. We describe an implementation of this algorithm and report some results of its performance on a sample molecule. We present a new iterative algorithm for the molecular distance geometry problem with inaccurate and sparse data, which is based on the solution of linear systems, maximum cliques, and a minimization of nonlinear least-squares function. Computational results with real protein structures are presented in order to validate our approach. We discuss the geometrical interpretation of a well-known smoothing operator applied to the Molecular Distance Geometry Problem (MDGP), and we then describe a heuristic approach based on Variable Neighbourhood Search on the smoothed and original problem. This algorithm often manages to find solutions having higher accuracy than other methods. This is important as small differences in the objective function value may point to completely different 3D molecular structures.  In this paper, we compare two different approaches to nonconvex global optimization. The first one is a deterministic spatial Branch-and-Bound algorithm, whereas the second approach is a Quasi Monte Carlo (QMC) variant of a stochastic multi level single linkage (MLSL) algorithm. Both algorithms apply to problems in a very general form and are not dependent on problem structure. The test suite we chose is fairly extensive in scope, in that it includes constrained and unconstrained problems, continuous and mixed-integer problems. The conclusion of the tests is that in general the QMC variant of the MLSL algorithm is generally faster, although in some instances the Branch-and-Bound algorithm outperforms it. We study the performance of the code for the solution of distance geometry problems with lower and upper bounds on distance constraints. The code uses only a sparse set of distance constraints, while other algorithms tend to work with a dense set of constraints either by imposing additional bounds or by deducing bounds from the given bounds. Our computational results show that protein structures can be determined by solving a distance geometry problem with and that the approach based on is significantly more reliable and efficient than multi-starts with an optimization code. Distance geometry problems arise in the determination of protein structure. We consider the case where only a subset of the distances between atoms is given and formulate this distance geometry problem as a global minimization problem with special structure. We show that global smoothing techniques and a continuation approach for global optimization can be used to determine global solutions of this problem reliably and efficiently. The global continuation approach determines a global solution with less computational effort than is required by a standard multistart algorithm. Moreover, the continuation approach usually finds the global solution from any given starting point, while the multistart algorithm tends to fail. We report on the theory and implementation of a global optimization solver for general constrained nonlinear programming problems based on Variable Neighbourhood Search, and we give comparative computational results on several instances of continuous nonconvex problems. Compared to an efficient multi-start global optimization solver, the VNS solver proposed appears to be significantly faster. 