Molecular structure determination by convex, global underestimation of local energy minima

The determination of a stable molecular structure can often be formulated in terms of calculating the global (or approximate global) minimum of a potential energy function. Computing the global minimum of this function is very difficult because it typically has a very large number of local minima which may grow exponentially with molecule size. The optimization method presented involves collecting a large number of conformers, each attained by finding a local minimum of the potential energy function from a random starting point. The information from these conformers is then used to form a convex quadratic global underestimating function for the potential energy of all known conformers. This underestimator is an L1 approximation to all known local minima, and is obtained by a linear programming formulation and solution. The minimum of this underestimator is used to predict the global minimum for the function, allowing a localized conformer search to be performed based on the predicted minimum. The new set of conformers generated by the localized search serves as the basis for another quadratic underestimation step in an iterative algorithm. This algorithm has been used to determine the structures of homopolymers of length n ≤ 30 with no sidechains. While it is estimated that there are O(3n) local minima for a chain of length n, this method requires O(n4) computing time on average. It is also shown that the global minimum potential energy values lie on a concave quadratic curve for n ≤ 30. This important property permits estimation of the minimum energy for larger molecules, and also can be used to accelerate the global minimization algorithm.