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

Euclidean Distance Matrices and Applications

“…Other algorithmic developments followed, including: a divide-and-conquer algorithm called ABBIE based on identifying rigid substructures [72]; an alternating projection approach [58]; a global smoothing continuation code called DGSOL [93,94]; a geometric build-up algorithm [48,49,116,117]; an extended recursive geometric build-up algorithm [50]; a difference of convex functions (d.c.) optimization algorithm [15]; a method based on rank-reducing perturbations of the distance matrix that maintain desired structures [52]; an algorithm for solving a distance matrix based, large-scale, bound constrained, non-convex optimization problem called StrainMin [68].…”

“…This model often includes upper and lower bounds on the distances; see, e.g., [4]. We have not specified the norm in (15). The Frobenius norm was used in [4], where small problems were solved; see also [5].…”

A continuous DC programming approach to the strategic supply chain design problem from qualified partner set

“…The DCA (see [1][2][3][4]16,17], and references therein) is an efficient method which has been successfully applied to a lot of various large-scale nonconvex programs. It is a descent method without linesearch, consisting of the construction of the two sequences {x k } and {y k }, (candidates for being primal and dual solutions, respectively), such that their corresponding limit points x 1 and y 1 satisfy local optimality conditions.…”

“…(i) Is a consequence of DCA's convergence theorem for a general DC program (see [1][2][3][4]16,17]). (ii) As mentioned above, if V(K) is contained in the feasible solution set of (GMIP) then the assertion is trivial with t 1 = 0.…”

On the computation of protein backbones by using artificial backbones of hydrogens

“…In order to overcome this problem, Moré and Wu [21,22] proposed to approximate the used penalty function with smoother functions converging to the original one. Another method for the MDGP makes use of a penalty function which can be seen as the difference of two convex functions [1], and particular techniques for d.c. optimization are used. Moreover, in [8], a Population Basin Hopping method is employed, in which basic concepts (such as the ones of funnel and funnel bottom) usually used in methods for finding molecular conformations are exploited.…”