Optimization Techniques for Phase Retrieval Based on Single-Crystal X-Ray Diffraction Data

Optimization techniques in molecular structure and function elucidation

“…In other words, solutions of MP do not guarantee that no two atoms will coincide. Nonetheless, recent progress on the phase problem suggests that the modeling of phases with integer variables can facilitate the use of linear and integer programming approaches to resolve the multiple minima difficulty (Vaia and Sahinidis, 2003, 2005; Smith et al, 2007; Smith, 2008), while the addition of constraints that enforce atomicity requirements via limiting electron density over a grid provides a complete formulation that leads to correct structures (Smith, 2008). Models and algorithms have been developed for centrosymmetric and non-centrosymmetric structures.…”

Domain reduction techniques for global NLP and MINLP optimization

“…MINLP is a very general representation for optimization problems and includes linear programming (LP), mixed-integer linear programming (MIP) and nonlinear programming (NLP) in its subclasses. A variety of applications in diverse fields are routinely formulated using this framework including water network design [94,62], hydro energy systems management [44], protein folding [143], robust control [18], trim loss [84], heat exchanger network synthesis [69], gas networks [128], transportation [68], chemical process operations [76], chemical process synthesis [77], crystallographic imaging [178], and seizure predictions [148]. Modelling via nonconvex objective functions or constraints is necessitated for many of these practical applications.…”