Semiglobal simplex optimization and its application to determining the preferred solvation sites of proteins

The classical simplex method is extended into the Semiglobal Simplex (SGS) algorithm. Although SGS does not guarantee finding the global minimum, it affords a much more thorough exploration of the local minima than any traditional minimization method. The basic idea of SGS is to perform a local minimization in each step of the simplex algorithm, and thus, similarly to the Convex Global Underestimator (CGU) method, the search is carried out on a surface spanned by local minima. The SGS and CGU methods are compared by minimizing a set of test functions of increasing complexity, each with a known global minimum and many local minima. Although CGU delivers substantially better success rates in simple problems, the two methods become comparable as the complexity of the problems increases. Because SGS is generally faster than CGU, it is the method of choice for solving optimization problems in which function evaluation is computationally inexpensive and the search region is large. The extreme simplicity of the method is also a factor. The SGS method is applied here to the problem of finding the most preferred (i.e., minimum free energy) solvation sites on a streptavidin monomer. It is shown that the SGS method locates the same lowest free energy positions as an exhaustive multistart Simplex search of the protein surface, with less than one-tenth the number of minizations. The combination of the two methods, i.e.. multistart simplex and SGS, provides a reliable procedure for predicting all potential solvation sites of a protein.