Pareto frontier

The ability of MoSELEGT to simultaneously optimize molecular weight and diversity was tested in the selection of 30 x 30 combinatorial subsets of a 10000-member virtual amide library. It was shown that as the selection progressed, there was an improvement in both the molecular weight and diversity, as well as a spread of nondominated solutions across the Pareto frontier. The 17 nondominated solutions found after 5000 iterations emphasized the competing nature of these two objectives - those with lower... 

A Pareto frontier is a limit for the solution along which the improvement in one objective can be achieved only at the cost of another objective. 

Messac, A., Ismail-Yahaya, A., Mattson, C. A. (2003) The normalized normal constraint method for generating the pareto frontier. StructMultidis Optim 25(2), 86-98. 

Wilson, B., Cappelleri, D., Simpson, T. W. and Frecker, M. (2001). Efficient Pareto frontier exploration using surrogate approximations, Optim,ization and Engineering 2, 1, pp. 31-50. 

Lotov, A. V., Bushenkov, V. A. and Kamenev, G. K. (2004). Interactive Decision Maps. Approximation and Visualization of Pareto Frontier (Kluwer Academic Publishers, Boston). 

The solution of an MO problem is said to be a set of Pareto solutions, or a Pareto frontier. A Pareto solution is one for which any improvement in one objective can only take place if at least one other objective worsens (Messac et al. 2003). 

Now, the procedure to And the non-extreme Pareto solutions so as to approximate the Pareto frontier is presented. Such a procedure is shown in Fig. A.5. Firstly, one needs to define the number A of Pareto solutions that are to be found along directions parallel to the axis representing Zp. The total number of non-extreme Pareto solutions to be generated is equal to N. The space between consecutive Pareto solutions Sp) along directions parallel to the axis Zp is determined using the next expression. 

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