A new scalarization and numerical method for constructing the weak Pareto front of multi-objective optimization problems

A numerical technique is presented for constructing an approximation of the weak Pareto front of nonconvex multi-objective optimization problems, based on a new Tchebychev-type scalarization and its equivalent representations. First, existing results on the standard Tchebychev scalarization, the weak Pareto and Pareto minima, as well as the uniqueness of the optimal value in the Pareto front, are recalled and discussed for the case when the set of weak Pareto minima is the same as the set of Pareto minima, namely, when weak Pareto minima are also Pareto minima. Of the two algorithms we present, Algorithm 1 is based on this discussion. Algorithm 2, on the other hand, is based on the new scalarization incorporating rays associated with the weights of the scalarization in the value (or objective) space, as constraints. We prove two relevant results for the new scalarization. The new scalarization and the resulting Algorithm 2 are particularly effective in constructing an approximation of the weak Pareto sections of the front. We illustrate the working and capability of both algorithms by means of smooth and nonsmooth test problems with connected and disconnected Pareto fronts.