Effectiveness of Weighted Aggregation of Objectives for Evolutionary Multiobjective Optimization: Methods, Analysis and Applications

Multiobjective optimization using the conventional weighted aggregation of the objectives method is known to have several drawbacks. In this paper, multiobjective optimization using the weighted aggregation method is approached with the help of evolutionary algorithms. It is shown through a number of test functions that a Pareto front can be achieved from one single run of evolutionary optimization by changing the weights during optimization, no matter whether the Pareto-optimal front is convex or concave. To establish a theoretical background that accounts for the success of the method, the phenomenon of global convexity in multiobjective optimization is investigated. Global convexity means that 1) most Pareto-optimal solutions are concentrated in a very small fraction of the parameter space, and 2) the solutions that are in the neighborhood on the Pareto front are also in the neighborhood in the parameter space. It is shown that all test functions studied in this paper do exhibit the global convexity property. In fact, for the two or three dimensional problems studied in the paper, the Pareto-optimal solutions can be defined by a piecewise linear function in the parameter space. Furthermore, the mapping of a normal distribution in the parameter space onto the fitness space is investigated. It is shown that the Pareto front is sometimes a local attractor even without the influence of selection. Finally, two application examples in design optimization are given to show the effectiveness of the evolutionary dynamic weighted aggregation method.