Properties of objective functions and search algorithms in multi-objective optimization problems

Objectives. A frequently used method for obtaining Pareto-optimal solutions is to minimize a selected quality index under restrictions of the other quality indices, whose values are thus preset. For a scalar objective function, the global minimum is sought that contains the restricted indices as penalty terms. However, the landscape of such a function has steep-ascent areas, which significantly complicate the search for the global minimum. This work compared the results of various heuristic algorithms in solving problems of this type. In addition, the possibility of solving such problems using the sequential quadratic programming (SQP) method, in which the restrictions are not imposed as the penalty terms, but included into the Lagrange function, was investigated.

Methods. The experiments were conducted using two analytically defined objective functions and two objective functions that are encountered in problems of multi-objective optimization of characteristics of analog filters. The corresponding algorithms were realized in the MATLAB environment.

Results. The only heuristic algorithm shown to obtain the optimal solutions for all the functions is the particle swarm optimization algorithm. The sequential quadratic programming (SQP) algorithm was applicable to one of the analytically defined objective functions and one of the filter optimization objective functions, as well as appearing to be significantly superior to heuristic algorithms in speed and accuracy of solutions search. However, for the other two functions, this method was found to be incapable of finding correct solutions.

Conclusions. A topical problem is the estimation of the applicability of the considered methods to obtaining Pareto-optimal solutions based on preliminary analysis of properties of functions that determine the quality indices.

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