Keywords: multi-objective optimization, curve fitting, regression, parameters estimation, evolutionary algorithms, optimization.

To the authors' knowledge there is no published work either on Pareto-based multi-objective curve fitting or Pareto-based multi-objective parameters estimation. The advantage of using a multi-objective formulation based on traditional single-objective optimization problems has been shown in [1] where the authors argue that multi-objectivization can bypass local minima and hence improve solvability of the given problem.

The potential multi-objectivization of curve fitting is different in comparison with the traditional multi-objective (MO) optimization because the objective functions in the curve fitting are usually in terms of some error measures that we would like to minimize, and therefore, all objectives are correlated.

Particular multi-objectivization is straightforward since different error measures can be defined and also several errors can be independently measured at different parts of the resulting curves to include the experimenter's knowledge of the problem solved. Moreover, this methodology permits one to intuitively include other criteria such as monotonicity or minimum curvature conditions. The single objective solution is usually implemented using the weighted aggregation approach and is often multi-modal and requires tuning of the weights which is another optimization task itself.

Our goal is to answer the question whether the single-objective (SO) optimization will be better than the multi-objective optimization based on the same evaluation criteria, i.e. the aggregated objective function. In other words, the aim is to investigate whether the multi-objective optimization wastes time by exploring a Pareto set or the single-objective optimization is lost somewhere in a local optima.

The comparison is demonstrated on two artificial curves with known mathematical formula to assess the accuracy. Then one real problem is solved, particularly the fitting of a curve describing the time-heat relationship during the cement paste hydration process.

Three approaches are discussed: a pure single-objective optimization, a multi-objective problem transformed into the single-objective one using a weighted sum method and a multi-objective optimization.

There is no clear conclusion from presented results. The accuracy, efficiency and computational time of all presented methods is very similar in all shown examples. However, a pure single-objective description of a problem in some cases is not sufficient.

1

J.D. Knowles, R.A. Watson, D.W. Corne, "Reducing local optima in single-objective problems by multi-objectivization", In Proceedings of the First International Conference on Evolutionary Multi-criterion Optimization (EMO'01), 269-283, 2001.

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