Efficient curve fitting: An application of multiobjective programming

Curve fitting is an interesting and important subject in mathematics and engineering. It has been studied extensively and a number of approaches, mostly based on polynomials and piecewise polynomials, have been employed. In the usual setting, some data points are given and one wants to find a polynomial function with the minimum violations measured by a norm in the given data points. In these approaches, norms are applied to aggregate all violations as a scalar.In this paper, the polynomial curve fitting problem is considered from the viewpoint of decision making. Taking into account some weaknesses of the norm-based approaches, a multiobjective programming model for curve fitting is given in which the violations are minimized simultaneously as a vector. This approach is more flexible for the curve fitting problem. Indeed, using the concept of efficiency in multiobjective programming, it enables us to impose some additional helpful secondary preferences. Especially, this approach can obtain a fitted curve with efficient violations and minimum average curvature or minimum average slope.