New Concepts in Graphic Visualization of Objective Functions

A method of inspecting the design space of multivariable objective functions is proposed. By scanning 1 or 2 of the variables at a constant step while partially minimizing or maximizing the function with respect to the remaining variables, sets of points are generated that can be further used in producing 2D or 3D diagrams. A number of examples are given for showing the usefulness of the method in studying the design space of objective functions and of the constraint activity. All graphs are produced with an in-house program that allows generation of logarithmically spaced level-curve diagrams and accurately truncating function surfaces over the z-axis at specified heights. INTRODUCTION Functions of one variable f(x) can be straightforwardly represented in the 2D paper space where one dimension can be associated with the variable and the other dimension with the function value. As an extension, functions of two variables f(x1, x2) can be visualized as families of curves x1=constant, x2=constant or z=constant (also called level curve diagrams). Another possibility is to plot the z=f(x1, x2) surface, which is equally intuitive since of the 3D Euclidean space, two dimensions can be associated with x1 and x2 and the third dimension with the function value. Numerous mathematical and operations research publications give examples of such representations, which allow for a better understanding of the properties of the respective functions, such as: monotonicity, convexity, smoothness, the existence of multiple minima, constraint activity etc. Functions of three variables can also be represented graphically, like plotting the relative proportions of three ingredients in a mixture [1] (although the 3 variables are not independent because the proportions sum must be unity). Other methods employing multiple-color plots and computer animation are described by Encarnação et al [2] and Jones [3]. One example of scanning two of the variables through a number of 3D representations generated for different values of the third variable in the case of a f(x1, x2, x3) function is available in [4]. It is obvious that the number of plots required for visualizing the same way functions of more than three variables is prohibitively large, and the method becomes ineffective (following the approach described in [4], where 15 surface plots were used for inspecting the design space of a function of 3 variables, a 4-variable objective function would require 15 2 =225 representations, and a 5-variable objective function 15 3 =3375 representations). In the field of artificial neural networks, in particular, have been proposed numerous techniques for the visualization of the multidimensional spaces in both the learning process and the prediction process [5-7]. Fluid flow data, radar range images, atmospheric and oceanographic measurements also generate large multidimensional, multivariate data sets, the visualization of which continues to be an active and promising research area [8]. The interested reader is suggested to visit the Web site: http://catt.bus.okstate.edu/jones98/ for an exhaustive list of publication on the use of visualization to support optimisation modelling, solution and analysis. In the present paper a method of inspecting graphically the design space of multivarite objective functions F(x1...xn), for which the configurations of the minimum domains are particularly of interest, will be presented. Since minF=-max(-F), only functions for which the optimum sought is a minimum will be discussed. The main idea of the method is to perform partial minimizations of the function with respect to all but 1 or 2 of the design variables, and to generate an appropriate number point-sets that will be further used for producing 2D or 3D graphical representations.