Visualization of an optimization problem (i.e the "design space") becomes complex when the number of independent variables of the problem increases beyond two. Unfortunately, realistic optimization problems and their design spaces are often greater than two dimensions and therefore difficult to visualize. In order to create and display in greater than three dimensions it is necessary to use color, size, or symbols to show added dimensions. With the complexity in a visualization that uses these extra dimensional features, an observer is often overloaded with data and it can be difficult to grasp a firm understanding of the relationships therein. Furthermore, this solution of adding dimensions greater than three can only increment to a few dimensions beyond three and cannot achieve higher dimensions. There are currently two general areas for visualizing a higher dimensional design space: dimensional reduction, and individual variable comparison. With either of these methods, it is possible to display the resulting design space, or portion thereof, in a viewable dimensionality such as two or three dimensions. Self-organizing contextual maps provide a solution to this visualization problem by utilizing the dimensionality reduction capability of self-organizing maps and the display capability of the contextual map.

Self-organizing maps (SOMs) are able to map a design space of varying dimensionality to a two dimensional neuron lattice. The SOM can then be provided contextual information to display the similarities between areas of the design space either in terms of alphanumerical labels or visuals. This method will organize the numerical objective values associated with a design space to apply labels to the contextual SOMs. These contextual self-organizing maps allow the user to observe the entire design space in a two dimensional representation.

The ability to view an entire design space in this manner provides many advantages such as an understanding of the characteristics of the design space and optimization problem. This thesis will explain the work completed to apply contextual self-organizing maps to the visualization of optimization design spaces by:

1. Providing a visualization of the design space in two dimensions.

2. Extract characteristics of the design space using the resulting contextual map.

The resulting visual representations are achieved by generating a typical self-organizing map, and applying the objective function values as labels to each winning node. With a set of labels on each node, it was possible to calculate the mean, standard deviation, and minimum value for each node and display the results visually in the representation. The hue saturation value coloring scheme was used to display these three statistical measures using a single color for each node. The visual display of this coloring system makes the optimal node the closest to a brighter colored and more vibrant green colored node than the rest of the nodes in the map.

The results from this work show that contextual self-organizing maps can display valuable information about the design space that can then be extracted and applied to the solution of the optimization problem. The primary characteristics identified in the results are the modality of the design space and the optimal region within the design space. The results of this research will improve optimization by decreasing the time needed to solve optimization problems by gaining an understanding of the design space prior to a solution run.