Braden Hancock and Dr. Christopher A. Mattson, Mechanical Engineering Department
Introduction
In October 2013, we proposed the development of a method to directly generate smart Pareto sets of solutions in multiobjective optimization problems. Thanks to the opportunity that we received to pursue this research through an ORCA grant, we were able to develop that technology and disseminate it in the scientific community. The results of this research include a peer-reviewed journal article, a professional conference paper, a regional student conference paper, and an Honors thesis.
Methodology
In multiobjective design scenarios, designers often seek smart Pareto sets—minimal sets of nondominated solutions that sufficiently represent the tradeoff properties of the solution space. Traditionally, these smart Pareto sets are produced by generating very large sets of equally distributed solutions and then filtering out those solutions which do not provide the designer with sufficiently unique information to merit a place in the final set. Striving to avoid the inherent inefficiency present in such an approach, we developed an approach for generating only those solutions that will remain in the minimal set that is presented to the user; i.e., directly generating the smart Pareto set. The result was the Smart Normal Constraint (SNC) method.
The SNC method incorporates the primary solution-generation mechanism of the well-known Normal Constraint (NC) method, which converts a multiobjective problem into a series of single objective problems with linear restrictions that target particular regions of the solution space. In the NC method, a hyperplane is constructed above the Pareto frontier and divided up into smaller regions of equal size. For each of these regions, linear restrictions are applied and a corresponding point is found on the Pareto front. Because the shape of the Pareto frontier is generally unknown before optimization, such a computationally exhaustive approach has seemed necessary in the past.
The key to improving efficiency lies in recognizing that each new solution provides additional information about the Pareto frontier. By updating a simple linear approximation of the Pareto frontier after each new point is discovered, the SNC method is able to predict which unexplored regions of the solution space are most likely to yield new solutions of interest. Using this information, the resolution with which the solution space is explored can be adjusted dynamically and by region. This results in not only a more efficient search, but also in more interesting potential solutions, because they tend to be found in what are known as “knee” regions of the Pareto frontier, where rapid tradeoff between objectives occurs (see Figure 1).
Results
The performance of the SNC method was compared to the industry standard approach of using the NC method with a smart Pareto filter. Both methods were applied to three challenging test problems from the literature with features known to cause difficulties for Pareto set generation algorithms. The results are shown below in Table 1. In every case, the SNC method outperformed the NC method, requiring fewer function calls to generate a nearly identical final set of smart Pareto solutions. Furthermore, the advantage of the SNC method over the NC method had a positive correlation with the level of difficulty of the test problems, up to a 99% decrease in the required number of function calls for the 5D test problem WATER.
Conclusion
From the results shown in Table 1, it is evident that the SNC method is an improvement over the predominant existing method for generating smart Pareto sets. By dynamically developing an approximation of the Pareto frontier over the course of the optimization routine, the SNC method is able to more efficiently and intelligently search the solution space for those solutions that will be presented to the designer in the final smart Pareto set.
Publications and Awards
The results from this research have been published or presented in the following venues:
- Hancock, B. J., Mattson, C. A., “The Smart Normal Constraint Method for Directly Generating a Smart Pareto Set,” Structural and Multidisciplinary Optimization, Published Online Apr. 2013, DOI 10.1007/s00158-013-0925-6
- Hancock, B. J., Mattson, C. A., “The Smart Normal Constraint Method for Directly Generating a Smart Pareto Set, ” 9th AIAA Multidisciplinary Design Optimization Specialist Conference, Apr. 2013 (MDO2013-1512152)
- Hancock, B. J., Mattson, C. A., “The Smart Normal Constraint Method for Directly Generating a Smart Pareto Set,” AIAA Region VI 2013 Student Conference*
- Hancock, B. J. “The Smart Normal Constraint Method for Directly Generating a Smart Pareto Set,” Brigham Young University, Department of Mechanical Engin