Approximation Model Generation and Optimization Methodology for Highly Non-Linear Multi-Objective Problems with Unevenly Spaced Data

Authors: NICHOLSON John, GOYAL Deepak

This post discusses an approximation model generation and optimization methodology for highly non-linear multi-objective problems with unevenly spaced data. The overall approach is given first, followed by approximation and optimization methodology.

Overall Approach

1. Generate various approximation models and compare error analysis data to select the best.
2. Perform Multi-Objective Genetic Algorithm (MOGA) optimization on the approximation model to generate a Pareto Front of optimal designs.
3. Perform gradient based optimization starting from select Pareto-Optimal designs to identify the final set of optimal designs.
4. Perform full analyses at each of the resulting optimal designs and select the best design.  If large errors exist between the predicted objective at the optimal design and the actual objective at the optimal design then take steps to refine the approximation model and go to step 1.

Approximation Model Generation and Error Analysis

The approximation model types and associated configuration options that typically give good approximation models of highly non-linear problems with unevenly spaced data are given in Table 1: 

Each of these models should be generated and the one that provides the best fit for the data should be selected.  It’s recommended that the following criteria be used in identifying the model that provides the best fit:

• Use R-Squared as the primary error analysis measure when comparing fitness of models (R-Squared of 1 means that the model fits the data perfectly at the sample points, R-Squared greater than 0.9 represents a good fit, R-Squared less than 0.8 represents a poor fit and steps should be taken to refine the model.  See Figures 1-3 for examples of various R-Squared values).

• Only use a higher order fit than 2 if the maximum and average errors do not increase (While a higher-order may fit the sample points better, it may not fit interpolated points located between the sample points well).
• Verify that the residual plot of the selected model does not contain a discernible pattern (If a pattern exists, it is an indication that there is a problem with using the selected model to approximate the data set.  See Figures 4-7 [Reference 1]).

Multi-Objective Genetic Algorithm (MOGA) Optimization to Generate Pareto Front

            Performing MOGA optimization using the NCGA technique ensures that the entire design space is searched and reduces the likelihood of converging to a local minimum.  Additionally, it provides the engineer with a set of Pareto-Optimal designs to choose from and optimize further using gradient based optimization techniques.  As shown below, a Pareto Front of Pareto-Optimal designs can be generated by plotting each objective vs. another objective.  In practice, it may be helpful to perform MOGA optimization on the two best approximation models obtained in the previous step.  For example, the two Pareto Fronts obtained below were generated by performing MOGA optimization on a response surface approximation model (Figure 8) and a radial basis function approximation model (Figure 9).  Each approximation model generates very different Pareto Fronts.  The engineer’s knowledge of the problem and how objectives should relate to one another may be used to identify which model more accurately represents the problem at hand.

Gradient Based Optimization of Select Pareto-Optimal Designs

            Multi-Objective Genetic Algorithms will terminate after a predetermined number of evaluations based on the population size and number of generations. The Pareto-Optimal design set found at termination may be near optimal but often not sufficiently so. Therefore, gradient based optimization techniques such as Nonlinear Programming by Quadratic Lagrangian (NLPQL) can be performed starting from the Pareto-Optimal designs identified by the engineer as best (using his or her intuition and judgment) to fine tune the designs to the desired level.  It is important to note that the chosen Pareto-Optimal designs were obtained by considering each objective separately and hence no scaling factors were employed. In contrast, gradient based optimization will require us to combine the objectives into one scalar function.  Therefore, to preserve the trade-off identified by the engineer as best, it is recommended that each objective be scaled by its current Pareto-Optimal value before performing gradient based optimization.  This will help ensure that the chosen trade-off is maintained and one objective does not dominate the optimization as much as possible. 

Perform Full Analyses and Take Steps to Refine the Model if Necessary

            Once a set of optimal designs has been obtained using the process outlined above, full analyses can be carried out and the best resulting design selected.  If large errors exist between the predicted objective at the optimal design and the actual objective at the optimal design then take the following steps to refine the approximation model and go to step 1:

• Try scaling sample point input variables to obtain a better approximation model (most of the approximation models involve the solution of a system of linear equations and poorly scaled input data may lead to a coefficient matrix with a large condition number that results in large errors in the calculated coefficients for an approximation model).  The following formula may be applied to each set of input variables, , to transform them into a new set of variables, , ranging between -1 and 1:

Here  max(xi) and min(xi)  refer to the maximum and minimum values, respectively, of the ith input variable over all sample points [Reference 2].

• Try narrowing the range of the input variables by removing select sample points (a more accurate approximation model may be obtainable if a smaller design space can be identified).
• Last, try collecting more sample points to improve the accuracy of the model.  It is recommended that this be done incrementally so that as few sample points as possible must be collected to achieve the desired accuracy.

References

[1] http://www.originlab.com/www/helponline/Origin/en/UserGuide/Graphic_Residual_Analysis.html accessed on 2/14/2013.

[2] Arora, Jasbir.  2012.  Introduction to Optimum Design.  3rd ed.  Massachusetts: Elsevier Inc.