Learn more about Model-Based Calibration

Setting Up Multiobjective Optimizations

On this page…
Overview of Setting Up Multiobjective Optimizations About the NBI (Normal Boundary Intersection) Algorithm

Overview of Setting Up Multiobjective Optimizations

CAGE Optimization contains an algorithm (NBI) to solve multiobjective optimization problems. For example, you could use this type of optimization to determine the optimal torque versus NOx emissions curve for an engine over the operating range of the engine. To solve this problem you must define two competing optimization objectives, to maximize torque while minimizing NOx emissions.

To set up a new multiobjective optimization:

1. Use the wizard for Creating Optimizations from Models to create your optimization. You can configure one of your objectives in the wizard. You must select the NBI algorithm to solve a multiobjective optimization.

   When you select NBI, the wizard automatically creates a second blank objective for you. When you finish the wizard and return to the Optimization view, you can configure the second objective (and add a third if desired).

2. You can add a boundary model constraint in the wizard. To apply other types of constraints you must use the Optimization view. You can apply linear, ellipsoid, 1-D table, 2-D table, and range constraints, and some constraints are specific to sum optimizations—sum constraints and table gradient constraints.

   See Edit Constraint for details of all these constraints.

3. You can use the wizard to choose the points where you want to run the optimization. You can select a suitable table grid, data set, point-by-point model operating points, or use the variable set points. You can also set up your optimization variable values in the Optimization view. See Editing Variable Values. You can enter values manually or import them from data sets, tables, or the output of existing optimizations.

4. Run the optimization using the procedure forRunning Optimizations.

   • Click Run Optimization in the toolbar to run the optimization with the default number of solutions (10 for two objectives. For more objectives, see NBI Options).

   • Click Set Up and Run Optimization to change the number of solutions before running. You can use the Optimization Parameters dialog to change how many tradeoff solutions you want the optimization to find per run. See NBI Optimization Parameters.

5. View the results (see Viewing Your Optimization Results). For descriptions of optimization output specific to multiobjective problems, see Tools for Multiobjective Optimizations.

Back to Top

About the NBI (Normal Boundary Intersection) Algorithm

To understand the options for the NBI algorithm, some limited understanding of the algorithm is required. For more information on the NBI algorithm, see the following reference:

Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems, I. Das and J.E. Dennis, SIAM J. on Optimization. 8(3), 631-657 (1998).

The NBI algorithm is performed in two steps. The first step is to find the global of each objective individually. This is called the shadow minima problem, and is a single-objective problem for each objective function. The MATLAB routine fmincon is used to find these . Once these are found, they can be plotted against each other. For example, consider an NBI optimization that simultaneously maximizes TQ and minimizes NOX emissions. A plot of the against each other might resemble the following.

The second step is to find the "best" set of tradeoff solutions between your objectives. To do this, the NBI algorithm spaces Npts start points in the (n-1) hypersurface, S, that connects the shadow . In the above example, S is the straight line that connects the points N and T. For each of the Npts points on S, the algorithm tries to maximize the distance along the normal away from this surface (this distance is labeled L in the following figure). This is called the NBI subproblem. For each of the points, the NBI subproblem is a single-objective problem and the algorithm uses the MATLAB fmincon routine to solve it. This is illustrated below for the TQ-NOX example.

The figure above shows spacing of the points between the along the (n-1) surface. The algorithm tries to maximize the distance L along the normal away from the surface. The following figure shows the final solution found by the NBI algorithm.

To see how the NBI settings are used in the Optimization Parameters dialog box, see NBI Optimization Parameters.

Back to Top

Setting Up Sum Optimizations

Editing Variable Values

© 1984-2009- The MathWorks, Inc.    -   Site Help   -   Patents   -   Trademarks   -   Privacy Policy   -   Preventing Piracy   -   RSS