Steps for Problem-Based Multiobjective Optimization

This topic shows how to set up a multiobjective optimization in the problem-based approach, and details the format of results and initial points. For an example, see Pareto Front for Multiobjective Optimization, Problem-Based.

Specify Multiple Objective Functions

Specify multiple objective functions in one of two ways:

Optimization expression — Give an optimization expression that has vector or array values. For example, this objective function returns a vector of three values:
```
prob.Objective = [sin(x),cos(x),1 - x.^2];
```
Structure — Give a structure of optimization expressions, each of which evaluates to a scalar. For example, this objective function returns a structure with three objective components:
```
prob.Objective.sin = sin(x);
prob.Objective.cos = cos(x);
prob.Objective.quad = 1 - x^2;
```

Specify Multiple Objective Senses (Maximize or Minimize)

Specify an objective function sense, meaning maximize or minimize, depending on how you specify the objective function.

Objective is an optimization expression — All objectives in the problem have the same objective sense. For example,
```
prob.ObjectiveSense = "max";
```
Objective is a structure — Each objective function can have its own sense. The prob.ObjectiveSense structure has the same fields as the prob.Objective structure. For example,
```
prob.ObjectiveSense.sin = "minimize";
prob.ObjectiveSense.cos = "maximize";
prob.ObjectiveSense.quad = "max";
```

The default sense is "minimize".

Data Format of Multiobjective Solutions

The returned sol output is a vector of OptimizationValues objects. Each object contains the values of the optimization variables and the objective functions at one point on the Pareto front. If the problem has nonlinear constraints, sol also contains the nonlinear constraint violations at each solution point.

The returned fval output is a matrix where each row represents one solution point and each column represents one objective function. The fval output is numeric, unlike the sol output. You can obtain the objective function values from the sol object. However, you can find the values more easily in fval.

You can plot the resulting Pareto front in two or three dimensions by calling paretoplot on sol. For an example, see Pareto Front for Multiobjective Optimization, Problem-Based.

Supply Initial Points for Multiobjective Problem

Specifying initial points for multiobjective problems is optional. However, you can sometimes obtain better solutions by doing so. For an example showing the benefit, see Pareto Front for Multiobjective Optimization, Problem-Based.

To specify initial points, create an OptimizationValues object using the optimvalues function. For examples, see the optimvalues reference page.

Hybrid Function

To obtain more accurate solutions, the gamultiobj solver can optionally call fgoalattain. For an example, see Design Optimization of a Welded Beam. To use this hybrid function in the problem-based workflow, set the HybridFcn option to "fgoalattain":

options = optimoptions('gamultiobj',HybridFcn="fgoalattain");

Include the solver and options arguments in the solve call:

[sol,fval,exitflag,output] = solve(prob,...
    Solver="gamultiobj",...
    Options=options);

View Pareto Set

To view the Pareto set in two or three dimensions while the solver proceeds, set a plot option.

For the gamultiobj function, set the PlotFcn option to 'gaplotpareto'.

options = optimoptions("gamultiobj",PlotFcn="gaplotpareto");
sol = solve(prob,Options=options)

For the paretosearch function, set the PlotFcn option to 'psplotparetof'.

To view the Pareto set after the solver finishes, call paretoplot on the solution.

sol = solve(prob);
paretoplot(sol)

For an example, see Pareto Front for Multiobjective Optimization, Problem-Based.

If you have more than three objectives, paretoplot allows you to choose which objectives to plot. See the paretoplot reference page for details.