## Pareto Front for Two Objectives

### Multiobjective Optimization with Two Objectives

This example shows how to find a Pareto set for a two-objective function of two variables. The example presents two approaches for minimizing: using the Optimize Live Editor task and working at the command line.

The two-objective function *f*(*x*), where
*x* is also two-dimensional, is

$$\begin{array}{l}{f}_{1}(x)={x}_{1}^{4}+{x}_{2}^{4}+{x}_{1}{x}_{2}-{(}^{{x}_{1}}-10{x}_{1}^{2}\\ {f}_{2}(x)={x}_{1}^{4}+{x}_{2}^{4}+{x}_{1}{x}_{2}-{(}^{{x}_{1}}.\end{array}$$

### Find Pareto Set Using Optimize Live Editor Task

Create a new live script by clicking the

**New Live Script**button in the**File**section on the**Home**tab.Insert an

**Optimize**Live Editor task. Click the**Insert**tab and then, in the**Code**section, select**Task > Optimize**.Click the

**Solver-based**task.For use in entering problem data, insert a new section by clicking the

**Section Break**button on the**Insert**tab. New sections appear above and below the task.In the new section above the task, enter the following code to define the number of variables and lower and upper bounds.

nvar = 2; lb = [0 -5]; ub = [5 0];

To place these variables into the workspace, run the section by pressing

**Ctrl+Enter**.**Specify Problem Type**In the

**Specify problem type**section of the task, click the**Objective > Nonlinear**button.Click the

**Constraints > Lower bounds**and**Upper bounds**buttons.Select

**Solver > gamultiobj - Multiobjective optimization using genetic algorithm**.**Select Problem Data**In the

**Select problem data**section, select**Objective function > Local function**, and then click the**New**button. The function appears in a new section below the task.Edit the resulting function definition to contain the following code.

function f = mymulti1(x) f(2) = x(1)^4 + x(2)^4 + x(1)*x(2) - (x(1)*x(2))^2; f(1) = f(2) - 10*x(1)^2; end

In the

**Select problem data**section, select the**Local function > mymulti1**function.Select

**Number of variables > nvar**.Select

**Lower bounds > From workspace > lb**and**Upper bounds > From workspace > ub**.**Specify Solver Options**Expand the

**Specify solver options**section of the task, and then click the**Add**button. To have a denser, more connected Pareto front, specify a larger-than-default populations by selecting**Population settings > Population size > 60**.To have more of the population on the Pareto front than the default settings, click the

**+**button. In the resulting options, select**Algorithm > Pareto set fraction > 0.7**.**Set Display Options**In the

**Display progress**section of the task, select the**Pareto front**plot function.**Run Solver and Examine Results**To run the solver, click the options button

**⁝**at the top right of the task window, and select**Run Section**. The plot appears in a separate figure window and in the task output area.The plot shows the tradeoff between the two components of

*f*, which is plotted in objective function space. For details, see the figure Figure 14-2, Set of Noninferior Solutions.

### Find Pareto Set at the Command Line

To perform the same optimization at the command line, complete the following steps.

Create the

`mymulti1`

objective function file on your MATLAB^{®}path.function f = mymulti1(x) f(2) = x(1)^4 + x(2)^4 + x(1)*x(2) - (x(1)*x(2))^2; f(1) = f(2) - 10*x(1)^2; end

Set the options and bounds.

options = optimoptions('gamultiobj','PopulationSize',60,... 'ParetoFraction',0.7,'PlotFcn',@gaplotpareto); lb = [0 -5]; ub = [5 0];

Run the optimization using the options.

`[solution,ObjectiveValue] = gamultiobj(@mymulti1,2,... [],[],[],[],lb,ub,options);`

Both the Optimize Live Editor task and the command line allow you to formulate and solve problems, and they give identical results. The command line is more streamlined, but provides less help for choosing a solver, setting up the problem, and choosing options such as plot functions. You can also start a problem using Optimize, and then generate code for command line use, as in Constrained Nonlinear Problem Using Optimize Live Editor Task or Solver.

### Alternate Views

You can view this problem in other ways. The following figure contains a plot of
the level curves of the two objective functions, the Pareto frontier calculated by
`gamultiobj`

(boxes), and the x-values of the true Pareto
frontier (diamonds connected by a nearly straight line). The true Pareto frontier
points are where the level curves of the objective functions are parallel. The
algorithm calculates these points by finding where the gradients of the objective
functions are parallel. The figure is plotted in parameter space; see Figure 14-1, Mapping from Parameter Space into Objective Function Space.

**Contours of objective functions, and Pareto frontier**

`gamultiobj`

finds the ends of the line segment, meaning it
finds the full extent of the Pareto frontier.