Example: Nonlinear Constrained Minimization

over the unit disk, i.e., the disk of radius 1 centered at the origin. In other words, find x that minimizes the function f(x) over the set . This problem is a minimization of a nonlinear function with a nonlinear constraint.

Note Rosenbrock's function is a standard test function in optimization. It has a unique minimum value of 0 attained at the point (1,1). Finding the minimum is a challenge for some algorithms since it has a shallow minimum inside a deeply curved valley.

Here are two views of Rosenbrock's function in the unit disk. The vertical axis is log-scaled; in other words, the plot shows log(1+f(x)). Contour lines lie beneath the surface plot.

Rosenbrock's function, log-scaled: two views.

Code for generating the figure

function createfigure

% Create figure
figure1 = figure('Position',[1 400 1200 600]);
colormap('gray');
axis square;
R=0:.002:1;
TH=2*pi*(0:.002:1); 
X=R'*cos(TH); 
Y=R'*sin(TH); 
Z=log(1+vrosenbrock(X,Y));

% Create subplot
subplot1 = subplot(1,2,1,'Parent',figure1);
view([124 34]);
grid('on');
hold('all');

% Create surface
surf(X,Y,Z,'Parent',subplot1,'LineStyle','none');

% Create contour
contour(X,Y,Z,'Parent',subplot1);

% Create subplot
subplot2 = subplot(1,2,2,'Parent',figure1);
view([234 34]);
grid('on');
hold('all');

% Create surface
surf(X,Y,Z,'Parent',subplot2,'LineStyle','none');

% Create contour
contour(X,Y,Z,'Parent',subplot2);

The function vrosenbrock, used in createfigure, calculates Rosenbrock's function at a whole vector of points:

function z=vrosenbrock(x,y)

z=100*(y - x.^2).^2 + (1-x).^2;

The function f(x) is called the objective function. This is the function you wish to minimize. The inequality is called a constraint. Constraints limit the set of x over which you may search for a minimum. You can have any number of constraints, which are inequalities or equations.

All Optimization Toolbox optimization functions minimize an objective function. To maximize a function f, apply an optimization routine to minimize –f. For more details about maximizing, see Maximizing an Objective.

Defining the Problem in Toolbox Syntax

To use Optimization Toolbox software, you need to

Define your objective function in the MATLAB language, as a function file or anonymous function. This example will use a function file.
Define your constraint(s) as a separate file or anonymous function.

Function File for Objective Function

A function file is a text file containing MATLAB commands with the extension .m. Create a new function file in any text editor, or use the built-in MATLAB Editor as follows:

At the command line enter:
```
edit rosenbrock
```
The MATLAB Editor opens.

In the editor enter:

function f = rosenbrock(x)
f = 100*(x(2) - x(1)^2)^2 + (1 - x(1))^2;

Save the file by selecting File > Save.

File for Constraint Function

Constraint functions must be formulated so that they are in the form c(x) ≤ 0 or ceq(x) = 0. The constraint needs to be reformulated as in order to have the correct syntax.

Furthermore, toolbox functions that accept nonlinear constraints need to have both equality and inequality constraints defined. In this example there is only an inequality constraint, so you must pass an empty array [] as the equality constraint function ceq.

With these considerations in mind, write a function file for the nonlinear constraint:

Create a file named unitdisk.m containing the following code:

function [c, ceq] = unitdisk(x)
c = x(1)^2 + x(2)^2 - 1;
ceq = [ ];

Save the file unitdisk.m.

Running the Optimization

There are two ways to run the optimization:

Using the Optimization Tool Graphical User Interface (GUI)
Using command line functions; see Minimizing at the Command Line.

Optimization Tool

Start the Optimization Tool by typing optimtool at the command line. The following GUI opens.

For more information about this tool, see Graphical Optimization Tool.
The default Solver fmincon - Constrained nonlinear minimization is selected. This solver is appropriate for this problem, since Rosenbrock's function is nonlinear, and the problem has a constraint. For more information about how to choose a solver, see Choosing a Solver.
In the Algorithm pop-up menu choose Active set—the default Trust region reflective algorithm doesn't handle nonlinear constraints.
For Objective function enter @rosenbrock. The @ character indicates that this is a function handle of the file rosenbrock.m.
For Start point enter [0 0]. This is the initial point where fmincon begins its search for a minimum.
For Nonlinear constraint function enter @unitdisk, the function handle of unitdisk.m.
Your Problem Setup and Results pane should match this figure.
In the Options pane (center bottom), select iterative in the Level of display pop-up menu. (If you don't see the option, click Display to command window.) This shows the progress of fmincon in the command window.
Click Start under Run solver and view results.

The following message appears in the box below the Start button:

Optimization running.
Objective function value: 0.045674808692966654
Local minimum possible. Constraints satisfied.

fmincon stopped because the predicted change in the objective function
is less than the default value of the function tolerance and constraints 
are satisfied to within the default value of the constraint tolerance.

Your objective function value may differ slightly, depending on your computer system and version of Optimization Toolbox software.

The message tells you that:

The search for a constrained optimum ended because the derivative of the objective function is nearly 0 in directions allowed by the constraint.
The constraint is very nearly satisfied.

Exit Flags and Exit Messages discusses exit messages such as these.

The minimizer x appears under Final point.

Minimizing at the Command Line

You can run the same optimization from the command line, as follows.

Create an options structure to choose iterative display and the active-set algorithm:
```
options = optimset('Display','iter','Algorithm','active-set');
```
Run the fmincon solver with the options structure, reporting both the location x of the minimizer, and value fval attained by the objective function:
```
[x,fval] = fmincon(@rosenbrock,[0 0],...
                    [],[],[],[],[],[],@unitdisk,options)
```
The six sets of empty brackets represent optional constraints that are not being used in this example. See the fmincon function reference pages for the syntax.

MATLAB outputs a table of iterations, and the results of the optimization:

Local minimum possible. Constraints satisfied.

fmincon stopped because the predicted change in the objective function
is less than the default value of the function tolerance and constraints 
are satisfied to within the default value of the constraint tolerance.

<stopping criteria details>

Active inequalities (to within options.TolCon = 1e-006):
  lower      upper     ineqlin   ineqnonlin
                                     1

x =
    0.7864    0.6177

fval =
    0.0457

The message tells you that the search for a constrained optimum ended because the derivative of the objective function is nearly 0 in directions allowed by the constraint, and that the constraint is very nearly satisfied. Several phrases in the message contain links that give you more information about the terms used in the message. For more details about these links, see Enhanced Exit Messages.

Interpreting the Result

The iteration table in the command window shows how MATLAB searched for the minimum value of Rosenbrock's function in the unit disk. This table is the same whether you use Optimization Tool or the command line. MATLAB reports the minimization as follows:

                                Max   Line search  Directional First-order 
Iter F-count        f(x)   constraint steplength   derivative  optimality Procedure 
   0      3            1           -1                                         
   1      9     0.953127      -0.9375      0.125         -2       12.5   
   2     16     0.808446      -0.8601     0.0625      -2.41       12.4   
   3     21     0.462347       -0.836       0.25      -12.5       5.15   
   4     24     0.340677      -0.7969          1      -4.07      0.811   
   5     27     0.300877      -0.7193          1     -0.912       3.72   
   6     30     0.261949      -0.6783          1      -1.07       3.02   
   7     33     0.164971      -0.4972          1     -0.908       2.29   
   8     36     0.110766      -0.3427          1     -0.833          2   
   9     40    0.0750939      -0.1592        0.5       -0.5       2.41   
  10     43    0.0580974    -0.007618          1     -0.284       3.19   
  11     47     0.048247    -0.003788        0.5      -2.96       1.41   
  12     51    0.0464333     -0.00189        0.5      -1.23      0.725   
  13     55    0.0459218   -0.0009443        0.5     -0.679      0.362   
  14     59    0.0457652   -0.0004719        0.5       -0.4      0.181   
  15     63    0.0457117   -0.0002359        0.5     -0.261     0.0905  Hessian modified  
  16     67    0.0456912   -0.0001179        0.5     -0.191     0.0453  Hessian modified  
  17     71    0.0456825  -5.897e-005        0.5     -0.156     0.0226  Hessian modified  
  18     75    0.0456785  -2.948e-005        0.5     -0.139     0.0113  Hessian modified  
  19     79    0.0456766  -1.474e-005        0.5      -0.13    0.00566  Hessian modified

This table might differ from yours depending on toolbox version and computing platform. The following description applies to the table as displayed.

The first column, labeled Iter, is the iteration number from 0 to 19. fmincon took 19 iterations to converge.
The second column, labeled F-count, reports the cumulative number of times Rosenbrock's function was evaluated. The final row shows an F-count of 79, indicating that fmincon evaluated Rosenbrock's function 79 times in the process of finding a minimum.
The third column, labeled f(x), displays the value of the objective function. The final value, 0.0456766, is the minimum that is reported in the Optimization Tool Run solver and view results box, and at the end of the exit message in the command window.
The fourth column, Max constraint, goes from a value of –1 at the initial value, to very nearly 0, –1.474e–005, at the final iteration. This column shows the value of the constraint function unitdisk at each iteration. Since the value of unitdisk was nearly 0 at the final iteration, there.

The other columns of the iteration table are described in Iterative Display.

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