Optimize Live Editor Task with fmincon Solver

This example shows how to use the solver-based Optimize Live Editor task with the fmincon solver to minimize a quadratic subject to linear and nonlinear constraints and bounds.

Consider the problem of finding [x₁, x₂] that solves

$$\underset{x}{\min }f(x)=x_1^2+x_2^2$$

subject to the constraints

$$\begin{array}{rr}\hfill 0.5\le x_1& \hfill \text{(bound)}\\ \hfill -x_1-x_2+1\le 0& \hfill \text{(linear inequality)}\\ \hfill \begin{array}{r}\hfill -x_1^2-x_2^2+1\le 0\\ \hfill -9x_1^2-x_2^2+9\le 0\\ \hfill -x_1^2+x_2\le 0\\ \hfill -x_2^2+x_1\le 0\end{array}\}& \hfill \text{(nonlinear inequality)}\end{array}$$

The starting point x0 for this problem is x₁ = 3 and x₂ = 1.

Start 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.

For this example, choose the solver-based task.

For later use in entering problem data, select Insert > Section Break. New sections appear above and below the task.

Enter Problem Data

1. Starting from the top of the task, enter the problem type and constraint types. Click the Objective > Quadratic button and the Constraints > Lower bounds, Linear inequality, and Nonlinear buttons. The task shows that the recommended solver is fmincon.

2. Objective Function

   The objective function is simple enough to represent as an anonymous function. Position the cursor in the section above the task and enter this code.

   fun = @(x)sum(x.^2);

3. Lower Bound

   The problem contains the lower bound x₁ ≥ 0.5. Express this bound as a variable lb. With the cursor at the end of the line defining the objective function, press Enter, and enter the following code to specify the lower bound.

   lb = [0.5 -Inf];

4. Initial Point

   With the cursor at the end of the line defining the lower bound, press Enter, and enter the following code to set the initial point.

   x0 = [3,1];

5. Linear Constraint

   With the cursor at the end of the line defining the initial point, press Enter, and enter the following code to set the linear constraint.

   A = [-1,-1];
   b = -1;

6. Run Section

   The top section now includes five parameters.

   

   Next, you need to run the section to place the parameters in the workspace as variables. To do so, click the left-most area of the section, which contains a bar of diagonal stripes. After you click this area, the bar becomes a solid bar, indicating the variables are now in the workspace. (Note: You can also press Ctrl+Enter to run the section.)

7. Set Problem Data

   Enter the variables in the Select problem data section of the task. To specify the objective function, select Objective function > Function handle and choose fun.

8. Set the initial point x0.

9. Select Lower bounds > From workspace and select lb.

10. Set the linear inequality constraint variables A and b in the Linear inequality area.

11. Now specify the nonlinear inequality constraints. In the Select problem data section, select Nonlinear > Local function, and then click the New button. The function appears in a new section below the task. Edit the resulting code to contain the following uncommented lines.

    function [c,ceq] = constraintFcn(x)
    % You can include commented code lines or not.
    % Be sure that just these uncommented lines remain:
    c = [-x(1)^2 - x(2)^2 + 1;
         -9*x(1)^2 - x(2)^2 + 9;
         -x(1)^2 + x(2);
         -x(2)^2 + x(1)];
    ceq = [];
    end

12. In the Select problem data section, select the constraintFcn function.

13. Monitor Progress

    In the Display progress section of the task, select Text display > Each iteration so you can monitor the solver progress. Select Objective value for the plot.

    Your setup looks like this:

    

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.

To see where the solution variables are returned, look at the top of the task.

The final point and its associated objective function value appear in the solution and objectiveValue variables in the workspace. View these values by entering this code in the live editor section below the task.

solution, objectiveValue

Press Ctrl+Enter to run the section.