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## Optimization App with the fmincon Solver

This example shows how to use the Optimization app with the `fmincon` solver to minimize a quadratic subject to linear and nonlinear constraints and bounds.

### Note

The Optimization app warns that it will be removed in a future release.

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

`$\underset{x}{\mathrm{min}}f\left(x\right)={x}_{1}^{2}+{x}_{2}^{2}$`

subject to the constraints

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

### Step 1: Write a file objecfun.m for the objective function.

```function f = objecfun(x) f = x(1)^2 + x(2)^2;```

### Step 2: Write a file nonlconstr.m for the nonlinear constraints.

```function [c,ceq] = nonlconstr(x) 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 = [];```

### Step 3: Set up and run the problem with the Optimization app.

1. Enter `optimtool` in the Command Window to open the Optimization app.

2. Select `fmincon` from the selection of solvers and change the Algorithm field to `Active set`.

3. Enter `@objecfun` in the Objective function field to call the `objecfun.m` file.

4. Enter `[3;1]` in the Start point field.

5. Define the constraints.

• Set the bound `0.5`x1 by entering `[0.5,-Inf]` in the Lower field. The `-Inf` entry means there is no lower bound on x2.

• Set the linear inequality constraint by entering ```[-1 -1]``` in the A field and enter `-1` in the b field.

• Set the nonlinear constraints by entering `@nonlconstr` in the Nonlinear constraint function field.

6. In the Options pane, expand the Display to command window option if necessary, and select `Iterative` to show algorithm information at the Command Window for each iteration.

7. Click the button as shown in the following figure.

8. When the algorithm terminates, under Run solver and view results the following information is displayed:

• The Current iteration value when the algorithm terminated, which for this example is `7`.

• The final value of the objective function when the algorithm terminated:

`Objective function value: 2.0000000268595803`
• The algorithm termination message:

```Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.```
• The final point, which for this example is

``` 1 1 ```
9. In the Command Window, the algorithm information is displayed for each iteration:

``` Max Line search Directional First-order Iter F-count f(x) constraint steplength derivative optimality Procedure 0 3 10 2 Infeasible start point 1 6 4.84298 -0.1322 1 -5.22 1.74 2 9 4.0251 -0.01168 1 -4.39 4.08 Hessian modified twice 3 12 2.42704 -0.03214 1 -3.85 1.09 4 15 2.03615 -0.004728 1 -3.04 0.995 Hessian modified twice 5 18 2.00033 -5.596e-05 1 -2.82 0.0664 Hessian modified twice 6 21 2 -5.326e-09 1 -2.81 0.000522 Hessian modified twice Active inequalities (to within options.ConstraintTolerance = 1e-06): lower upper ineqlin ineqnonlin 3 4 Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.```

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