This example shows how to use the Optimization app to solve a constrained least-squares problem.

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

The problem in this example is to find the point on the plane *x*_{1} + 2*x*_{2} + 4*x*_{3} = 7 that is closest to the origin. The easiest way to solve this
problem is to minimize the square of the distance from a point *x* = (*x*_{1},*x*_{2},*x*_{3}) on the plane to the origin, which returns the same optimal point
as minimizing the actual distance. Since the square of the distance from an
arbitrary point (*x*_{1},*x*_{2},*x*_{3}) to the origin is $${x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}$$, you can describe the problem as follows:

$$\underset{x}{\mathrm{min}}f(x)={x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2},$$

subject to the constraint

*x*_{1} + 2*x*_{2} + 4*x*_{3} = 7.

The function *f*(*x*) is called the
*objective function* and *x*_{1} + 2*x*_{2} + 4*x*_{3} = 7 is an *equality constraint*. More complicated
problems might contain other equality constraints, inequality constraints, and upper
or lower bound constraints.

This section shows how to set up the problem with the
`lsqlin`

solver in the Optimization app.

Enter

`optimtool`

in the Command Window to open the Optimization app.Select

`lsqlin`

from the selection of solvers. Use the`Interior point`

algorithm.Enter the following to create variables for the objective function:

In the

**C**field, enter`eye(3)`

.In the

**d**field, enter`zeros(3,1)`

.

The

**C**and**d**fields should appear as shown in the following figure.Enter the following to create variables for the equality constraints:

In the

**Aeq**field, enter`[1 2 4]`

.In the

**beq**field, enter`7`

.

The

**Aeq**and**beq**fields should appear as shown in the following figure.Click the

**Start**button as shown in the following figure.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`1`

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

Objective function value: 2.333333333333334

The exit message:

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

0.333 0.667 1.333