## Write Objective Function for Problem-Based Least Squares

To specify an objective function for problem-based least squares, write the objective either explicitly as a sum of squares or as the square of a norm of an expression. By explicitly using a least-squares formulation, you obtain the most appropriate and efficient solver for your problem. For example,

```t = randn(10,1); % Data for the example x = optimvar('x',10); obj = sum((x - t).^2); % Explicit sum of squares prob = optimproblem("Objective",obj); % Check to see the default solver opts = optimoptions(prob)```
```opts = lsqlin options: ...```

Equivalently, write the objective as a squared norm.

```obj2 = norm(x-t)^2; prob2 = optimproblem("Objective",obj2); % Check to see the default solver opts = optimoptions(prob2)```
```opts = lsqlin options: ...```

In contrast, expressing the objective as a mathematically equivalent expression gives a problem that the software interprets as a general quadratic problem.

```obj3 = (x - t)'*(x - t); % Equivalent to a sum of squares, % but not interpreted as a sum of squares prob3 = optimproblem("Objective",obj3); % Check to see the default solver opts = optimoptions(prob3)```
```opts = quadprog options: ...```

Similarly, write nonlinear least-squares as a square of a norm or an explicit sums of squares of optimization expressions. This objective is an explicit sum of squares.

```t = linspace(0,5); % Data for the example A = optimvar('A'); r = optimvar('r'); expr = A*exp(r*t); ydata = 3*exp(-2*t) + 0.1*randn(size(t)); obj4 = sum((expr - ydata).^2); % Explicit sum of squares prob4 = optimproblem("Objective",obj4); % Check to see the default solver opts = optimoptions(prob4)```
```opts = lsqnonlin options: ...```

Equivalently, write the objective as a squared norm.

```obj5 = norm(expr - ydata)^2; % norm squared prob5 = optimproblem("Objective",obj5); % Check to see the default solver opts = optimoptions(prob5)```
```opts = lsqnonlin options: ...```

The most general form that the software interprets as a least-squares problem is a square of a norm or else a sum of expressions Rn of this form:

`${R}_{n}={a}_{n}+{k}_{1}\sum \left({k}_{2}\sum \left({k}_{3}\sum \left(...{k}_{j}{e}_{n}^{2}\right)\right)\right)$`
• en is any expression. If multidimensional, en should be squared term-by-term using `.^2`.

• an is a scalar numeric value.

• The kj are positive scalar numeric values.

• Instead of multiplying by kj, you can instead divide by kj, which is equivalent to multiplying by 1/kj.

Each expression Rn must evaluate to a scalar, not a multidimensional value. For example,

```x = optimvar('x',10,3,4); y = optimvar('y',10,2); t = randn(10,3,4); % Data for example u = randn(10,2); % Data for example a = randn; % Coefficient k = abs(randn(5,1)); % Positive coefficients % Explicit sums of squares: R1 = a + k(1)*sum(k(2)*sum(k(3)*sum((x - t).^2,3))); R2 = k(4)*sum(k(5)*sum((y - u).^2,2)); R3 = 1 + cos(x(1))^2; prob = optimproblem('Objective',R1 + R2 + R3); options = optimoptions(prob) ```
```options = lsqnonlin options: ...```