## 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
*R _{n}* of this form:

$${R}_{n}={a}_{n}+{k}_{1}{\displaystyle \sum \left({k}_{2}{\displaystyle \sum \left({k}_{3}{\displaystyle \sum \left(\mathrm{...}{k}_{j}{e}_{n}^{2}\right)}\right)}\right)}$$

*e*is any expression. If multidimensional,_{n}*e*should be squared term-by-term using_{n}`.^2`

.*a*is a scalar numeric value._{n}The

*k*are positive scalar numeric values._{j}Instead of multiplying by

*k*, you can instead divide by_{j}*k*, which is equivalent to multiplying by 1/_{j}*k*._{j}

Each expression *R _{n}* 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: ...