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
solver = solvers(prob)

solver = 
"lsqlin"

Equivalently, write the objective as a squared norm.

obj2 = norm(x-t)^2;
prob2 = optimproblem(Objective=obj2);
solver2 = solvers(prob2)

solver2 = 
"lsqlin"

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);
solver3 = solvers(prob3)

solver3 = 
"quadprog"

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);
solver4 = solvers(prob4)

solver4 = 
"lsqnonlin"

Equivalently, write the objective as a squared norm.

obj5 = norm(expr - ydata)^2; % norm squared
prob5 = optimproblem(Objective=obj5);
solver5 = solvers(prob5)

solver5 = 
"lsqnonlin"

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:

$$R_n=a_n+k_1\sum (k_2\sum (k_3\sum (\dots k_j{e}_n^2)))$$

• $$e_n$$ is any expression. If multidimensional, $$e_n$$ should be squared term-by-term using .^2.

• $$a_n$$ is a scalar numeric value.

• The $$k_j$$ are positive scalar numeric values.

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

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;
prob6 = optimproblem(Objective=R1 + R2 + R3);
solver6 = solvers(prob6)

solver6 = 
"lsqnonlin"