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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 = 

Equivalently, write the objective as a squared norm.

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

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 = 

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 = 

Equivalently, write the objective as a squared norm.

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

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:


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

See Also


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