Writing Vector and Matrix Objective Functions
What Are Vector and Matrix Objective Functions?
Some solvers, such as
lsqcurvefit, have objective functions that are vectors or
matrices. The main difference in usage between these types of objective functions
and scalar objective
functions is how you write their derivatives. The first-order partial
derivatives of a vector-valued or matrix-valued function is called a Jacobian; the
first-order partial derivatives of a scalar function is called a gradient.
For information on complex-valued objective functions, see Complex Numbers in Optimization Toolbox Solvers.
Jacobians of Vector Functions
If x is a vector of independent variables and F(x) is a vector function, the Jacobian J(x) is
If F has m components and x has k components, J is an m-by-k matrix.
For example, if
then J(x) is
The function file associated with this example is:
function [F jacF] = vectorObjective(x) F = [x(1)^2 + x(2)*x(3); sin(x(1) + 2*x(2) - 3*x(3))]; if nargout > 1 % need Jacobian jacF = [2*x(1),x(3),x(2); cos(x(1)+2*x(2)-3*x(3)),2*cos(x(1)+2*x(2)-3*x(3)), ... -3*cos(x(1)+2*x(2)-3*x(3))]; end
To indicate to the solver that your objective function includes a Jacobian, set
SpecifyObjectiveGradient option to
options = optimoptions('lsqnonlin','SpecifyObjectiveGradient',true);
Jacobians of Matrix Functions
To define the Jacobian of a matrix F(x), change the matrix to a vector, column by column. For example, rewrite the matrix
as a vector f
The Jacobian of F is defined in terms of the Jacobian of f,
If F is an m-by-n matrix, and x is a k-vector, the Jacobian is an mn-by-k matrix.
For example, if
then the Jacobian of F is
Jacobians with Matrix-Valued Independent Variables
If x is a matrix, define the Jacobian of F(x) by changing the matrix x to a vector, column by column. For example, if
then the gradient is defined in terms of the vector
and f having the vector form of F, the Jacobian of F(X) is defined as the Jacobian of f(x):
So, for example,
If F is an m-by-n matrix and x is a j-by-k matrix, then the Jacobian is an mn-by-jk matrix.