Solve nonnegative least-squares constraints problem
x = lsqnonneg(C,d)
x = lsqnonneg(C,d,options)
x = lsqnonneg(problem)
[x,resnorm] = lsqnonneg(...)
[x,resnorm,residual] = lsqnonneg(...)
[x,resnorm,residual,exitflag] = lsqnonneg(...)
[x,resnorm,residual,exitflag,output] = lsqnonneg(...)
[x,resnorm,residual,exitflag,output,lambda] = lsqnonneg(...)
x = lsqnonneg(C,d,options) minimizes with the optimization parameters specified in the structure options. You can define these parameters using the optimset function. lsqnonneg uses these options structure fields:
Level of display. 'off' displays no output; 'final' displays just the final output; 'notify' (default) displays output only if the function does not converge.
Termination tolerance on x.
x = lsqnonneg(problem) finds the minimum for problem, where problem is a structure with the following fields:
|Options structure created using optimset|
Indicates that the function converged to a solution x.
Indicates that the iteration count was exceeded. Increasing the tolerance (TolX parameter in options) may lead to a solution.
The number of iterations taken
[x,resnorm,residual,exitflag,output,lambda] = lsqnonneg(...) returns the dual vector (Lagrange multipliers) lambda, where lambda(i)<=0 when x(i) is (approximately) 0, and lambda(i) is (approximately) 0 when x(i)>0.
Compare the unconstrained least squares solution to the lsqnonneg solution for a 4-by-2 problem:
C = [ 0.0372 0.2869 0.6861 0.7071 0.6233 0.6245 0.6344 0.6170]; d = [ 0.8587 0.1781 0.0747 0.8405]; [C\d lsqnonneg(C,d)] = -2.5627 0 3.1108 0.6929 [norm(C*(C\d)-d) norm(C*lsqnonneg(C,d)-d)] = 0.6674 0.9118
The solution from lsqnonneg does not fit as well (has a larger residual), as the least squares solution. However, the nonnegative least squares solution has no negative components.
lsqnonneg uses the algorithm described in . The algorithm starts with a set of possible basis vectors and computes the associated dual vector lambda. It then selects the basis vector corresponding to the maximum value in lambda in order to swap out of the basis in exchange for another possible candidate. This continues until lambda <= 0.
 Lawson, C.L. and R.J. Hanson, Solving Least Squares Problems, Prentice-Hall, 1974, Chapter 23, p. 161.