Solve nonnegative least-squares constraint 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(problem) finds the minimum for problem, where problem is a structure described in Input Arguments.
Create the structure problem by exporting a problem from Optimization app, as described in Exporting Your Work.
Function Arguments contains general descriptions of arguments passed into lsqnonneg. This section provides function-specific details for options and problem:
Level of display:
Termination tolerance on x, a positive scalar. The default is 10*eps*norm(C,1)*length(C).
|Options structure created using optimset|
Function Arguments contains general descriptions of arguments returned by lsqnonneg. This section provides function-specific details for exitflag, lambda, and output:
Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated.
Function converged to a solution x.
Number of iterations exceeded options.MaxIter.
Vector containing the Lagrange multipliers: lambda(i)≤0 when x(i) is (approximately) 0, and lambda(i) is (approximately) 0 when x(i)>0.
Structure containing information about the optimization. The fields are
Number of iterations taken
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)] ans = -2.5627 0 3.1108 0.6929 [norm(C*(C\d)-d), norm(C*lsqnonneg(C,d)-d)] ans = 0.6674 0.9118
The solution from lsqnonneg does not fit as well as the least-squares solution. However, the nonnegative least-squares solution has no negative components.
The nonnegative least-squares problem is a subset of the constrained linear least-squares problem. Thus, when C has more rows than columns (i.e., the system is overdetermined),
[x,resnorm,residual,exitflag,output,lambda] = ... lsqnonneg(C,d)
is equivalent to
[m,n] = size(C); [x,resnorm,residual,exitflag,output,lambda_lsqlin] = ... lsqlin(C,d,-eye(n,n),zeros(n,1));
except that lambda = -lambda_lsqlin.ineqlin.
For problems greater than order 20, lsqlin might be faster than lsqnonneg; otherwise lsqnonneg is generally more efficient.
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 it 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, Chapter 23, p. 161, 1974.