lsqnonneg (MATLAB Functions)

Linear least squares with nonnegativity constraints

Syntax

x = lsqnonneg(C,d)
x = lsqnonneg(C,d,x0)
x = lsqnonneg(C,d,x0,options)
[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(...)

Description

x = lsqnonneg(C,d) returns the vector x that minimizes norm(C*x-d) subject to x >= 0. C and d must be real.

x = lsqnonneg(C,d,x0) uses x0 as the starting point if all x0 >= 0; otherwise, the default is used. The default start point is the origin (the default is used when x0==[ ] or when only two input arguments are provided).

x = lsqnonneg(C,d,x0,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:

Display
Level of display. 'off' displays no output; 'final' displays just the final output; 'notify' (default) displays output only if the function does not converge.

TolX
Termination tolerance on x.

`Display`	Level of display. `'off'` displays no output; `'final'` displays just the final output; `'notify'` (default) displays output only if the function does not converge.
`TolX`	Termination tolerance on `x.`

[x,resnorm] = lsqnonneg(...) returns the value of the squared 2-norm of the residual: norm(C*x-d)^2.

[x,resnorm,residual] = lsqnonneg(...) returns the residual, d-C*x.

[x,resnorm,residual,exitflag] = lsqnonneg(...) returns a value exitflag that describes the exit condition of lsqnonneg:

>0
Indicates that the function converged to a solution x.

0
Indicates that the iteration count was exceeded. Increasing the tolerance (TolX parameter in options) may lead to a solution.

>0	Indicates that the function converged to a solution `x.`
0	Indicates that the iteration count was exceeded. Increasing the tolerance (`TolX` parameter in `options`) may lead to a solution.

[x,resnorm,residual,exitflag,output] = lsqnonneg(...) returns a structure output that contains information about the operation:

output.algorithm
The algorithm used

output.iterations
The number of iterations taken

`output.algorithm`	The algorithm used
`output.iterations`	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.

Examples

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.

Algorithm

lsqnonneg uses the algorithm described in [1]. 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.

See Also

The arithmetic operator \, optimset

References

[1] Lawson, C.L. and R.J. Hanson, Solving Least Squares Problems, Prentice-Hall, 1974, Chapter 23, p. 161.

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