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lsqnonneg

Solve nonnegative least-squares constraints problem

Equation

Solves nonnegative least-squares curve fitting problems of the form

Syntax

x = lsqnonneg(C,d)
x = lsqnonneg(C,d,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,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.

[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.

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

 algorithm 'active-set' iterations The number of iterations taken message Exit message

[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.

expand all

Algorithms

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.

References

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