*> Hi!*

> In a least squares fitting application, I'm trying to determine the parameters x_1, x_2

> In a least squares fitting application, I'm trying to determine the parameters x_1, x_2

*> and x_3 so that the function*

*> f(x_1, x_2, x_3) = \sum_{i=1}^N (a_i*x_1 + b_i*x_2 + c_i*x_3 - d_i)^2*

*> is minimized under the constrains x_1, x_2, x_3 >= 0 (LaTeX notation)*

> I was pointed to the keyword "nonnegative least squares" (NNLS), but wasn't able to

> I was pointed to the keyword "nonnegative least squares" (NNLS), but wasn't able to

*> find out much about this: Lawson, Hanson: "Solving Least Squares Problems", Ch. 23*

*> Briggs: "High Fidelity Deconvolution of Moderately Resolved Sources", Ch. 4.3*

> Can you recommend further literature or web resources on this topic, ideally with examples

> Can you recommend further literature or web resources on this topic, ideally with examples

*> on how to solve this type of problem?*

Any NLP solver can solve these type of problems quite efficiently. See e.g.

http://www.princeton.edu/~rvdb/ampl/nlmodels/nnls/index.html

A GAMS version can look like:

` 1 set`

2 i /1*1000/

3 j /1*300/

4 ;

5

6 parameters

7 b(i)

8 A(i,j)

9 ;

10

11 a(i,j)$(uniform(0,1) < 0.21) = 10*(uniform(0,1)-1);

12 parameter x0(j);

13 x0(j) = uniform(0,1);

14 b(i) = sum(j, a(i,j)*x0(j)) + 100*(uniform(0,1)-0.5)

15

16 positive variable x(j);

17 variable sum_sqs;

18

19 equation esum_sqs;

20

21 esum_sqs.. sum_sqs =e= sum(i, sqr[ b(i) - sum(j, A(i,j)*x[j]) ]);

22

23 model m /all/;

24 solve m minimizing sum_sqs using nlp;

25

26

27 *conopt : 0.702 seconds

28 *minos : 1.264

29 *snopt : 0.343

30 *knitro : 7.385

31 *ipopt : 1.088