We will test the APBB algorithms on various examples using randomly generated data and the real life ORL database  and CBCL database  and compare them with the Lee-Seung, Lin, and HALS/RRI algorithms.
We then incorporate them into an ANLS framework and obtain four APBB algorithms for nsNMF.
As mentioned in the introduction, the APBB algorithms alternately solve the subproblems for nsNMF using a projected Barzilai-Borwein method.
In this paper, we will use the second set, that is, conditions (2.19) and (2.20) to terminate our APBB algorithms.
As mentioned earlier, each of our APBB algorithms for the nsNMF problem 1.2 is based on a PBB method for solving Subproblems (1.14) and (1.15), alternately.
We are now ready to present the APBB algorithms for nsNMF which are based on the four PBBNLS methods respectively.