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For the details of Karush-Kuhn-Tucker complementary slackness conditions, you can refer to:
[1] Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2009.

The derivation from Equation 6 to 7 is similar to the one in:
Chris Ding, Tao Li, Wei Peng, and Haesun Park. Orthogonal nonnegative matrix t-factorizations for clustering. In KDD’06, pages 126–135. ACM, 2006.