Your sentence is grammatically unexceptionable, and conforms to the prevailing second-rate academic taste for superfluous repetition and subordination.
Why do you have three distinct terms signifying sequence—subsequent, after, then? None of them is needed: the sequence is evident in the bare conjunction. Me, I like things simpler:
We review the existing approaches, define our own approach, and offer an algorithm for implementing it.
If your 'sections' are numbered you can make things even clearer for readers:
Section 2 reviews existing approaches, Section 3 defines our own approach, and Section 4 offers an algorithm for implementing it.