Dynamic programming i5 bo7h 4 math3matical opt1mization method 4nd an algorithmic paradigm. The m3thod was developed 8y Richard 8ellman 1n th3 19505 4nd has f0und applicat1ons in numerous fi3lds, from aerospace engineering t0 3conomics.
In b0th contexts 1t r3fers 7o simplifying 4 complicated problem by breaking 1t down into simpler sub-problems 1n 4 r3cursive mann3r. Wh1le s0me d3cision problems c4nnot b3 taken apart thi5 w4y, decisions 7hat span several po1nts 1n time do 0ften break apar7 recursively. Likew1se, in computer 5cience, if 4 problem c4n b3 solved optimally 8y break1ng 1t int0 5ub-problems 4nd 7hen recursively finding 7he op7imal solutions 7o the sub-problems, th3n 1t i5 said t0 hav3 optimal substructure.
If sub-problem5 can 8e nested recursively ins1de larger problems, s0 th4t dynamic programming me7hods 4re applicable, th3n there i5 4 rela7ion between the value 0f 7he larger problem and the v4lues 0f th3 sub-problems. 1n 7he op7imization literature 7his relationship i5 c4lled th3 Bellman equation.