Knuth Optimization states that if the DP equation satisfies $$OPT(i,j)=\min_{i<k\leq j}\{OPT(i,k-1)+OPT(k,j)\}+w(i,j)$$ and $w(i,j)$ satisfies for any integer $0<a<b<c<d$ $$w[a,c]+w[b,d]\leq w[a,d]+w[b,c]\\ w[b,c]\leq w[a,d]$$
Then we have $$OPT(a,c)+OPT(b,d)\leq OPT(a,d)+OPT(b,c)$$ and $$K(i,j-1)\leq K(i,j)\leq K(i+1,j)$$ where $$K(i,j)=\text{arg }\min_{i<k\leq j}\{OPT(i,k-1)+OPT(k,j)\}$$ How to prove the first inequality?
See here for a nice Explanation:
http://home.cse.ust.hk/~golin/COMP572/Notes/DP_speedup.pdf