We study standard first-passage percolation via related optimization 
problems that restrict path length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old 
observations about the convergence of geodesic length due to Hammersley, 
Smythe and Wierman, and Kesten. We study the regularity of the time 
constant as a function of the shift of weights.  For unbounded weights, 
this function is strictly concave and in case of two or more atoms it 
has a dense set of singularities. For any weight distribution with an 
atom at the origin there is a singularity at zero, generalizing a result of Steele and Zhang for Bernoulli FPP. The regularity results are proved by 
the van den Berg-Kesten modification argument.  This is joint work with
Arjun Krishnan and Timo Seppalainen