Given a graph G: (V, E, w), initial vertex a, and goal vertex z, find the length of the shortest path between a and z. The shortest path is given by a labeling of each vertex v by L, L(v), satisfying the following constraints:

L(a) = 0
for each vertex v, L(v) is equal to L(u) + w(u, v) for some adjacent vertex u that minimizes the value.

In other words,

L(a) = 0
for each vertex v, L(v) = min { L(u) + w(u, v) : {u, v} in E }.

One way of solving this system of constraints for L(z) is via linear programming:

maximize L(z)
subject to
    L(a) = 0
  & for all {u, v} in E, L(u) <= L(v) + w(u, v)

Maximizing L(z) at the same time that each vertex's L-value is upper bounded has the same effect as using min. Dijkstra's algorithm essentially solves this particular form of linear program. Each iteration improves the estimate of lengths.