A polynomial local optimality condition for the concave piecewise linear network flow problem

This paper studies the local optimality condition for the widely applied concave piecewise linear network flow problem (CPLNFP). Traditionally, for CPLNFP the complexity of checking the local optimality condition is exponentially related to the number of active arcs (i.e., arcs in which the flow is at the breakpoints). When the scale of CPLNFP is large, even local optimality is unverifiable and the corresponding local algorithms are inefficient. To overcome this shortcoming, a new local optimality condition is given. This new condition is based on the concavity and piecewise linearity of the cost function and makes full use of the network structure. It is proven that the complexity of the new condition is polynomial. Therefore, using the new condition to verify the local optimality is far superior to using the traditional condition, especially when there are many active arcs.