Multiplicative resource-price dynamics (${\dot{\mu }}_j={\kappa }_j{\mu }_j[(Ax{)}_j-C_j]$)

For route prices $p=A^T\mu$, logarithmic users demand $x_r=w_r/p_r$, while resource prices adjust asThis model raises prices under excess demand and lowers them under spare capacity. Positive capacities and nonempty routes are standard well-definedness assumptions. Positive initial prices approach the dual optimum; zero prices are invariant and can create boundary equilibria of multiplicative price dynamics that overload a resource.

For a complementary-slackness optimum ${\mu }^{\ast }$, defineinterpreting a zero-reference summand as ${\mu }_j/{\kappa }_j$. This is a nonnegative generalized reverse-relative-entropy expression for positive resource prices. Along multiplicative resource-price dynamics, with $p=A^T\mu$ and $g=Ax-C$,Its sublevel bounds keep all route prices away from zero and permit the LaSalle invariance principle, including optima with some zero individual prices.

For multiplicative resource-price dynamics with positive capacities, nonempty routes and strictly positive initial prices, full row rank of the link-route incidence matrix makesstrictly concave. There is a unique optimizing price vector, possibly on the boundary, and the relative-entropy Lyapunov function for resource prices proves convergence to it. Row-rank deficiency still gives unique optimal route prices and rates, but resource prices may depend on the initial state. Full column rank alone and unrestricted zero initial prices do not yield the same uniqueness theorem.