= Relative-entropy Lyapunov function for resource prices
{title2=$L(\mu)$}
For a complementary-slackness optimum $\mu^*$, define
$$
L(\mu)=\sum_j\kappa_j^{-1}[\mu_j-\mu_j^*-\mu_j^*\log(\mu_j/\mu_j^*)],
$$
interpreting 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 $g=Ax-C$,
$$
\dot L=-\sum_r\frac{w_r(p_r-p_r^*)^2}{p_rp_r^*}+\mu^Tg(\mu^*)\leq0.
$$
Its sublevel bounds keep all route prices away from zero and permit the <LaSalle invariance principle>, including optima with some zero individual prices.
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