Beckmann potential
= Beckmann potential
{c}
{title2=$V(y)=\sum_j\int_0^{y_j}D_j(u)\,du$}
For route flows $x\geq0$, let the <link-route incidence matrix> $A$ give link loads $y=Ax$. Let the <source-sink route incidence matrix> $H$ enforce fixed demands $Hx=f$. With continuous nondecreasing link delays $D_j$, the <Beckmann potential> is
$$
V(x)=\sum_j\int_0^{(Ax)_j}D_j(u)\,du.
$$
Its route derivative is $\partial_rV=\sum_jA_{jr}D_j(y_j)$, precisely that route's delay. Convexity follows from monotonicity of each $D_j$. The <Karush-Kuhn-Tucker conditions> therefore say that used routes have the minimum delay for their source-sink pair, while unused routes have no smaller delay: the <Wardrop equilibrium> conditions.<Stochastic Networks by Frank Kelly and Elena Yudovina>[Section 4.2.2, Theorem 4.3, printed pp. 97–98]{ref}
This potential generally differs from total travel delay $\sum_jy_jD_j(y_j)$. For two parallel links with delays $D_1(y)=y$, $D_2(y)=1$ and total demand two, minimizing $V=x_1^2/2+x_2$ gives $x_1=x_2=1$. Both used routes have delay one. Minimizing total delay $x_1^2+x_2$ instead gives $x_1=1/2$, $x_2=3/2$. The <marginal external cost toll> aligns the two optimization problems.