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The Beckmann potential integrates each link's delay function. Its gradient with respect to route flows is the corresponding vector of route delays, making Wardrop equilibrium a convex optimization problem.

Let $R_s$ be the finite, nonempty set of available routes for source-sink pair $s$, and write $x_r$ for a route flow. The link-route incidence matrix has $A_{jr}=1$ when route $r$ uses link $j$, and zero otherwise. The source-sink route incidence matrix has $H_{sr}=1$ when $r\in R_s$. Thus $y=Ax$ gives link throughputs and $f=Hx$ gives aggregate source-sink flows. A route's delay isA Wardrop equilibrium uses only routes of minimum delay for their source-sink pair:An infinitesimal user cannot improve its delay by changing route.

For fixed $f_s\ge 0$, the feasible route flows form a nonempty compact convex set: for each $s$, $x_r\ge 0$ and ${\sum }_{r\in R_s}{x}_r=f_s$. Define the Beckmann potentialIt is continuous, so the extreme value theorem gives a minimizer. Since every $D_j$ is strictly increasing, $V$ is a strictly convex function of $y$. Its route-flow gradient is $\partial V(Ax)/\partial x_r=d_r(x)$.

If an occupied route $r\in R_s$ has greater delay than another route $q\in R_s$, moving a small amount of flow from $r$ to $q$ gives a negative directional derivative, contradicting minimality. Conversely, at a Wardrop equilibrium, for any feasible $\tilde{x}$,The first-order condition for convex optimization therefore proves optimality. ConsequentlyIf two minimizers had different $y$, their midpoint would give a strictly smaller Beckmann potential. Thus the equilibrium link throughputs are unique. The route flows need not be unique, since $V(Ax)$ need not be a strictly convex function of $x$.

For a concrete example, take two successive stages of parallel links: $a,b$ in the first stage and $c,d$ in the second, with one source-sink pair and four routes $ac,ad,bc,bd$. Put $D_j(y)=y$ on every link and total flow $f=2$. All allocationshave the same link throughputs $(1,1,1,1)$ and every route has delay two. These distinct allocations are all Wardrop equilibria.

For elastic demand, use the standard physical convention that $D_j\ge 0$ and $B_s:[0,\infty )\to [0,\infty )$ is finite, continuous, and strictly decreasing. Set $M_s=B_s(0)$ and ${\ell }_s={\lim }_{\lambda \to \infty }{B}_s(\lambda )$, and define the inverse demand function $P_s=B_s^{-1}$ on $({\ell }_s,M_s]$. It is continuous and strictly decreasing, with $P_s(M_s)=0$ and $P_s(u)\to \infty$ as $u\downarrow {\ell }_s$. Choose any reference value $f_s^0\in ({\ell }_s,M_s)$ and setEach summand of $-G$ is a strictly convex function. The reference values only add a constant to the objective. In the usual case ${\ell }_s=0$, one may use a lower limit zero whenever the improper integral is finite; that integrability is not guaranteed merely by continuity of $B_s$.

Minimize $\Psi$ over the compact feasible set $x\ge 0$, ${\ell }_s\le (Hx{)}_s\le M_s$, interpreting the objective at $f_s={\ell }_s$ by its one-sided limit, which can be $+\infty$. This extension is sequentially lower semicontinuous, and an interior-demand feasible point has finite value, so a minimizer exists. It cannot have $f_s={\ell }_s$: adding a small amount on any route for $s$ has bounded marginal link cost, whereas its marginal demand benefit $P_s(f_s)$ tends to infinity. Thus $f_s>{\ell }_s$.

For ${\ell }_s<f_s<M_s$, the route-flow partial derivative is $d_r-P_s(f_s)$. The first-order conditions for convex optimization giveIf $f_s=M_s$, reducing any occupied route cannot improve the objective only if that route has zero delay; nonnegative link delays then give ${\lambda }_s=0=P_s(M_s)$. Therefore the conditions in every case areConversely these conditions give the nonnegative directional derivative for every feasible competitor and hence minimize the convex function $\Psi$. Allowing $f$ explicitly yields the stated form with $Hx=f$ and $Ax=y$, with inverse-demand endpoint conventions understood. In particular, $G$ is defined on its demand range, rather than evaluating $B_s^{-1}$ outside that range.

Strict convexity in the pair $(y,f)$ proves that the equilibrium aggregate source-sink flows and link throughputs are unique. Individual route flows may still be nonunique. Nonnegative, finite demand, nonnegative delays, finite route sets, and a usable route for each source-sink pair are the implicit modeling assumptions needed for the existence assertions.