Dynamic User Optimum (DUO) in Traffic Models

• Dynamic User Optimum (DUO) is defined as a dynamic equilibrium where all chosen route–departure-time pairs incur the minimal generalized travel cost.
• DUO employs various formulations such as variational inequalities, complementarity systems, and fixed-point methods to express user equilibrium rigorously.
• Practical implementations highlight computational challenges in large-scale dynamic traffic assignments and showcase the impact of behavioral assumptions on network performance.

Dynamic User Optimum (DUO) denotes the dynamic analogue of Wardrop’s user equilibrium or user optimal principle. In the continuous-time simultaneous route-and-departure-time formulations used in dynamic traffic assignment, DUO is synonymous with dynamic user equilibrium (DUE): for each origin–destination pair, all used path–departure-time choices have equal and minimal generalized cost, and no traveler can reduce cost by unilaterally changing route or departure time (Han et al., 2018, Han et al., 2014). In more local dynamic loading models, the same term is also used for an “instantaneous” rule in which travelers select currently shortest paths from their current node on the basis of instantaneous predicted travel times, a notion that is simpler than a full Friesz-type DUE but can be embedded directly in a differentiable Link Transmission Model (LTM) (Seo, 13 Apr 2026).

1. Definition and generalized travel cost

In the classical simultaneous route-and-departure-time setting, the basic decision variable is the path departure rate $h_p(\cdot)$, a nonnegative function on a commuting period $[t_0,t_f]$. Collecting all path departure rates gives

$$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$

The feasible set is determined by fixed origin–destination demand volumes $Q_{ij}>0$: $$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$ Generalized travel cost is represented by an effective delay operator $\Psi$. In the formulation of Han, Szeto, and Friesz, the effective path delay is

$$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$

where $D_p(t,h)$ is path travel time, $T_A$ is the desired arrival time, and $f(\cdot)$ is an arrival penalty function. The minimum effective cost for an origin–destination pair is defined through an essential infimum: $[t_0,t_f]$0 The SRDT DUE, or equivalently DUO, is then

$[t_0,t_f]$1

This definition encodes perfect rationality: any used path–departure-time pair must attain the minimal generalized cost for its origin–destination pair (Han et al., 2014, Han et al., 2018).

An equivalent many-to-one formulation in destination-arrival-time coordinates expresses the same behavioral rule through link and origin variables. In that representation, a DUE state is “a state in which each user chooses a departure time and route such that their experienced travel cost is minimal, and no alternative route or departure time provides a lower cost, given the congestion caused by all other users.” The generalized cost of a user arriving at the destination at time $[t_0,t_f]$2 on route $[t_0,t_f]$3 is

$[t_0,t_f]$4

with free-flow times $[t_0,t_f]$5, queueing delays $[t_0,t_f]$6, schedule delay cost $[t_0,t_f]$7, and value of time $[t_0,t_f]$8 (Sakai et al., 10 Aug 2025).

2. Variational, complementarity, and fixed-point formulations

A central feature of DUO research is the coexistence of several mathematically equivalent formulations. In the Hilbert-space setting, the DUE condition is equivalent to the variational inequality

$[t_0,t_f]$9

or, written explicitly,

$$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$0

This $$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$1 form is the standard analytical expression of dynamic user optimality in the continuous-time SRDT literature (Han et al., 2014, Han et al., 2018).

The same condition can be restated as a nonlinear complementarity system: $$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$2 together with complementarity expressions for the demand constraints. A differential variational inequality version introduces cumulative departures $$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$3 and writes the feasible set as a two-point boundary-value system,

$$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$4

thereby linking DUO to optimal-control-style state equations (Han et al., 2018).

A computationally convenient form is the fixed-point representation

$$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$5

for boundedly rational models, or

$$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$6

for classical DUE. Here $$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$7 or $$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$8 is a projection onto the feasible set. This formulation underlies projected fixed-point algorithms in both the exact-rationality and bounded-rationality settings (Han et al., 2014, Han et al., 2018).

In the Lagrangian-like coordinate system used for route-and-departure-time DUE assignment, DUO is written as a mixed linear complementarity problem in variables $$h(\cdot)=\{h_p(\cdot):p\in\mathcal{P}\}\in \big(L_+^2[t_0,t_f]\big)^{|\mathcal{P}|}.$$9. Route choice is expressed by

$Q_{ij}>0$0

and departure-time choice by

$Q_{ij}>0$1

These complementarity relations enforce that used links lie on minimum-cost routes and used arrival times attain the minimum generalized trip cost $Q_{ij}>0$2 for each origin (Sakai et al., 10 Aug 2025).

3. Behavioral variants: perfect rationality, bounded rationality, and instantaneous DUO

The perfect-rationality DUO condition can be relaxed in two distinct ways in the cited literature. The first is bounded rationality. In the boundedly rational DUE, travelers accept any used path–departure-time choice whose experienced cost lies within an indifference band: $Q_{ij}>0$3 The variable-tolerance extension replaces $Q_{ij}>0$4 by path- and flow-dependent tolerances $Q_{ij}>0$5, yielding

$Q_{ij}>0$6

Han, Szeto, and Friesz formulate these models as $Q_{ij}>0$7 and $Q_{ij}>0$8, prove an existence result for VT-BR-DUE under assumptions weaker than those required for mere DUE models, and show that the solution set is non-empty, compact, non-convex in general, and may have multiple connected components (Han et al., 2014).

The second relaxation appears in the differentiable LTM simulator. There, DUO is not a full dynamic user equilibrium with experienced travel times and variational inequality constraints, but an “instantaneous” dynamic user optimum aligned with Ran et al. (1993) and Kuwahara and Akamatsu (2001). At each route choice update time, every traveler chooses the currently shortest path in terms of instantaneous predicted travel time from the current node to the destination. Route choice is implemented locally at nodes through per-destination shortest-path trees $Q_{ij}>0$9, per-destination weights $$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$0, and diverge ratios

$$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$1

Per-destination outflows are then propagated through a FIFO rule,

$$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$2

This formulation is myopic or reactive: travelers do not anticipate future changes in congestion, but they do have perfect perception of current link travel times as implied by the LTM state (Seo, 13 Apr 2026).

The same paper introduces a logit extension,

$$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$3

where $$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$4 is a logit scale parameter and $$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$5 is the shortest-path cost from link $$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$6 to destination $$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$7. As $$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$8, the probabilities converge to deterministic DUO; for finite $$\Lambda \doteq \left\{h(\cdot)\ge 0: \sum_{p\in\mathcal{P}_{ij}}\int_{t_0}^{t_f} h_p(t)\,dt = Q_{ij}\quad\forall(i,j)\in\mathcal{W}\right\}.$$9, route choice is smooth in costs and provides non-zero gradients for optimization (Seo, 13 Apr 2026).

4. Dynamic network loading and the physical realization of DUO

DUO is not defined solely by behavioral conditions; it also depends on the dynamic network loading model that maps departures into experienced costs. In the LWR-based formulation, link density $\Psi$0 satisfies

$\Psi$1

with a triangular fundamental diagram

$\Psi$2

and demand–supply boundary conditions at junctions. The paper reformulates the resulting dynamic network loading problem as a system of differential algebraic equations based on cumulative counts, link entry and exit time functions, path fractions, and a junction model $\Psi$3. Path travel time is then computed through a composition of link exit-time mappings,

$\Psi$4

This DAE-based DNL captures the formation, propagation and dissipation of physical queues as well as vehicle spillback on networks (Han et al., 2018).

The differentiable LTM uses a different state representation. Each link $\Psi$5 is described by cumulative entries $\Psi$6 and cumulative exits $\Psi$7, with demand and supply written as $\Psi$8-based expressions and cumulative counts updated by

$\Psi$9

Instantaneous link travel times are derived either from an average-density approximation or from a segmented within-link approximation based on Newell’s formula, and shortest-path trees are recomputed every $$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$0 timesteps (Seo, 13 Apr 2026).

A third state representation is the Lagrangian-like destination-arrival-time coordinate used in the queue-replacement approach. There the variables $$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$1, $$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$2, $$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$3, and $$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$4 are indexed by destination arrival time rather than by the local Eulerian time on a link. This choice is justified by the property that, in DUE, users with the same destination arrival time follow the same node-passing times along their chosen shortest-cost paths (Sakai et al., 10 Aug 2025). These formulations differ in state coordinates and equilibrium encoding, but each supplies the network-performance operator needed to state DUO precisely.

5. Algorithms, exactness, and large-scale computation

Classical DUO computation in the SRDT setting often relies on fixed-point iteration. The MATLAB implementation of large-scale DUE uses the update

$$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$5

which requires one full DNL evaluation per iteration and a projection step that solves, for each origin–destination pair, a scalar equation in a dual variable $$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$6. The package is demonstrated on Nguyen, Sioux Falls, Anaheim, and Chicago Sketch; for Chicago Sketch, the reported configuration has 2,950 links, 933 nodes, 387 zones, 86,179 OD pairs, and 250,000 paths, with 69 iterations and a total time of about 4.8 hours (Han et al., 2018).

For BR-DUE and VT-BR-DUE, three algorithmic families are developed: a fixed-point method, a self-adaptive projection method, and a proximal point method. The fixed-point method is fastest per iteration because it needs one DNL per iteration; the self-adaptive projection method and PPM require multiple DNL evaluations per iteration but have weaker theoretical monotonicity requirements. Numerical studies on a 7-arc, 19-arc, and Sioux Falls network show that all three methods converge empirically, and larger tolerances generally result in faster convergence for the fixed-point and proximal point methods (Han et al., 2014).

A different computational direction is the generalized queue replacement principle (GQRP). In that framework, the original DUE with route and departure time choice is decomposed into two linear programs: a cost LP that determines a candidate cost pattern $$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$7, and a flow LP that checks whether those costs are dynamically consistent with the original queueing dynamics. The GQRP holds if and only if the flow LP is feasible and has optimal value zero; when that condition is met, the resulting pattern is an exact DUE solution. Numerical examples include Braess, Nguyen, Sioux Falls, and Eastern Massachusetts networks (Sakai et al., 10 Aug 2025).

The differentiable simulator introduces yet another algorithmic regime. Because the LTM is composed of arithmetic, $$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$8, and logit functions acting on continuous cumulative counts, the simulator computes exact gradients via reverse-mode automatic differentiation in a single backward pass regardless of the parameter dimension. On the Chicago-Sketch dataset with around 2500 links and 1 million vehicles and 15 000 decision variables, the model solves a dynamic congestion toll optimization problem; one simulation run and gradient derivation takes about $$\Psi_p(t,h)=D_p(t,h) + f\big(t + D_p(t,h) - T_A\big),$$9 second, and 10,000 iterations take about 2 hours. In the detailed application, the network has 927 nodes, 2,557 links, 17,963 OD pairs, demand of about 1.17 million vehicles, and 15,320 toll variables; total travel time is reduced from 1,120,159 veh-h to 496,366 veh-h under logit-DUO with $D_p(t,h)$0 and route choice updates every 300 s (Seo, 13 Apr 2026).

6. DUO, system optimum, and recognized limitations

Several cited works analyze DUO relative to dynamic system optimum (DSO). In the corridor-network model, the DSO is formulated as a no-queue optimization problem with dynamic toll multipliers $D_p(t,h)$1, while the DUE permits queueing delays $D_p(t,h)$2. Under conditions on the schedule delay function, the paper proves a queue–toll equivalence: $D_p(t,h)$3 so that queuing delay at a bottleneck in a DUE solution is equal to an optimal toll that eliminates the queue in a DSO solution. Under those conditions, DUE and DSO equilibrium costs coincide, and imposing dynamic prices equal to queuing delays yields system-optimal flows without increasing any commuter’s generalized cost (Fu et al., 2021).

The queue-replacement approach reaches a closely related conclusion. Under an additional structural assumption and valid GQRP, the DSO cost solution coincides with the DUE cost solution, with optimal congestion tolls $D_p(t,h)$4 equal to equilibrium queueing delays $D_p(t,h)$5. The same framework also states its own validity domain explicitly: homogeneous users, many-to-one origin–destination pattern, point queues, and no spillback (Sakai et al., 10 Aug 2025).

The literature also identifies several limitations of current DUO models. The instantaneous DUO in the differentiable LTM is myopic and does not capture strategic departure time choice or anticipatory routing; under pure deterministic DUO, route-choice gradients are zero almost everywhere, which motivates logit-DUO; and the Bellman–Ford shortest-path computation is treated as non-differentiable, so gradients pass through costs and flows but not through the combinatorial structure of paths (Seo, 13 Apr 2026). In large-scale SRDT DUE, the effective delay operator $D_p(t,h)$6 is available only via numerical simulation, and continuity and generalized monotonicity properties needed for convergence proofs are not established, so convergence of fixed-point schemes is empirical rather than guaranteed (Han et al., 2018). In boundedly rational formulations, tolerances enlarge the equilibrium set substantially, producing extensive multiplicity of equilibria and convex local families of solutions around certain interior points (Han et al., 2014).

Taken together, these formulations show that DUO is not a single model class but a family of equilibrium concepts. At one end is classical SRDT DUE, expressed through variational inequalities, complementarity systems, and fixed points over path departure rates. At the other end are instantaneous or smoothed node-based route-choice rules designed for differentiable simulation and high-dimensional control. The common element is individual optimality under endogenous congestion; what changes across formulations is the time scale of choice, the state representation, the admissible behavioral deviation from minimum cost, and the computational mechanism used to enforce equilibrium (Han et al., 2018, Seo, 13 Apr 2026).