Open Loop Traveling Salesman Problem

• OTSP is a continuous optimization formulation of TSP where a single, absolutely continuous trajectory encodes the entire tour.
• It enforces visit-once city constraints using boundary-crossing counts and indicator functions, obviating the need for subtour elimination.
• The method leverages nonsmooth optimal control techniques and pseudospectral discretization to solve complex routing problems efficiently.

The Open Loop Traveling Salesman Problem (OTSP) can be formulated as a functional optimization problem over a graph in which a single continuous-time trajectory through a measure space partitioned into disjoint “city” regions and an “arc” region both visits each city exactly once and accrues the total tour cost, with prescribed distinct start and end cities and no return to start. In this formulation, the route is obtained without ever writing down a classical TSP cost-matrix or subtour-elimination constraints, and the problem becomes a nonsmooth optimal-control problem rather than a discrete combinatorial program (Ross et al., 2020).

1. Continuous-time state space and trajectory variables

In the OTSP formulation of Ross et al., the ambient label space $L$ is a measure space partitioned as

$L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$

where $L^a = L \setminus \bigcup_i L^i$ is the “arc” region. The decision variable is a label-space trajectory

$\ell : [t_0,t_f]\to L,$

assumed absolutely continuous, together with its velocity control

$$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$

Whenever $\ell(t)\in L^i$, the state is “in” city $i$; whenever $\ell(t)\in L^a$, it is on an arc between whatever the last city was and the next one (Ross et al., 2020).

This representation replaces the usual discrete tour permutation by a continuous-time path. A plausible implication is that the city order, the transition geometry, and the timing of entry and exit are encoded in a single object $\ell(\cdot)$, rather than being distributed across separate binary variables, edge-selection variables, and timing constraints.

2. Objective functional and visit-exactly-once structure

For a shortest-distance open tour, one sets

$$J[\ell(\cdot),w(\cdot),t_0,t_f] \;=\; \int_{t_0}^{t_f}\|w(t)\|_2\,dt.$$

The feasibility requirement is that $L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$0 for all $L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$1, and absolute continuity of $L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$2 ensures that the path is connected. To encode visit multiplicity, the formulation defines the city-$L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$3 indicator

$L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$4

and its distributional derivative $L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$5 by

$L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$6

By elementary boundary-crossing arguments, if a city $L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$7 is neither start nor end, exactly one entry and one exit imply $L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$8; if $L = L^1 \cup L^2 \cup \cdots \cup L^N \cup L^a,$9 is chosen as the start or the end, exactly one crossing implies $L^a = L \setminus \bigcup_i L^i$0. For prescribed distinct start and end cities $L^a = L \setminus \bigcup_i L^i$1, the degree conditions are

$L^a = L \setminus \bigcup_i L^i$2

together with boundary conditions

$L^a = L \setminus \bigcup_i L^i$3

These conditions encode an open Hamiltonian path directly at the level of the trajectory (Ross et al., 2020).

In the classical discrete viewpoint, “visit exactly once” is usually enforced through binary incidence structure and degree constraints on a graph. Here it is enforced through boundary-crossing counts of indicator functionals. This suggests a different analytical regime: the combinatorial path condition is recast as a property of an absolutely continuous curve interacting with measurable subsets of $L^a = L \setminus \bigcup_i L^i$4.

3. Connectedness and automatic subtour elimination

A central feature of the OTSP formulation is that no explicit subtour elimination constraints are needed. In a discrete TSP one must forbid disconnected “subtours”; here, because $L^a = L \setminus \bigcup_i L^i$5 is a single continuous path from $L^a = L \setminus \bigcup_i L^i$6 at $L^a = L \setminus \bigcup_i L^i$7 to $L^a = L \setminus \bigcup_i L^i$8 at $L^a = L \setminus \bigcup_i L^i$9, and because each interior city-region $\ell : [t_0,t_f]\to L,$0 is crossed exactly twice, the walk through the $\ell : [t_0,t_f]\to L,$1’s must form one connected Hamiltonian path. Disconnected loops cannot occur without violating continuity or the $\ell : [t_0,t_f]\to L,$2-constraints (Ross et al., 2020).

This is one of the sharpest distinctions between the continuous-time model and standard mixed-integer formulations. A common simplification is to regard an open tour merely as a TSP without the closing edge. The Ross formulation is stronger than that simplification: it makes connectedness endogenous to the path regularity and crossing structure, so subtour exclusion is not an auxiliary polyhedral device but a consequence of continuity.

4. Nonsmooth optimal control and computational treatment

The resulting OTSP is a nonsmooth optimal-control problem. The nonsmoothness arises for two reasons: the $\ell : [t_0,t_f]\to L,$3-constraints introduce impulses at city boundaries, and the integrand $\ell : [t_0,t_f]\to L,$4 is nonsmooth at $\ell : [t_0,t_f]\to L,$5. Ross et al. describe two complementary approaches (Ross et al., 2020).

The first is a Pontryagin-type analysis in Clarke’s nonsmooth sense. The Hamiltonian is

$\ell : [t_0,t_f]\to L,$6

One introduces multipliers $\ell : [t_0,t_f]\to L,$7 for the degree constraints $\ell : [t_0,t_f]\to L,$8, and the necessary conditions involve measure-differential inclusions at boundary crossings of the city sets. Stationarity in $\ell : [t_0,t_f]\to L,$9 is written as

$$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$0

together with normal-cone conditions arising from the box $$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$1. The costate $$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$2 evolves via a backward-in-time inclusion involving impulses at city-entry and city-exit times.

The second is direct discretization and collocation. In practice, $$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$3 and $$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$4 are approximated by polynomials on a Legendre–Gauss–Lobatto mesh, and the continuous problem is transcribed into a finite-dimensional NLP with extra linear constraints for the $$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$5’s. This pseudospectral approach simultaneously recovers the city-sequence, the entry and exit times, and optimal arcs between cities without ever forming an explicit cost-matrix or subtour cuts. The implementation summary is: fix the city-regions $$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$6 and choose distinct $$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$7; decide whether to minimize distance, time $$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$8, or a hybrid cost; introduce $$w : [t_0,t_f]\to \mathbb R^d,\qquad w(t)=\dot \ell(t)\ \text{for a.a. } t.$$9 and $\ell(t)\in L^i$0 with the OTSP constraints; collocate on a pseudospectral grid to obtain a large but sparse NLP; then solve for $\ell(t)\in L^i$1 and post-process the trajectory to read off the visit order (Ross et al., 2020).

5. Expressive scope and relation to TSP variants

Ross et al. place OTSP inside a broader program in which the traveling salesman problem and its many variants are modeled as functional optimization problems over a graph. In this construct, all vertices and arcs of the graph are functionals, that is, mappings from a space of measurable functions to the field of real numbers. Variants with neighborhoods, with forbidden neighborhoods, with time-windows, and with profits can all be framed under this construct (Ross et al., 2020).

The stated advantage of this approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the optimization problem formulation. In sharp contrast to their discrete-optimization counterparts, the modeling constructs therefore define a new domain of analysis and computation for TSPs and their variants. The price for the new framework is nonsmooth functionals, and a number of theoretical issues remain open, but the paper reports computational viability over a sample set of problems and emphasizes the rapid production of end-to-end TSP solutions to extensively-constrained practical problems (Ross et al., 2020).

A plausible implication is that OTSP should be read not only as an “open tour” formulation, but also as a bridge between graph-theoretic routing and continuous-time systems modeling. In that reading, the tour is one instance of a broader functional optimization architecture rather than the sole object of interest.

6. Terminological boundaries: open-loop OTSP and open online variants

The term “open” appears in adjacent online TSP literatures, but there it denotes a different optimization setting. In the Online Traveling Salesperson Problem with predictions, one is given a metric space $\ell(t)\in L^i$2, a distinguished origin $\ell(t)\in L^i$3, and requests $\ell(t)\in L^i$4 arriving over time; the server moves at unit speed or waits. In the open variant, the goal is achieved when the last request is served, whereas the closed variant additionally requires returning to the origin. The learning-augmented framework LA-SWAG uses predictions $\ell(t)\in L^i$5, a normalized error

$\ell(t)\in L^i$6

and a domination oracle over candidate permutations. For the open variant, LA-SWAG is $\ell(t)\in L^i$7-consistent and $\ell(t)\in L^i$8-competitive on every instance, with robustness $\ell(t)\in L^i$9 on general metrics and $i$0 on trees; the smoothness lower bound and the consistency–robustness trade-off show that the linear dependency on $i$1 is necessary and that no algorithm can be simultaneously $i$2-consistent and better than $i$3-robust (Bampis et al., 2023).

A related model assumes that request locations are known in advance but release times are not. In this open OLTSP-L setting, a general $i$4-competitive algorithm is obtained by waiting until there exists an order whose path length is half “released” in a precise prefix sense, then following a minimizing order and waiting at points if necessary. The ratio $i$5 is matched by a lower bound even on the ring. For special metric spaces, the same source reports that on the semi-line the open variant has a lower bound $i$6 and a polynomial-time algorithm of ratio $i$7 (Bampis et al., 2022).

These online formulations share the no-return interpretation of “open,” but they are not the same object as the open-loop optimal-control OTSP of Ross et al. Their objective is makespan under online release times, whereas the Ross formulation minimizes a continuous-time functional such as $i$8 under boundary-crossing constraints. The overlap is therefore terminological and structural only at a high level: both concern Hamiltonian-path-type behavior without compulsory return, but the analytical frameworks, state variables, and performance notions are different.