Monotone Causality in Opportunistically Stochastic Shortest Path Problems

TL;DR

This work defines a class of Opportunistically Stochastic Shortest Path (OSSP) problems and deriving sufficient conditions for applicability of noniterative label-setting methods, and uses OSSPs to derive causality conditions for semi-Lagrangian discretizations of anisotropic Hamilton-Jacobi equations.

Abstract

When traveling through a graph with an accessible deterministic path to a target, is it ever preferable to resort to stochastic node-to-node transitions instead? And if so, what are the conditions guaranteeing that such a stochastic optimal routing policy can be computed efficiently? We aim to answer these questions here by defining a class of Opportunistically Stochastic Shortest Path (OSSP) problems and deriving sufficient conditions for applicability of non-iterative label-setting methods. The usefulness of this framework is demonstrated in two very different contexts: numerical analysis and autonomous vehicle routing. We use OSSPs to derive causality conditions for semi-Lagrangian discretizations of anisotropic Hamilton-Jacobi equations. We also use a Dijkstra-like method to solve OSSPs optimizing the timing and urgency of lane change maneuvers for an autonomous vehicle navigating road networks with a heterogeneous traffic load.

Figures (13)

• Figure 1: Transition digraphs for two simple SSP examples. The target node is marked by $\bm{t}$, and the possibility of a transition from node $\bm{x}_i$ to a successor node $\bm{x}_j$ is indicated by a dashed arrow.
• Figure 1: The geometric interpretation of condition \ref{['eq:original_MC_cond']} for $m = 2$ with $\delta = 0$ (panel (a)) and $\delta = 0.3$ (panel (b)) for several examples of transition cost functions. The solid purple and solid gold curves are smooth and convex. The solid green curve is smooth and nonconvex, and the green-dashed curve is its convexified version. In all cases, $K(p)$ is monotone ($\delta$-)causal provided that the curve stays entirely above the two restriction lines on the interval $[0, 1]$. In (a), the purple curve violates the condition \ref{['eq:original_MC_cond']} for $r = 1$ even with $\delta = 0$; so, it cannot be monotone causal. While the other two curves satisfy \ref{['eq:original_MC_cond']} with $\delta = 0$, only the nonconvex curve (along with its convexified version) is monotone $\delta$-causal for $\delta \leq 0.3$. The smooth gold convex curve can only be monotone causal (with $\delta = 0$), as the restriction lines imposed by \ref{['eq:original_MC_cond']} coincide with its tangent lines at $p = 0$ and $p = 1$. \newlabelfig:mc_geometric_illustration0
• Figure 1: Two computational stencils based on a uniform Cartesian grid in $\mathbb{R}^2$. The value function is computed at $\bm{x}$, and $\tilde{\bm{x}}_{\bm{\xi}}$ is the new position after traveling in a straight line along the chosen direction of motion $\bm{a}_{\bm{\xi}}$ with speed $f(\bm{x}, \bm{a}_{\bm{\xi}})$. In panel (a), the modes are essentially quadrants: $\mathcal{S}(\bm{x}) = \{ (\bm{x}_1, \bm{x}_3), (\bm{x}_3, \bm{x}_5), (\bm{x}_5, \bm{x}_7), (\bm{x}_7, \bm{x}_1) \}.$ In panel (b), each quadrant is split into two modes: $\mathcal{S}(\bm{x}) = \{ (\bm{x}_1, \bm{x}_2), (\bm{x}_2, \bm{x}_3), (\bm{x}_3, \bm{x}_4), (\bm{x}_4, \bm{x}_5), (\bm{x}_5, \bm{x}_6), (\bm{x}_6, \bm{x}_7), (\bm{x}_7, \bm{x}_8), (\bm{x}_8, \bm{x}_1) \}.$ \newlabelfig:4pt_and_8pt_stencils0
• Figure 1: Panel (a): Example lane-level road network representation of a three-lane highway. Lanes are discretized into cells of length $D$ meters, and each node marks the center of a cell. The vehicle travels from the starting point $\bm{s}$ to the destination $\bm{t}$ via a series of planned LSMs. Panel (b): Actions available at node $\bm{x}$ in mode $s_1 = \{\bm{x}_j, \bm{x}_k\} \in \mathcal{S}(\bm{x})$. The vehicle may continue driving in the current lane and directly transition to $\bm{x}_j$ (solid purple arrow), forcefully switch lanes and directly transition to $\bm{x}_k$ (solid red arrow), or attempt a tentative lane change (dashed blue arrow). The other mode available at $\bm{x}$ is $s_2 = \{\bm{x}_j, \bm{x}_i\}$, encoding a possible switch to another lane. \newlabelfig:lane_level_network0
• Figure 2: Example of deterministic (panel (a)) and probabilistic (panel (b)) pruning of the action set when $m = 2$. Depicted action choices at the node $\bm{x}_1$ in Figure \ref{['fig:ssp_examples']}(b), with transition probabilities $\mathbb{P}(\bm{x}_1 \rightarrow \bm{x}_5 \mid \bm{a}) = p(\bm{a})$ and $\mathbb{P}(\bm{x}_1 \rightarrow \bm{x}_2 \mid \bm{a}) = 1-p(\bm{a}).$ The specific cost function $K(p(\bm{a}))$ is chosen for the sake of illustration only. Many application-motivated examples of cost functions are considered in §\ref{['section:ossp_and_ahj']} and §\ref{['section:lane_change_formulation']}. Panel (a): The transition probabilities associated with each action in $A_1 = [\underline{\bm{a}}, \bar{\bm{a}}] \cup \{\bm{a}_1, \ldots, \bm{a}_4\}$ are indicated in green on the $p$-axis. During the deterministic pruning process, all actions $\bm{a} \in A_1$ that are transition-equivalent to another action $\tilde{\bm{a}} \in A_1$ with $K(p(\bm{a})) \ge K(p(\tilde{\bm{a}}))$ are removed along with the corresponding portion of the transition cost curve (crossed in red). Panel (b): Following the deterministic pruning, we obtain $\check{K}(p)$ by taking the lower convex envelope of $K$ (solid and dashed purple curve), and the resulting $A^{co}_1$ is indicated in orange above the $p$-axis. Replaceable actions $\bm{a}'$ and $\bm{a}_1$ are also removed at this stage. The transition probabilities associated with the remaining useful pure actions $A_1^u$ are indicated on the $p$-axis in solid purple, and their corresponding transition cost values are also marked in solid purple along $\check{K}$. \newlabelfig:cost_pruning0

Theorems & Definitions (30)

• Definition 1: Transition-equivalent Relaxed Actions
• Definition 2: Dominated, Replaceable, and Useful Actions
• proof 1
• Definition 3: OSSP
• Definition 4
• Theorem 1: Sufficient Monotone $\delta$-Causality Condition in OSSPs
• proof 2
• Theorem 2: Simplified Monotone $\delta$-Causality Condition in OSSPs
• proof 3
• Remark 1