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GPU-Accelerated SPOCK for Scenario-Based Risk-Averse Optimal Control Problems

Ruairi Moran, Pantelis Sopasakis

TL;DR

This work delivers a GPU-accelerated SPOCK solver for scenario-based risk-averse optimal control problems by marrying the Chambolle-Pock proximal framework with Anderson acceleration in a SuperMann setting. It reformulates multistage RAOCPs on scenario trees into a conic program using epigraphical relaxation, enabling SOC constraints and efficient parallelization on GPUs. The approach achieves competitive solve times and markedly lower memory usage compared to state-of-the-art interior-point solvers, and its performance scales with tree width and problem size. The results indicate strong potential for real-time control in high-dimensional, uncertain environments, supported by detailed parallelization strategies, preconditioning, and practical case studies.

Abstract

This paper presents a GPU-accelerated implementation of the SPOCK algorithm, a proximal method designed for solving scenario-based risk-averse optimal control problems. The proposed implementation leverages the massive parallelization of the SPOCK algorithm, and benchmarking against state-of-the-art interior-point solvers demonstrates GPU-accelerated SPOCK's competitive execution time and memory footprint for large-scale problems. We further investigate the effect of the scenario tree structure on parallelizability, and so on solve time.

GPU-Accelerated SPOCK for Scenario-Based Risk-Averse Optimal Control Problems

TL;DR

This work delivers a GPU-accelerated SPOCK solver for scenario-based risk-averse optimal control problems by marrying the Chambolle-Pock proximal framework with Anderson acceleration in a SuperMann setting. It reformulates multistage RAOCPs on scenario trees into a conic program using epigraphical relaxation, enabling SOC constraints and efficient parallelization on GPUs. The approach achieves competitive solve times and markedly lower memory usage compared to state-of-the-art interior-point solvers, and its performance scales with tree width and problem size. The results indicate strong potential for real-time control in high-dimensional, uncertain environments, supported by detailed parallelization strategies, preconditioning, and practical case studies.

Abstract

This paper presents a GPU-accelerated implementation of the SPOCK algorithm, a proximal method designed for solving scenario-based risk-averse optimal control problems. The proposed implementation leverages the massive parallelization of the SPOCK algorithm, and benchmarking against state-of-the-art interior-point solvers demonstrates GPU-accelerated SPOCK's competitive execution time and memory footprint for large-scale problems. We further investigate the effect of the scenario tree structure on parallelizability, and so on solve time.
Paper Structure (41 sections, 2 theorems, 45 equations, 9 figures, 4 algorithms)

This paper contains 41 sections, 2 theorems, 45 equations, 9 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem statement
  4. Dynamical system
  5. Scenario trees
  6. Dynamics.
  7. Constraints.
  8. Quadratic-plus-linear costs.
  9. Risk measures
  10. Average-value-at-risk.
  11. Conic representation.
  12. Conditional risk mappings
  13. Original problem
  14. Reformulation
  15. The Chambolle-Pock method
  16. ...and 26 more sections

Key Result

Proposition 1

Let $Q\in\mathbb{S}^n_{+}$ with $\operatorname{\mathbf{rank}} Q=p \leq n$, $q\in{\rm I\space R}^n$ and define the function $\ell(z) = z^\intercal Q z + q^\intercal z$. Let $S\in{\rm I\space R}^{n\times p}$ be such that $(\ker Q)^{\perp} = \operatorname{range} S$, so every $z\in{\rm I\space R}^n$ can where $\mathcal{G}_{S, Q, q}$ is the linear map

Figures (9)

  • Figure 1: Discrete problem described by a scenario tree. Each node at stage $t+1$ is a possible realization of an event $w_t$ at stage $t$. The probability of each event is $\pi^i$. The dynamics $h^{i}$ and stage costs $\ell^{i}$ are a function of the ancestor state and input, and the event $w_t$ at the ancestor stage $t$. The terminal costs $\ell_{N}$ are a function of the ancestor state and the event $w_t$ at the ancestor stage $t$. The risk mapping $\rho$ is a many-to-one operator from children to their ancestor. Inset (1a) is a space efficient representation of the tree. The root node at stage $0$ is at the center and each consecutive stage is the next concentric circle outwards. This representation will be used later for depicting the branching structure of trees.
  • Figure 2: The value-at-risk at level $\gamma$ of a continuously distributed random cost $Z$, denoted $\operatorname{V@R}_{\gamma}[Z]$, is the lowest cost $x$ such that $\mathrm{P}[Z\geq x]=\gamma$. The average-value at risk at level $\gamma$, $\operatorname{AV@R}_{\gamma}[Z]$, is the expectation of the tail of $Z$ that lies above $\operatorname{V@R}_\gamma[Z]$rockafellar2000optimization (shaded area).
  • Figure 3: Fork-join chain of streams of main parallelized processes in a CP iteration.
  • Figure 4: Cost vs Stage plot for an instance of the problem in bodard2023spock with a horizon of 10.
  • Figure 5: Speedup factor of GPU-accelerated SPOCK compared to serial SPOCK for the problem in bodard2023spock, where $n_x$ is the number of states and also the number of inputs, i.e., $n_u = n_x$.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Definition 3: Approximate optimality