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Fully Distributed GNE Algorithms for Multi-Robot Placement without Consensus on Multipliers
Shao-An Yin, Mingyi Hong, Nicola Elia
TL;DR
Shared-constraint GNEPs are difficult to solve distributively because existing methods often exchange multipliers and focus on v-GNEs. The paper proposes a fully distributed primal–dual approach without multiplier exchange, proves convergence to GNEs, and validates discrete-time variants on multi-robot placement. This reduces communication while retaining v-GNE recovery through identical multiplier initialization.
Problem
Distributed GNEP methods for strongly monotone games commonly require multiplier consensus, limiting them to v-GNEs and increasing information exchange under shared constraints.
Method
The paper develops a fully distributed continuous-time primal–dual algorithm for shared linear equality constraints, together with discrete-time discretizations.
Results
Every continuous-time trajectory converges to a GNE, and the diminishing-step-size discrete-time scheme converges to a GNE under its additional assumptions.
Takeaways & Limitations
Eliminating multiplier exchange reduces communication overhead while allowing convergence to the broader GNE set; identical multiplier initialization still recovers v-GNEs.
Abstract
from arXiv · showhide
Recent machine learning research has increasingly focused on equilibrium analysis in non-cooperative games rather than solely on optimal solutions. Many such problems involve shared constraints and can be formulated as Generalized Nash Equilibrium Problems (GNEPs). For strongly monotone games, existing methods compute consensus-based variational GNEs (v-GNEs) by exchanging Lagrange multipliers. We propose a fully distributed continuous-time algorithm for shared linear equality constraints that converges without multiplier exchange and reaches any GNE, reducing communication overhead and improving privacy. Discrete-time schemes are also provided, and the method is validated on a multi-robot placement task.
I. INTRODUCTION
GNEPs capture non-cooperative decisions under shared constraints, but centralized and consensus-based distributed methods face scalability, communication, privacy, and equilibrium-selection limits. The paper proposes a fully distributed algorithm that avoids multiplier consensus and can reach general GNEs.
- GNEPs model agents whose feasible decisions depend on others, often because of shared resources or safety constraints.
- Centralized penalty approaches are difficult to distribute because their nonsmooth inner problems require high computational and communication costs.
- Existing distributed methods typically enforce multiplier consensus and therefore compute only variational GNEs.
- The proposed method targets shared linear equality constraints and converges to general GNEs without exchanging Lagrange multipliers.
- The method reduces information exchange to one-third per iteration compared with existing approaches for the considered problem class.
A. Notations
The paper establishes notation for Euclidean spaces, real-number domains, indexed sets, vectors, matrices, graphs, projections, and function sublevel sets.
- R^m denotes the m-dimensional Euclidean space, while R≥0 and R>0 denote nonnegative and positive reals.
- The index set [m] is defined as {1, 2, . . . , m}, and 1 and 0 denote all-ones and all-zeros vectors of appropriate dimensions.
- For vectors and matrices, superscript ⊤ denotes transpose, a⊤b denotes an inner product, and inequalities are component-wise.
- A function sublevel set V^-1(≤β) collects points x satisfying V(x) ≤ β for β > 0.
B. Generalized Nash Equilibrium Problems (GNEPs)
A GNEP lets each agent optimize a local cost subject to individual and shared constraints that may depend on other agents’ decisions. Under the stated convexity and constraint qualifications, GNEs are characterized by KKT conditions.
- Each agent’s cost depends on its own decision and potentially on other agents’ decisions, inducing a cost-function graph G_f.
- Agents also face individual constraints and coupled constraints whose feasible sets depend on other agents’ decisions, inducing a constraint graph G_g.
- The paper assumes globally shared linear equality constraints represented by A and c, although each agent observes only its local constraint component.
- A GNE is an equilibrium where no agent can lower its cost by unilaterally changing its decision within its feasible set.
- Continuous differentiability, convexity, local Lipschitz gradients, and LICQ support equivalence between GNEs and the stacked KKT conditions.
C. Variational GNE (v-GNE)
Variational GNEs arise from a variational inequality over the shared feasible set and use identical multipliers for shared constraints. Strong monotonicity gives a unique VI solution, while the proposed method instead targets the broader GNE set.
- The VI feasible set combines local feasible sets with global shared constraints, which are unknown in full to individual agents.
- Solving the shared-constraint VI yields a variational GNE, and every v-GNE belongs to the GNE solution set.
- Identical multipliers for shared constraints link VI solutions to v-GNEs, whereas general GNEs may use agent-specific multipliers.
- Strong monotonicity guarantees a unique VI solution and thereby ensures existence of a GNE, even when the game admits multiple equilibria.
- The proposed algorithm converges to KKT points and hence to any GNE without requiring multiplier consensus.
D. Projected Dynamical System (PDS)
The projected dynamical system projects a vector field onto a closed convex set, enabling stability analysis despite discontinuous dynamics through a generalized Lie derivative and invariance principles.
- D. Projected Dynamical System (PDS): A projected dynamical system combines a mapping with projection onto a closed convex feasible set.The projection selects the point in the set minimizing Euclidean distance to the input vector.
- D. Projected Dynamical System (PDS): Because projected dynamics are discontinuous, the paper defines the Lie derivative of a continuously differentiable function along them.This definition adapts derivative-based analysis to the projected setting.
- D. Projected Dynamical System (PDS): The generalized Lie-derivative construction enables standard stability analysis and application of invariance principles.
III. PROPOSED FULLY DISTRIBUTED ALGORITHM
The proposed algorithm uses two dependency graphs and agent-local primal–dual dynamics to solve shared-constraint GNEPs while exchanging only decision variables, without multiplier consensus.
- III. PROPOSED FULLY DISTRIBUTED ALGORITHM: The distributed setup uses a cost-function graph and a shared-constraint graph, with each agent knowing only its local functions.The graphs capture dependencies in costs and shared constraints, respectively.
- III. PROPOSED FULLY DISTRIBUTED ALGORITHM: The algorithm’s objective is to converge to a game solution while limiting information exchange and preserving data privacy.The setup allows both dependency graphs to be complete in the worst case.
- III. PROPOSED FULLY DISTRIBUTED ALGORITHM: Algorithm 1 defines continuous-time dynamics for each agent’s decisions and dual variables using local information and communicated decisions.The dynamics update decisions through the local Lagrangian gradient and update shared-constraint multipliers from constraint values.
- III. PROPOSED FULLY DISTRIBUTED ALGORITHM: Agents exchange only decision variables, whereas existing v-GNE methods also exchange Lagrangian multipliers and auxiliary variables.Decision variables are communicated over the cost-function and constraint graphs.
- III. PROPOSED FULLY DISTRIBUTED ALGORITHM: One-third communication cost is achieved when the two graphs coincide and each edge transmits a vector of dimension d.Consensus-based methods require an additional 2|E_g|d messages per fixed time for multipliers and auxiliary variables.
A. Convergence Analysis
The convergence analysis interprets the algorithm as a projected dynamical system, establishes trajectory properties through Lyapunov analysis, and proves convergence to GNEs, including v-GNE recovery under identical multiplier initialization.
- A. Convergence Analysis: Algorithm 1 implements a projected dynamical system over decisions, unrestricted equality multipliers, and nonnegative inequality multipliers.The projected set imposes nonnegativity only on the inequality-multiplier variables.
- A. Convergence Analysis: Reference-state deviations remain constant over time and parameterize the equilibrium selected by the initial multiplier configuration.The construction uses one reference state for each shared constraint component.
- A. Convergence Analysis: For any prescribed initial deviations, the algorithm has a unique equilibrium, and every equilibrium satisfies the GNEP’s KKT conditions.Agents generally cannot predict that equilibrium because they lack other agents’ initial multipliers and private functions and constraints.
- A. Convergence Analysis: The Lyapunov function has a nonpositive Lie derivative along all trajectories, supporting the convergence proof.The analysis combines this property with trajectory existence, uniqueness, continuity, and invariance arguments.
- A. Convergence Analysis: Every trajectory converges to a GNE, while identical multiplier initialization recovers the same v-GNE without multiplier exchange.The equilibrium set of Algorithm 1 is globally stable.
B. Discretization for Discrete-Time Algorithms
The paper develops discrete-time forward and forward–backward schemes for the continuous algorithm, with convergence guaranteed under additional assumptions and appropriate step-size or learning-rate conditions.
- B. Discretization for Discrete-Time Algorithms: Discretization introduces approximation errors, so each proposed scheme requires additional assumptions beyond the standard GNEP assumptions.The required conditions depend on the chosen discretization technique.
- B. Discretization for Discrete-Time Algorithms: The forward scheme uses a standard diminishing step size that is independent of agents’ private data.Agents update decisions, equality multipliers, and inequality multipliers using the discretized dynamics.
- B. Discretization for Discrete-Time Algorithms: Convergence of the forward scheme to a GNE follows under the additional assumptions, including global Lipschitz continuity and compact individual constraint sets.The theorem specifically assumes a diminishing step size and Assumptions 4 (1) and (2).
- B. Discretization for Discrete-Time Algorithms: The forward–backward hybrid scheme converges to a fixed point corresponding to a GNE under Assumptions 4 (1) and (3) and suitable inverse learning-rate matrices.The scheme is presented as an alternative to forward discretization.
- B. Discretization for Discrete-Time Algorithms: Both discrete-time algorithms receive other agents’ decision variables through the relevant communication graphs.The implementation retains the distributed exchange pattern of the continuous-time method.
IV. MULTI-ROBOTS PLACEMENT PROBLEM
The multi-robot placement problem is formulated as a distributed GNEP with convex individual constraints and shared linear equality constraints. Experiments show convergence to different GNEs from varying initializations, while the proposed discretization achieves similar convergence with substantially lower communication.
- Shared constraints: For each task, the positions of assigned robots must collectively center at the task location cj through a shared equality constraint.Each robot accesses only the subset of shared constraints associated with its assigned tasks.
- Problem formulation: Each robot balances proximity to its anchor with proximity to task-assigned robots through a private penalty parameter ρv.The anchor constraint is local, while coordination objectives couple robots assigned to common tasks.
- Continuous-time results: Different initial conditions lead the continuous algorithm to different equilibria, and the reported convergence points are all GNEs.Figure 2 shows trajectories from six random initializations.
- Communication efficiency: Communication exchange is reduced by two-thirds per iteration because Gf coincides with Gg, while convergence remains experimentally similar.The proposed method therefore uses one-third of the communication cost of the consensus-based alternatives in this setting.
- Problem setup: The placement instance contains 12 robots and 4 tasks connected by a bipartite robot–task assignment structure.Robots, anchor points, local constraint regions, and task locations are distinguished in Figure 1.
- Discrete-time results: Both Algorithm 3 and the consensus-based comparison converge experimentally to a v-GNE from zero initialization with learning rate 0.01.The comparison is shown in Figure 3.
V. CONCLUSION
The paper develops distributed continuous- and discrete-time methods for GNEPs with shared constraints without multiplier consensus. The approach reaches general GNEs, reduces communication, and is validated on multi-robot placement, while global inequality constraints remain an open extension.
- Conclusion: The proposed continuous-time algorithm solves GNEPs with shared linear equality constraints without exchanging Lagrange multipliers.Discrete-time variants are also introduced and validated on multi-robot placement.
- Conclusion: The method converges to the broader set of GNEs, while identical multiplier initialization can recover v-GNEs.The convergence claim is stated for the KKT points of the formulated GNEP.
- Conclusion: Future work includes global inequality constraints and determining how sparsity and graph structure affect communication efficiency.Formal convergence guarantees for global inequality constraints remain open.