A continuous-time violation-free multi-agent optimization algorithm and its applications to safe distributed control

TL;DR

This paper develops a continuous-time, violation-free distributed optimization framework for constraint-coupled problems on networks by introducing auxiliary variables to decouple costs and coupling constraints. A local subgradient-based update law leverages only neighboring information, ensuring all coupling constraints are satisfied during evolution and converging to the centralized optimum (or optimal set). The approach extends to sparse couplings and to control barrier function-induced QPs, with finite-time convergence guarantees for slowly time-varying parameters, enabling safe, real-time distributed control. The authors demonstrate the methodology on static resource allocation and multi-agent safe coordination tasks, showing improved memory/communication efficiency and practical safety guarantees for MAS control.

Abstract

In this work, we propose a continuous-time distributed optimization algorithm with guaranteed zero coupling constraint violation and apply it to safe distributed control in the presence of multiple control barrier functions (CBF). The optimization problem is defined over a network that collectively minimizes a separable cost function with coupled linear constraints. An equivalent optimization problem with auxiliary decision variables and a decoupling structure is proposed. A sensitivity analysis demonstrates that the subgradient information can be computed using local information. This then leads to a subgradient algorithm for updating the auxiliary variables. A case with sparse coupling constraints is further considered, and it is shown to have better memory and communication efficiency. For the specific case of a CBF-induced time-varying quadratic program (QP), an update law is proposed that achieves finite-time convergence. Numerical results involving a static resource allocation problem and a safe coordination problem for a multi-agent system demonstrate the efficiency and effectiveness of our proposed algorithms.

Figures (7)

• Figure 1: Illustration of the constraint decomposition scheme. Here only one coupling constraint is considered and the superscript is neglected for simplicity. By introducing an auxiliary variable $\bm{y}$, one coupling constraint becomes $N$ local constraints. Proposition \ref{['prop:equivalent_problem']} shows that when $\bm{y}$ is a decision variable, the two class of constraints represent the same feasible region in $\bm{x}$. Our proposed algorithm locally updates $y_i$ to obtain an "optimal" constraint decomposition in the sense that $\bm{a}_i^\top \bm{x}_i^\star + b_i + \sum_{j\in \mathcal{N}_i} (y_i - y_j) \leq 0$ for all $i\in \mathcal{I}$, where $\bm{x}_i^\star$ is the optimal solution to the original optimization problem.
• Figure 2: An illustration of a communication graph satisfying Assumption \ref{['ass:sparsity']}. In this scenario, the agent index sets for each constraint are $\mathcal{I}_1 =\{1,2,3\}, \mathcal{I}_2 =\{1,2,4\}$, the constraint index sets for each agent are $\mathcal{M}_1 = \mathcal{M}_2 = \{1,2\}$, $\mathcal{M}_3 = \{1\}, \mathcal{M}_4 = \{2\}$, and the sets of neighboring nodes with relevant constraint involvement are $N_1^1 = \{ 2,3\}, N_1^2 = \{ 2\}, N_2^1 = \{ 1,3\}, N_2^2 = \{ 1,4\}, N_3^1 = \{ 1,2\}, N_4^2 = \{ 2\}$, respectively.
• Figure 3: Communication graph
• Figure 4: Numerical results involving $9$ agents solving a static optimization problem.
• Figure 5: Safe distributed control for coordinating $9$ agents.

Theorems & Definitions (26)

• Proposition 1: Equivalence
• proof
• Proposition 2: Constraint satisfaction
• proof
• Proposition 3: Subdifferential
• proof
• Remark 1
• Theorem 1
• proof
• Remark 2