Source-linked AI summary
Reaching an Optimal Consensus: Dynamical Systems that Compute Intersections of Convex Sets
Guodong Shi, Karl Henrik Johansson, Yiguang Hong
TL;DR
The paper asks how multi-agent systems can jointly solve distributed convex optimization and reach consensus when information is local and communications vary over time. It converts the problem into convex-set intersection computation and proposes a decentralized continuous-time control law. Under jointly connected graphs, the system achieves consensus within the global solution set, while the analysis also identifies a limitation when the true optimum lacks a simple set representation.
Problem
The paper addresses distributed optimization in which each node knows only one objective component, with few existing continuous-time approaches for achieving both consensus and optimality.
Method
The paper represents each local optimum by a convex set, assumes their nonempty intersection equals the group optimum, and uses a continuous-time law combining consensus and projection terms.
Results
Under jointly connected graphs and time-varying communications, the system achieves consensus within the global solution set of the summed objective.
Takeaways & Limitations
The proposed decentralized dynamical system solves the corresponding convex intersection computation problem while supporting directed and bidirectional communication analyses.
Takeaways & Limitations
When the real optimal solution set has no simple expression, the paper states that the convergence argument may fail and only suboptimal convergence is conjectured.
Abstract
from arXiv · showhide
In this paper, multi-agent systems minimizing a sum of objective functions, where each component is only known to a particular node, is considered for continuous-time dynamics with time-varying interconnection topologies. Assuming that each node can observe a convex solution set of its optimization component, and the intersection of all such sets is nonempty, the considered optimization problem is converted to an intersection computation problem. By a simple distributed control rule, the considered multi-agent system with continuous-time dynamics achieves not only a consensus, but also an optimal agreement within the optimal solution set of the overall optimization objective. Directed and bidirectional communications are studied, respectively, and connectivity conditions are given to ensure a global optimal consensus. In this way, the corresponding intersection computation problem is solved by the proposed decentralized continuous-time algorithm. We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non-smooth analysis.
1 Introduction
The paper develops a distributed continuous-time approach for converting a convex distributed optimization problem into an intersection computation problem. It establishes optimal consensus under time-varying communications and derives connectivity conditions for directed and bidirectional networks.
- Motivation: Distributed optimization is challenging because each node knows only one component of the sum objective, while consensus and optimality must be achieved together.Existing work is predominantly discrete-time, where weights, step sizes, and graph connectivity must be carefully selected.
- Problem transformation: The paper assumes each node observes a convex solution set X_i and that their intersection is nonempty, making the group optimum equal to that intersection.This converts the optimization task into distributed computation of an intersection of convex sets.
- Proposed approach: The proposed control law combines a consensus component with a projection component to solve the convex intersection problem in continuous time.The projection term can be interpreted as special subgradient information, linking the dynamics to earlier distributed optimization algorithms.
- Main results: The method achieves optimal consensus under time-varying communications, meaning agents agree within the global optimal solution set.Both directed and bidirectional communication cases are analyzed.
- Main results: The paper derives sharp connectivity conditions and shows that optimal consensus generally fails under weaker connectedness assumptions.The communication weights may depend on time or system state.
2 Preliminaries
The preliminaries define the graph, convex-analysis, distance, projection, Dini-derivative, and invariant-set concepts used to analyze the continuous-time multi-agent system.
- Graph-theoretic preliminaries: A digraph is strongly connected when every ordered pair of nodes has directed paths in both directions; bidirectionality requires reciprocal arcs.These definitions distinguish directed and bidirectional communication models.
- Graph-theoretic preliminaries: A time-varying graph is represented by a piecewise-constant graph signal, and its joint graph is the union of edges over a specified time interval.Joint connectivity properties are used to characterize communication across changing topologies.
- Convex analysis: The preliminaries introduce convex sets and convex hulls as foundational objects for the intersection and projection arguments.Convexity ensures that line segments between points in a set remain in the set.
- Convex analysis: For a closed convex set K, the distance |x|_K measures the Euclidean distance from x to K, and the projector P_K(x) is its unique nearest point in K.Projection and distance properties support the convergence analysis.
- Dynamical-systems tools: Upper Dini derivatives provide a way to test whether continuous functions are non-increasing along trajectories, while positively invariant sets contain all future solutions from their initial points.These tools are used to establish convergence and trajectory containment.
3 Problem Formulation and Main Results
The paper formulates distributed optimal consensus as convex-set intersection for first-order agents with time-varying communication graphs. A projection-based continuous-time control law achieves convergence under directed or bidirectional connectivity conditions.
- 3.1 Multi-agent Model: Each agent is modeled as a first-order integrator, and communication occurs over a time-varying graph with dwell-time switching.Neighbor sets and switching graphs define the information available to each agent over time.
- 3.1 Multi-agent Model: The agents cooperatively minimize a sum of locally known objectives while reaching consensus, assuming each component has a nonempty bounded convex solution set.The intersection X0 is assumed nonempty and becomes the optimal solution set of the group objective.
- 3.1 Multi-agent Model: The intersection X0 is convex, compact, and equal to the optimal solution set of the group objective under the stated assumptions.This follows from convexity of the local sets, their nonempty intersection, and boundedness.
- 3.2 Distributed Control: The distributed control combines neighbor-consensus terms with local projection corrections toward each agent’s observed set Xi.The projection term can be obtained locally when Xi is observable, and the weights may vary with time and state without requiring global information.
- 3.2 Distributed Control: Unlike alternating projected-consensus steps, the proposed continuous-time law uses consensus and projection information simultaneously.The paper identifies this simultaneous use as an essential difference from the compared projected consensus algorithm.
- 3.2 Distributed Control: Global optimal consensus requires both convergence of every agent to X0 and asymptotic agreement among all agent states.The paper defines optimal consensus as the conjunction of global optimal-set convergence and global consensus.
- 3.3 Main Results: For directed communications, UJSC guarantees global optimal consensus, while bidirectional communications require IJC and can avoid a uniform bound on connectivity-interval lengths.The bidirectional condition is stated as generally necessary as well as sufficient.
- 3.3 Main Results: The connectivity conditions are sharp for consensus across all possible local sets, but are not necessary for every particular optimization problem.Removing the nonempty-intersection assumption also removes the simple expression for the true optimal set, so the stated convergence analysis no longer applies directly.
4 Optimal Set Convergence
The paper analyzes convergence of all agents to the global optimal solution set using distance functions and convex-analysis arguments. Under suitable connectivity, the system achieves optimal set convergence for directed and bidirectional communications.
- Distance-based convergence analysis: The analysis studies distances between agent states and the global optimal solution set X0.The convergence proof uses a distance function and its Dini derivative along system trajectories.
- Distance-based convergence analysis: The maximum distance d(t) from the agents to X0 is nonincreasing and converges to a constant d*≥0.Optimal solution set convergence is equivalent to d*=0.
- Distance-based convergence analysis: Without connectivity assumptions, Lemmas 4.1 and 4.2 establish key distance properties, including convergence of each agent to its own solution set under stated conditions.The paper explicitly notes that these lemmas do not require communication-graph connectivity.
- Connectivity-dependent convergence: Under UJSC directed graphs or IJC bidirectional communications, the agents converge to the global optimal solution set.Propositions 4.1 and 4.2 give the corresponding connectivity conditions.
- Connectivity-dependent convergence: The bidirectional proof uses connected limiting graphs and invariant convex sets to show that the limiting distance d* cannot remain positive.The contradiction argument combines distance estimates, projections, and constructed invariant sets.
5 Global Consensus
The consensus analysis constructs positively invariant convex sets and bounds the spread of agent states. It proves global consensus under uniformly jointly strongly connected directed graphs and connected jointly connected bidirectional graphs.
- Invariant-set construction: The paper introduces multi-projection functions and defines ΔK as the convex hull of their images over a convex set K.This construction supports the invariant-set analysis used to establish agreement.
- Invariant-set construction: The constructed set ΔN is positively invariant, so trajectories remain within the relevant convex hull structure over time.Figure 2 illustrates constructing an invariant set from K = co{y1, y2}.
- Consensus criterion: Consensus is characterized by the condition limt→∞H(t)=0, where H(t) is the difference between the maximum and minimum agent values.The analysis tracks this spread through successive communication intervals.
- Directed communications: System (10) achieves global consensus when Gσ(t) is UJSC.The proof propagates bounds across the network and obtains H(t)→0.
- Bidirectional communications: With bidirectional communications, system (10) achieves global consensus when the communication graph is jointly connected.The proof expands connected node groups over time and concludes limt→∞H(t)=0.
6 Conclusions
The paper converts distributed sum optimization into an intersection computation problem using locally observed convex solution sets. Its simple distributed continuous-time control achieves optimal consensus under time-varying communications, while future work includes randomized and event-based methods.
- Problem and formulation: The paper addresses optimal consensus for multi-agent systems minimizing a sum of objective functions.Each component is associated with a particular node.
- Main result: The system achieves both consensus and agreement within the global solution set over jointly connected graphs.The control laws are described as simple and distributed.
- Problem and formulation: The optimization problem is converted into intersection computation when each agent observes projection information onto its own convex solution set and the total intersection is nonempty.These are explicit assumptions of the formulation.
- Main result: The results show that a global optimization problem can be solved over a multi-agent network with time-varying communications and limited interactions.This conclusion is stated for the proposed distributed approach.
- Future work: Future work includes randomization in nodes’ decision-making and event-based methods in optimization algorithm design.These directions mark the stated scope beyond the presented method.
A.1 Proof of Lemma 4.2
The proof establishes that each distance d_i(t) converges to the common value d*, with the zero case immediate and the positive case handled through eventual bounds and contradiction.
- For d* > 0, the argument uses definitions, boundedness, compactness, and relations from earlier equations to obtain the required estimates.
- lim t→+∞ d_i(t) = d* for every node i.
- For any ε > 0, all distances eventually satisfy d_i(t) ∈ [d*−ε, d*+ε].
- When d* = 0, the conclusion follows directly from |x_i(t)|_Xi ≤ |x_i(t)|_X0.
- The proof proceeds by contradiction after assuming the conclusion fails for some node and constant M0 > 0.
A.2 Proof of Lemma 4.3
The proof shows that the distance bounds propagate across the network under both directed and bidirectional communications. Uniform joint strong connectivity and dwell-time arguments yield the desired conclusion.
- Conclusion: The proof concludes after establishing the contradiction and recording the relevant bounded-distance condition.
- Case analysis: The proof splits into directed and bidirectional communication cases.
- Directed Case: In the directed case, uniform joint strong connectivity supplies successive arcs that transmit the estimate from one node to the next.
- Directed Case: The directed propagation uses intervals of length T0 = T + 2τD and recursively defined bounds m1 and subsequent estimates.
- Directed Case: Repeating the propagation across additional nodes leads to a contradiction for sufficiently small ε.
- Bidirectional Case: In the bidirectional case, the argument tracks the first times when new neighbors connect to the growing node set.
- Bidirectional Case: Dwell time ensures the connecting links persist long enough for analogous estimates to propagate through the network.
- Bidirectional Case: Because the graph is jointly connected, the propagation can continue until all nodes are included, producing a contradiction with the definition of d*.