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Distributed Convex Optimization with State-Dependent (Social) Interactions over Random Networks

Seyyed Shaho Alaviani, Atul Kelkar

TL;DR

This work addresses distributed convex optimization under a novel combination of state-dependent interactions and random arbitrary networks, introducing a state-dependent weighted random operator that is quasi-nonexpansive. It develops a gradient-based, discrete-time algorithm that blends local descent with a random operator and proves almost sure and mean-square convergence to the global optimum, even in totally asynchronous settings and under periodic synchronous graphs. A key contribution is removing the need for a priori topology distributions by exploiting the quasi-nonexpansive property, thereby broadening applicability to general switched networks, including robot networks with distribution-dependent communication. The results advance distributed optimization by enabling robust performance on networks with state-dependent weights and stochastic connectivity, with practical validation via a warehouse-robot example and extensions to distributed implementations without stringent connectivity assumptions.

Abstract

This paper aims at distributed multi-agent convex optimization where the communications network among the agents are presented by a random sequence of possibly state-dependent weighted graphs. This is the first work to consider both random arbitrary communication networks and state-dependent interactions among agents. The state-dependent weighted random operator of the graph is shown to be quasi-nonexpansive; this property neglects a priori distribution assumption of random communication topologies to be imposed on the operator. Therefore, it contains more general class of random networks with or without asynchronous protocols. A more general mathematical optimization problem than that addressed in the literature is presented, namely minimization of a convex function over the fixed-value point set of a quasi-nonexpansive random operator. A discrete-time algorithm is provided that is able to converge both almost surely and in mean square to the global solution of the optimization problem. Hence, as a special case, it reduces to a totally asynchronous algorithm for the distributed optimization problem. The algorithm is able to converge even if the weighted matrix of the graph is periodic and irreducible under synchronous protocol. Finally, a case study on a network of robots in an automated warehouse is given where there is distribution dependency among random communication graphs.

Distributed Convex Optimization with State-Dependent (Social) Interactions over Random Networks

TL;DR

This work addresses distributed convex optimization under a novel combination of state-dependent interactions and random arbitrary networks, introducing a state-dependent weighted random operator that is quasi-nonexpansive. It develops a gradient-based, discrete-time algorithm that blends local descent with a random operator and proves almost sure and mean-square convergence to the global optimum, even in totally asynchronous settings and under periodic synchronous graphs. A key contribution is removing the need for a priori topology distributions by exploiting the quasi-nonexpansive property, thereby broadening applicability to general switched networks, including robot networks with distribution-dependent communication. The results advance distributed optimization by enabling robust performance on networks with state-dependent weights and stochastic connectivity, with practical validation via a warehouse-robot example and extensions to distributed implementations without stringent connectivity assumptions.

Abstract

This paper aims at distributed multi-agent convex optimization where the communications network among the agents are presented by a random sequence of possibly state-dependent weighted graphs. This is the first work to consider both random arbitrary communication networks and state-dependent interactions among agents. The state-dependent weighted random operator of the graph is shown to be quasi-nonexpansive; this property neglects a priori distribution assumption of random communication topologies to be imposed on the operator. Therefore, it contains more general class of random networks with or without asynchronous protocols. A more general mathematical optimization problem than that addressed in the literature is presented, namely minimization of a convex function over the fixed-value point set of a quasi-nonexpansive random operator. A discrete-time algorithm is provided that is able to converge both almost surely and in mean square to the global solution of the optimization problem. Hence, as a special case, it reduces to a totally asynchronous algorithm for the distributed optimization problem. The algorithm is able to converge even if the weighted matrix of the graph is periodic and irreducible under synchronous protocol. Finally, a case study on a network of robots in an automated warehouse is given where there is distribution dependency among random communication graphs.
Paper Structure (9 sections, 3 theorems, 63 equations, 6 figures)

This paper contains 9 sections, 3 theorems, 63 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Algorithm and Its Convergence
  5. Almost Sure Convergence
  6. Mean Square Convergence
  7. Distributed Optimization
  8. Numerical Example
  9. Conclusions and Future Work

Key Result

Proposition 1

quasi11 If $C$ is a closed convex subset of a Hilbert space $\mathcal{H}$ and $T:C \longrightarrow C$ is quasi-nonexpansive, then $Fix(T)$ is a nonempty closed convex set.

Figures (6)

  • Figure 1: Variables $x_{i}^{1}, i=1, \hdots,20,$ of the robotic agents with weights of the form (\ref{['cuckerexample1']}). This figure shows that the variables are getting consensus when the robots communicate for one realization of random network with distribution dependency.
  • Figure 2: Variables $x_{i}^{2}, i=1, \hdots,20,$ of the robotic agents with weights of the form (\ref{['cuckerexample1']}). This figure shows that the variables are getting consensus when the robots communicate for one realization of random network with distribution dependency.
  • Figure 3: Two-dimensional (2D) plot of variables $x^{1}$ and $x^{2}$, in Figures \ref{['fig1']} and \ref{['fig2']}, where the initial positions of agents are shown with 'o', and the final position is shown with 'x'.
  • Figure 4: The error in Example 1 with weights of the form (\ref{['cuckerexample1']}) for one realization of the random network with distribution dependency.
  • Figure 5: Variables $x_{i}^{1}, i=1, \hdots,20,$ of the robotic agents with weights of the form (\ref{['weightlog']}). This figure shows that the variables are getting consensus when the robots communicate for one realization of random network with distribution dependency.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Proposition 1
  • Definition 4
  • Remark 2
  • Remark 3
  • Theorem 2
  • Remark 4
  • ...and 2 more