A cutting-surface consensus approach for distributed robust optimization of multi-agent systems

TL;DR

This work tackles distributed robust convex optimization with semi-infinite uncertainty-induced constraints over time-varying unbalanced directed networks. It introduces a fully distributed cutting-surface consensus framework that first solves a tractable ADRCP via a distributed projected gradient with an epigraphic reformulation, then iteratively tightens and augments constraints with cutting-surfaces to locate locally feasible, consensus solutions for the original DRCP. The approach guarantees finite-time termination and local feasibility for each agent, supported by convergence and rate results, and is demonstrated through a numerical case study. The method enables robust, distributed decision-making with guaranteed feasibility in settings where uncertainty affects constraints, and it provides practical termination criteria and privacy-preserving information exchange between agents.

Abstract

A novel and fully distributed optimization method is proposed for the distributed robust convex program (DRCP) over a time-varying unbalanced directed network under the uniformly jointly strongly connected (UJSC) assumption. Firstly, an approximated DRCP (ADRCP) is introduced by discretizing the semi-infinite constraints into a finite number of inequality constraints to ensure tractability and restricting the right-hand side of the constraints with a positive parameter to ensure a feasible solution for (DRCP) can be obtained. This problem is iteratively solved by a distributed projected gradient algorithm proposed in this paper, which is based on epigraphic reformulation and gradient projected operations. Secondly, a cutting-surface consensus approach is proposed for locating an approximately optimal consensus solution of the DRCP with guaranteed local feasibility for each agent. This approach is based on iteratively approximating the DRCP by successively reducing the restriction parameter of the right-hand constraints and adding the cutting-surfaces into the existing finite set of constraints. Thirdly, to ensure finite-time termination of the distributed optimization, a distributed termination algorithm is developed based on consensus and zeroth-order stopping conditions under UJSC graphs. Fourthly, it is proved that the cutting-surface consensus approach terminates finitely and yields a feasible and approximate optimal solution for each agent. Finally, the effectiveness of the approach is illustrated through a numerical example.

Figures (8)

• Figure 1: Graphic illustration of Algorithm \ref{['alg:DRCP']}. The steps in yellow and blue only require computation, information update and storage within the agent. The green steps require information exchange between each agent on this basis.
• Figure 2: Network Topology: time-varying unbalanced directed graphs, where the black and blue arrows denote the communication among agents at time $t$ and $t+1$. $\mathcal{G}(t:t+1)$ is strongly connected.
• Figure 3: Iterative process of Algorithm \ref{['alg:DRCP']}. The blue solid line indicates the evolution of $F=\sum^m_{i=1}f_i(z^k_i)$ with respect to iteration $k$. The red dashed line presents the global optimal value of the problem (\ref{['npf_1']}), where $x^*=[0,\sqrt{7}/4]^\top$, $F^*\approx 38.687746$. The horizontal coordinates of the black and red dotted-dashed lines represent the number of iterations required by the finite-time consensus termination algorithm in xie2017stop and the termination algorithm developed in this paper, respectively.
• Figure 4: Iterative process of the DPG algorithm, when the (${\rm ADRCP^k}$) is feasible ($\epsilon^k_i=0.1$ and $Y^k_i=[1]$ for all agents). The six solid lines indicates the evolution of $\Vert x^k_i(t)-x^{k,*}\Vert$ with respect to $t$, where $x^k_i(t)$ is agent $i$'s estimate generated by DPG algorithm at time $t$. The red dashed line presents the value $\Vert x^k_i(t)-x^{k,*}\Vert=0$. The horizontal coordinates of the black and red dotted-dashed lines represent the number of iterations required by the finite-time consensus termination algorithm in xie2017stop and the proposed termination algorithm, respectively. The zoomed-in part shows our optimality gap is smaller than that achieved by xie2017stop.
• Figure 5: Iterative process of the DPG algorithm, when the (${\rm ADRCP^k}$) is infeasible ($\epsilon^k_i=5$ and $Y^k_i=[1]$ for all agents). The six solid lines indicates the evolution of $F(x^k_i(t))$ with respect to $t$, where $x^k_i(t)$ denotes the agent $i$'s estimate generated by DPG algorithm at time $t$.

Theorems & Definitions (35)

• Remark 1
• Remark 2
• Remark 3: Dealing with Unbalance
• Remark 4
• Theorem 1: Convergence of DPG Algorithm
• Theorem 2: Convergence Rate of DPG Algorithm
• Definition 1: Local $\epsilon_1$-consensus xie2017stop
• Proposition 1
• proof
• Corollary 1