Multi-Agent Maximization of a Monotone Submodular Function via Maximum Consensus

In recent years, optimal multi-agent sensor placement/dispatch to cover areas for sampling/observation objectives has been of great interest in the robotic community to solve various coverage, exploration, and monitoring problems. Depending on the problem setting, the planning space can be continuous [1, 2, 3, 4] or discrete [5, 6, 7, 8]. In this paper, our interest is in planning in discrete space. Many optimal discrete policy-making for multi-agent sensor placement problems appear in the form of maximizing a utility for the team [9, 10, 11, 12, 13, 5]. A particular subset of these problems that we are interested in is in the form of a set-valued function maximization problem described by

where $\mathcal{I}\subset 2^{\mathcal{P}}$ is the admissible set of subsets originated from the ground policy set $\mathcal{P}$, which is the union of the local policy set of the agents; see Fig. 1 for an example.

Problem (1) is known to be NP hard [14]. However, when the objective function is monotone increasing and submodular set function, it is well-known that the sequential greedy algorithm [15] delivers a polynomial-time suboptimal solution with guaranteed optimality bound. For large scale submodular maximization problems, reducing the size of the problem through approximations [16] or using several processing units leads to a faster sequential greedy algorithm, however, with some sacrifices on the optimality bound [17, 18, 19, 20].

When the set that the agents can choose their policy from is the same for all agents and the agents are homogeneous, the feasible set $\mathcal{I}$ of (1) can be spanned by a uniform matroid constraint¹¹1see Section 3 for a formal definition of uniform matroid constraint.. For problems with uniform matroid constraint, the sequential greedy algorithm delivers a suboptimal solution whose utility value is no worse than $1-\frac{1}{\textup{e}}$ times the optimal utility value [14]. On the other hand, when the local policy set of the agents is disjoint and/or the agents are heterogeneous, and each agent has to choose one policy from its local set, the feasible set $\mathcal{I}$ of (1) can be spanned by the so-called partitioned matroid constraint²²2see Section 3 for a formal definition of partitioned matroid constraint.. For problems with partitioned matroid constraint, the sequential greedy algorithm delivers a suboptimal solution whose utility value is no worse than $\frac{1}{2}$ times the optimal utility value [15]. For problems with partitioned matroid constraint, more recently, a suboptimal solution with a better optimality gap is proposed in [21, 22, 23, 24] using the multilinear continuous relaxation of a submodular set function. The multilinear continuous relaxation of a submodular set function defined on the vertices of the $n$-dimensional hypercube facilitates achieves better optimality bounds for the maximization problem (1) by allowing to use a continuous gradient-based optimization algorithm. That is, the relaxation transforms the main discrete problem into a continuous optimization problem with linear constraints. The optimality bound achieved by the algorithm proposed in [21] is $1-\frac{1}{\textup{e}}$ bound for a partitioned matroid constraint, which outperforms the sequential greedy algorithm whose optimally bound is $1/2$. However, this solution requires a central authority to find the optimal solution. In sensor placement problems, when the agents are self-organizing autonomous mobile agents with communication and computation capabilities, it is desired to solve the optimization problem 1 in a distributed way, without involving a central authority. A distributed multilinear extension based continuous algorithm that uses an average consensus scheme between the agents to solve (1) subject to partitioned matroid constraint is proposed in [25]. However, the proposed algorithm assumes that access to the exact multilinear extension of the utility function is available, whose calculation is exponentially hard.

In this paper, motivated by the improved optimality gap of multilinear continuous relaxation based algorithms, we develop a distributed implementation of the algorithm of [21] over a connected undirected graph to obtain a suboptimal solution for (1) subject to a partitioned matroid constraint. In this algorithm, to manage the computational cost of constructing the multilinear extension of the utility function, we use a sampled based evaluation of the multilinear extension and propose a gradient-based algorithm, which uses a maximum consensus message passing scheme over the communication graph. Our algorithm uses a fully distributed rounding technique to compute the final solution of each agent. Through rigorous analysis, we show that our proposed distributed algorithm achieves, with a known probability, a $1-\frac{1}{\textup{e}}-O(1/T)$ optimality bound, where $T$ is the number of times agents communicated over the network. A numerical example demonstrates our results.

The organization of this paper is as follows. In Section 2 we introduce the notation used throughout the paper. Section 3 provides the necessary background on submodularity and submodular functions. We introduce our proposed algorithm in Section 5. A demonstration of our results is provided in Section 6. Appendix A gives the proof of our main results in Section 5. Appendix B gives the auxiliary results that we use in establishing the proof of our main results.

We denote the vectors with bold small font. The $p^{\textup{th}}$ element of vector $\boldsymbol{\mathbf{x}}$ is denoted by $x_{p}$. We denote the inner product of two vectors $\boldsymbol{\mathbf{x}}$ and $\boldsymbol{\mathbf{y}}$ with appropriate dimensions by $\boldsymbol{\mathbf{x}}.\boldsymbol{\mathbf{y}}$. Furthermore, we use $\boldsymbol{\mathbf{1}}$ as a vector of ones with appropriate dimension. We denote sets with capital curly font. For a $\mathcal{P}=\{1,\cdots,n\}$ and a vector $\boldsymbol{\mathbf{x}}\in{\mathbb{R}}_{\geq 0}^{n}$ with $0\leq x_{p}\leq 1$, the set $\mathcal{R}_{\boldsymbol{\mathbf{x}}}$ is a random set where $p\in\mathcal{P}$ is in $\mathcal{R}_{\boldsymbol{\mathbf{x}}}$ with the probability $x_{p}$. Hence, we call such vector $\boldsymbol{\mathbf{x}}$ as membership probability vector. Furthermore, for $\mathcal{S}\subset\mathcal{P}$, $\boldsymbol{\mathbf{1}}_{\mathcal{S}}\in\{0,1\}^{n}$ is the vector whose $p^{\textup{th}}$ element is $1$ if $p\in\mathcal{S}$ and $0$ otherwise; we call $\boldsymbol{\mathbf{1}}_{\mathcal{S}}$ the membership indicator vector of set $\mathcal{S}$. $|x|$ is the absolute value of $x\in$. By overloading the notation, we also use $|\mathcal{S}|$ as the cordiality of set $\mathcal{S}$. We denote a graph by $\mathcal{G}(\mathcal{A},\mathcal{E})$ where $\mathcal{A}$ is the node set and $\mathcal{E}\subset\mathcal{A}\times\mathcal{A}$ is the edge set. $\mathcal{G}$ is undirected if and only $(i,j)\in\mathcal{E}$ means that agents $i$ and $j$ can exchange information. An undirected graph is connected if there is a path from each node to every other node in the graph. We denote the set of the neighboring nodes of node $i$ by $\mathcal{N}_{i}=\{j\in\mathcal{A}|(i,j)\in\mathcal{E}\}$. We also use $d(\mathcal{G})$ to show the diameter of the graph.

Given a set $\mathcal{F}\subset\mathcal{X}\times$ and an element $(p,\alpha)\in\mathcal{X}\times$ we define the addition operator $\oplus$ as $\mathcal{F}{\oplus}\{(p,\alpha)\}=\{(u,\gamma)\in\mathcal{X}\times\,|\,(u,\gamma)\in\mathcal{F},u\not=p\}\cup\{(u,\gamma+\alpha)\in\mathcal{X}\times\,|\,(u,\gamma)\in\mathcal{F},u=p\}\cup\{(p,\alpha)\in\mathcal{X}\times\,|\,(p,\gamma)\not\in\mathcal{F}\}$. Given a collection of sets $\mathcal{F}_{j}\in\mathcal{X}\times$, $j\in\mathcal{N}$, we define the max-operation over these collection as $\underset{j\in\mathcal{N}}{\textup{MAX}}\,\,\mathcal{F}_{j}=\{(u,\gamma)\in X\times|(u,\gamma)\in\bar{\mathcal{F}}\text{~{}s.t.~{}}\gamma=\underset{(u,\alpha)\in\bar{\mathcal{F}}}{\textup{max}}\alpha\}$, where $\bar{\mathcal{F}}=\bigcup_{j\in\mathcal{N}}\mathcal{F}_{j}.$

A set function $f:2^{\mathcal{P}}\to{\mathbb{R}}_{\geq 0}$ is monotone increasing if for sets $\mathcal{P}_{1}\subset\mathcal{P}_{2}\subset\mathcal{P}$,

holds. Furthermore, defining $\Delta_{f}(p|\mathcal{S})=f(\mathcal{S}\cup\{p\})-f(\mathcal{S}),\,\,\mathcal{S}\subset\mathcal{P}$, the set function $f$ is submoular if for $p\in\mathcal{P}\setminus\mathcal{P}_{2}$,

holds, which shows the diminishing return of a set function. Furthermore, a set function is normalized if $f(\emptyset)=0$.

In what follows without loss of generality we assume that the ground set $\mathcal{P}$ is equal to $\{1,\cdots,n\}$. Submodular functions are functions assigning values to all subsets of a finite set $\mathcal{P}$. Using the set membership indicator, equivalently, we can regard them as functions on the boolean hypercube, $f:\{1,0\}^{n}\to$. For a submodular function $f:2^{\mathcal{P}}\to{\mathbb{R}}_{\geq 0}$, its multilinear extension $F:[0,1]^{n}\to{\mathbb{R}}_{\geq 0}$ in the continuous space is

The multilinear extension $F$ of set function $f$ is a unique multilinear function agreeing with f on the vertices of the hypercube. The multilinear extension function has the following properties.

An alternative way to perceive $F$ is to observe that it is indeed equivalent to

where $\mathcal{R}_{\boldsymbol{\mathbf{x}}}\subset\mathcal{P}$ is a set where each element $p\in\mathcal{R}_{\boldsymbol{\mathbf{x}}}$ appears independently with the probabilities $x_{p}$. Then, it can be shown that taking the derivatives of $F(\boldsymbol{\mathbf{x}})$ yields

and

Constructing $F(\boldsymbol{\mathbf{x}})$ in (3) requires the knowledge of $f(\mathcal{R})$ for all $\mathcal{R}\subset\mathcal{P}$, which becomes computationally intractable when the size of ground set $\mathcal{P}$ increases. However, with the stochastic interpretation (4) of the multilinear extension and its derivatives, if enough samples are drawn according to $\boldsymbol{\mathbf{x}}$, we can obtain an estimate of $F(\boldsymbol{\mathbf{x}})$ with a reasonable computational cost. We can use Chernoff-Hoeffding’s inequality to quantify the quality of this estimation given the number of samples.

Problems such as sensor placement with a limited number of sensors and ample places of possible sensor placement can be formulated as a set value optimization (1) where the feasible set constraint $\mathcal{I}$ is in the form of matroid. A matroid is defined as the pair $\mathcal{M}=\{\mathcal{P},\mathcal{I}\}$ with $\mathcal{I}\subset 2^{\mathcal{P}}$ such that

• •

  $\mathcal{B}\in\mathcal{I}$ and $\mathcal{A}\subset\mathcal{B}$ then $\mathcal{A}\in\mathcal{I}$

• •

  $\mathcal{A},\mathcal{B}\in\mathcal{I}$ and $|\mathcal{B}|>|\mathcal{A}|$, then there exists $x\in\mathcal{B}\setminus\mathcal{A}$ such that $\mathcal{A}\cup{x}\in\mathcal{I}$

One of the known matroids is uniform matroid $\mathcal{I}=\{\mathcal{R}\subset\mathcal{P},|\mathcal{R}|\leq k\}.$ A sensor placement scenario where there are $k$ sensors to be placed in locations selected out of possible locations $\mathcal{P}$ can be described by a uniform matroid. On the other hand, when there is a set of agents $\mathcal{A}$, each with multiple sensors $k_{i}$, $i\in\mathcal{A}$, that they can place in a set of specific and non-overlapping part $\mathcal{P}_{i}$ of a discrete space, this constrained can be described by the partitioned matroid $\mathcal{M}=\{\mathcal{P},\mathcal{I}\}$, where $\mathcal{I}=\{\mathcal{R}\subset\mathcal{P},|\mathcal{R}\cap\mathcal{P}_{i}|\leq k_{i}\}$ with $\bigcup\limits_{i=1}^{N}\mathcal{P}_{i}=\mathcal{P}$ and $\mathcal{P}_{i}\cap\mathcal{P}_{j}=\emptyset,\,\,i\neq j$.

A matroid polytop is a convex hull defined as

with $r_{\mathcal{M}}$ to be the rank function of the matroid defined as

For the partition matroid with $k_{i}=1$ which is the focus of this paper, the convex hull is such that for $\boldsymbol{\mathbf{x}}=[\boldsymbol{\mathbf{x}}_{1}^{\top},...,\boldsymbol{\mathbf{x}}_{N}^{\top}]^{\top}$ with $\boldsymbol{\mathbf{x}}_{i}\in{\mathbb{R}}_{\geq 0}^{|\mathcal{P}_{i}|}$ associated with membership probability of policies in $\mathcal{P}_{i}$, we have $\boldsymbol{\mathbf{1}}^{\top}\boldsymbol{\mathbf{x}}_{i}\leq 1$, i.e.,

We consider a group of $\mathcal{A}=\{1,...,N\}$ with communication and computation capabilities interacting over a connected undirected graph $\mathcal{G}(\mathcal{A},\mathcal{E})$. These agents aim to solve in a distributed manner the optimization problem

where utility function $f:2^{\mathcal{P}}\to_{\geq 0}$ is monotone increasing and submodular set function over the discrete policy space $\mathcal{P}=\bigcup\limits_{i=1}^{N}\mathcal{P}_{i}$, with $\mathcal{P}_{i}$ being the policy space of agent $i\in\mathcal{A}$, which is only known to agent $i$. In this paper, we work in the value oracle model where the only access to the utility function is through a black box returning $f(\mathcal{R})$ for a given set $\mathcal{R}$. Every agent can obtain the value of the utility function at any subset $\mathcal{R}\in\mathcal{P}$. The constraint (8b) is the partitioned matroid $\mathcal{M}=\{\mathcal{P},\mathcal{I}\}$, which ensures that only one policy per agent is selected from each local policy set $\mathcal{P}_{i}$, $i\in\mathcal{A}$. An example application scenario of our problem of interest is shown in Fig. 1. Without loss of generality, to simplify notation, we assume that the policy space is $\mathcal{P}=\{1,...,n\},$ and is sorted agent-wise with $1\in\mathcal{P}_{1}$ and $n\in\mathcal{P}_{N}$.

Finding the optimal solution $\mathcal{R}^{\star}\in\mathcal{I}$ of (8) even in central form is NP-Hard [28]. The computational time of finding the optimizer set increases exponentially with $N$ [29]. The well-known sequential greedy algorithm finds a suboptimal solution $\mathcal{R}_{\textup{SG}}$ for (8) with the optimality bound of $f(\mathcal{R}_{\textup{SG}})\geq\frac{1}{2}f(\mathcal{R}^{\star})$, i.e., a $1/2$-approximation at worst case [28]. More recently, by using the multilinear extension of the submodular utility functions, Vondrák [21] developed a randomized centralized continuous greedy algorithm which achieves a $(1-1/\text{e})-O(1/T)$-approximate for set value optimization (8) in the value oracle model, see Algorithm 1³³3Pipage rounding is a polynomial time algorithm which moves a fractional point $\boldsymbol{\mathbf{x}}$ on a hypercube to a integral point $\hat{\boldsymbol{\mathbf{x}}}$ on the same hypercube such that $f(\hat{\boldsymbol{\mathbf{x}}})\geq f(\boldsymbol{\mathbf{x}})$. Our objective in this paper is to develop a distributed implementation of Algorithm 1 to solve (8) for when agents interact over a connected graph $\mathcal{G}$. Recall that in our problem setting, every agent $i\in\mathcal{A}$ can evaluate the utility function for a given $\mathcal{R}\subset\mathcal{P}$ but it has access only to its own policy space $\mathcal{P}_{i}$.

Our proposed distributed multi-agent randomized continuous greedy algorithm to solve the set value optimization problem (8) over a connected graph $\mathcal{G}$ is given in Algorithm 2, whose convergence guarantee and suboptimality gap is given in Theorem 5.1 below. To provide the insight into construction of Algorithm 2, we first review the idea behind the central suboptimal solution via Algorithm 1 following [21]. We also provide some intermediate results that will be useful in establishing our results.

Consider the multi-agent sensor placement problem introduced in Fig. 1 for $5$ agents for which $\mathcal{B}_{i}=\mathcal{B}$, i.e., each agent is able to move to any of the sensor placement points. This sensor placement problem is cast by optimization problem (8). The field is a 6 unit by 6 unit square and the feasible sensor locations are the 6 by 6 grid in the center square of the field, see Fig. 2. The points of interest are spread around the map (small red dots in Fig. 2) in the total number of $900$. The sensing zone of the agents $\mathcal{A}=\{a,b,c,d,e\}$ are circles with radii of respectively $\{0.5,0.6,0.7,0.8,1.5\}$. The agents communicate over a ring graph as shown in Fig. 2. We first solve this problem using our proposed distributed Algorithm 2. The results of the simulation for different iteration and sampling numbers are shown in Table 1. The algorithm produces good results at a modest number of iteration and sampling numbers (e.g. see $T=20$ and $K=500$). Fig. 2(g) shows the result of the deployment using Algorithm 2. Next, we solve this algorithm using the sequential greedy algorithm [29] in a decentralized way by first choosing a route $\mathtt{SEQ}={\scriptsize\framebox{1}\to\framebox{2}\to\framebox{3}\to\framebox{4}\to\framebox{5}}$ that visits all the agents, and then giving $\mathtt{SEQ}$ to the agents so they follow $\mathtt{SEQ}$ to share their information in a sequential manner. Figure 2(a)-(f) gives 6 possible $\mathtt{SEQ}$ denoted by the semi-circular arrow inside the networks. The results of running the sequential greedy algorithm over the sequences in Fig. 2(a)-(f) is shown in Table 2.

What stands out about the sequential greedy algorithm is that the choice of sequence can affect the outcome of the algorithm significantly. We can attribute this inconsistency to the heterogeneity of the sensors’ measurement zone. We can see that when sensor $e$ is given the last priority to make its choice, the sequential greedy algorithm acts poorly. This can be explained by agents with smaller sensing zone picking high-density areas but not being able to cover it fully, see Fig. 3(h) which depicts the outcome of a sequential greedy algorithm using the sequence in Case 1. A simpler example justifying this observation is shown in Fig. 3 with the two disjoint clusters of points and two sensors. One may suggest to sequence the agents from high to low sensing zone order, however this is not necessarily the best choice as we can see in Table 2; the utility of case 6 is less than case 5 (the conjecture of sequencing the agents from strongest to weakest is not valid). Moreover, this ordering may lead to a very long $\mathtt{SEQ}$ over the communication graph. Interestingly, this inconsistency does not appear in solutions of Algorithm 2 where the agents intrinsically are overcoming the importance of a sequence by deciding the position of the sensors over a time horizon of communication and exchanging their information set.

We proposed a distributed suboptimal algorithm to solve the problem of maximizing an monotone increasing submodular set function subject to a partitioned matroid constraint. Our problem of interest was motivated by optimal multi-agent sensor placement problems in discrete space. Our algorithm was a practical decentralization of a multilinear extension based algorithm that achieves $(1-1/\textup{e}-O(1/T))$ optimally gap, which is an improvement over $1/2$ optimality gap that the well-known sequential greedy algorithm achieves. In our numerical example, we compared the outcome obtained by our proposed algorithm with that of a decentralized sequential greedy algorithm which is constructed from assigning a priority sequence to the agents. We showed that the outcome of the sequential greedy algorithm is inconsistent and depends on the sequence. However, our algorithm’s outcome due to its iterative nature intrinsically tended to be consistent, which in some ways also explains its better optimally gap over the sequential greedy algorithm. Our future work is to study the robustness of our proposed algorithm to message dropout.

• Proof of Lemma 5.1

  Since $f$ is a monotone increasing and submodular set function, we have $f(\mathcal{R}_{\boldsymbol{\mathbf{x}}_{i}(t)}\cup\{p\})-f(\mathcal{R}_{\boldsymbol{\mathbf{x}}_{i}(t)}\setminus\{p\})\geq 0$ and hence $\widetilde{\nabla F}(\boldsymbol{\mathbf{x}}_{i}(t))$ has positive entries $\forall i\in\mathcal{A}$. This results in the optimization (12) subject to vector space (5.2) to output vector $\widetilde{\boldsymbol{\mathbf{v}}}_{i}(t)\in P_{i}(\mathcal{M})$ that has entries greater or equal to zero. Hence, according to the propagation and update rule (16) and (17), we can conclude that $\boldsymbol{\mathbf{x}}_{ii}(t)$ has increasing elements and only agent $i$ can update it and other agents only copy this value as $\hat{\boldsymbol{\mathbf{x}}}_{ji}(t)$. Therefor, we can conclude that $\hat{x}_{jip}(t)\leq x_{iip}(t)$ for all $p\in\mathcal{P}_{i}$ which concludes the proof. $\Box$

• Proof of Proposition 5.1

  $f$ is a monotone increasing and submodular set function therefor $f(\mathcal{R}_{\boldsymbol{\mathbf{x}}_{i}(t)}\cup\{p\})-f(\mathcal{R}_{\boldsymbol{\mathbf{x}}_{i}(t)}\setminus\{p\})\geq 0$ and hence $\widetilde{\nabla}F(\boldsymbol{\mathbf{x}}_{i}(t))$ has positive entries $\forall i\in\mathcal{A}$. Then, because $\widetilde{\boldsymbol{\mathbf{v}}}_{j}(t)\in P_{j}(\mathcal{M})$, it follows from (12) that $\widetilde{\boldsymbol{\mathbf{v}}}_{j}(t)$ has non-negative entries, $\widetilde{{v}}_{jp}(t)\geq 0$, which satisfy $\sum_{p\in\mathcal{P}_{j}}\widetilde{{v}}_{jp}(t)=1$. Therefore, it follows from (16) and Lemma 5.1 that

  $\displaystyle\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{jj}(t+1)=\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{jj}(t)+\frac{1}{T},\quad j\in\mathcal{A}.$

  (25)

  Using (25), we can also write

  $\displaystyle\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{jj}(t)=\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{jj}(t-d(\mathcal{G}))+\frac{1}{T}d(\mathcal{G}),\quad j\in\mathcal{A}.$

  (26)

  Furthermore, it follows from Lemma (5.1) that for all $\forall p\in\mathcal{P}_{j}$ and any $i\in\mathcal{A}\backslash\{j\}$ we can write

  $\displaystyle x_{jjp}(t)\geq\hat{x}_{ijp}(t).$

  (27)

  Also, since every agent $i\in\mathcal{A}\backslash\{j\}$ can be reached from agent $j\in\mathcal{A}$ at most in $d(\mathcal{G})$ hops, it follows from the propagation and update laws (16) and (17), for all $\forall p\in\mathcal{P}_{j}$, for any $i\in\mathcal{A}\backslash\{j\}$ that

  $\displaystyle\hat{x}_{ijp}(t)\geq x_{jjp}(t-d(\mathcal{G})).$

  (28)

  Thus, for any $j\in\mathcal{A}$ and $i\in\mathcal{A}\backslash\{j\}$, (27) and (28) result in

  $\displaystyle\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{jj}(t)\geq\boldsymbol{\mathbf{1}}.\hat{\boldsymbol{\mathbf{x}}}_{ij}(t)\geq\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{jj}(t-d(\mathcal{G})).$

  (29)

  Next, we can use (26) and (29) to write

  $\displaystyle\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{jj}(t)\geq\boldsymbol{\mathbf{1}}.\hat{\boldsymbol{\mathbf{x}}}_{ij}(t)\geq\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{jj}(t)-\frac{1}{T}d(\mathcal{G}),$

  (30)

  for $j\in\mathcal{A}$ and $i\in\mathcal{A}\backslash\{j\}$. Using (30) for any $i\in\mathcal{A}$ we can write

  $\displaystyle\sum_{j\in\mathcal{A}}\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{jj}(t)\geq\boldsymbol{\mathbf{x}}_{ii}(t)+\sum_{j\in\mathcal{A}\backslash\{i\}}\boldsymbol{\mathbf{1}}.\hat{\boldsymbol{\mathbf{x}}}_{ij}(t)\geq\sum_{j\in\mathcal{A}}\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{jj}(t)-\frac{1}{T}Nd(\mathcal{G}).$

  (31)

  Then, using Lemma 5.1, from (31) we can write

  $\displaystyle\boldsymbol{\mathbf{1}}.\bar{\boldsymbol{\mathbf{x}}}(t)\geq\boldsymbol{\mathbf{1}}.\boldsymbol{\mathbf{x}}_{i}(t)\geq\boldsymbol{\mathbf{1}}.\bar{\boldsymbol{\mathbf{x}}}(t)-\frac{1}{T}Nd(\mathcal{G}),$

  which ascertains (19a). Next, note that from Lemma 5.1, we have $\boldsymbol{\mathbf{x}}_{jj}(t)=\boldsymbol{\mathbf{x}}_{jj}^{-}(t)$ for any $j\in\mathcal{A}$. Then, using (16) and invoking Lemma 5.1, we obtain (19b),which, given (25), also ascertains (19c). $\Box$