1st-Order Dynamics on Nonlinear Agents for ResourceAllocation over Uniformly-Connected Networks

Mohammadreza Doostmohammadian, Alireza Aghasi, Themistoklis Charalambous,

Abstract— In this paper, a general nonlinear 1st-orderconsensus-based solution for distributed constrained convexoptimization is considered for applications in network resourceallocation. The proposed continuous-time solution is used to op-timize continuously-differentiable strictly convex cost functionsover weakly-connected undirected multi-agent networks. Thesolution is anytime feasible and models various nonlinearitiesto account for imperfections and constraints on the (physicalmodel of) agents in terms of their limited actuation capabilities,e.g., quantization and saturation constraints among others.Moreover, different applications impose specific nonlinearitiesto the model, e.g., convergence in fixed/finite-time, robustnessto uncertainties, and noise-tolerant dynamics. Our proposeddistributed resource allocation protocol generalizes such non-linear models. Putting convex set analysis together with theLyapunov theorem, we provide a general technique to proveconvergence (i) regardless of the particular type of nonlinearity(ii) with weak network-connectivity requirement (i.e., uniform-connectivity). We simulate the performance of the protocolin continuous-time coordination of generators, known as theeconomic dispatch problem (EDP).

Keywords: Network resource allocation, graph theory, span-ning tree, convex optimization.

I. INTRODUCTION

Consensus [1], [2], [3], [4], [5], [6], [7], [8], [9] hasbeen infiltrated into different signal processing, control, andmachine learning literature with applications in distributedestimation [10], [11], optimization [12], [13], [14], andresource allocation [15], [16]. Distributed resource allocation(or network resource allocation), as the focus of this paper, isthe problem of allocating a constrained amount of resources(or utilities) among a fixed group of agents to minimize thetotal cost of resources. This arises, for example, in distributedeconomic dispatch over smart grids [17], distributed coveragecontrol/allocation [18], [19], [20], congestion control in data-sharing networks [21], and distributed load balancing inedge/fog computing [22]. In general, the distributed resourceallocation might be subject to physical constraints on theagents, communications, and sensors/actuators, leading tovarious nonlinearities in the system dynamics and affectingits stability/convergence. In this direction, this work for-mulates a general nonlinear solution to account for suchconstraints in the model.

Related literature: The literature on network resourceallocation spans from preliminary linear [15], [16] andaccelerated linear [23] solutions to more recent sign-basedconsensus [24], Newton-based [25], derivative-free swarm-based [26], Lagrangian-based [27], predictive online saddle-point method [28], 2nd-order autonomous dynamics [29],

[30], [31], [32], distributed mechanism over local message-passing networks [33], multi-objective [34], and projectedproximal sub-gradient algorithm [35] among others. None ofthe existing works cover the general nonlinear dynamics thatcan be accommodated with different nonlinear constraintsdue to, e.g., uncertain computation and constrained actuationcapacity of agents. Such imperfections may significantlyaffect the convergence or degrade the performance of theaforementioned resource allocation.

Contributions: In this work, we propose a general 1st-order consensus-based dynamics to solve the network re-source allocation problem. The proposed distributed solutiongeneralizes many nonlinear constraints on the agents includ-ing, but not limited to, (i) saturation and (ii) quantization.Further, some specific constraints, e.g., on the convergenceor robustness, impose nonlinearities on the agents’ dynamics.For example, it might be practical in application to design(iii) fixed-time and finite-time convergent solutions, and/or(iv) robust protocols to impulsive noise and uncertainties.Our proposed dynamics generalizes many similar symmetricsign-preserving nonlinearities, and therefore, suits many real-world resource allocation applications subject to, e.g., thementioned nonlinear contributions (i)-(iv) or their composi-tions. Using convex analysis and Lyapunov theorem theory,the uniqueness of the optimal solution, anytime feasibility,and convergence of the general dynamics is proved oversparse time-varying undirected networks which are not nec-essarily always connected (referred as uniform-connectivity).The objective function is not needed to be necessarily twice-differentiable, which allows for incorporating smooth exactpenalty functions to address the so-called box constraints onthe agents’ states. The main contribution and challenge is toprove convergence (to the optimal allocation), irrespectiveof the type of nonlinearity. Putting together concepts fromconvex sets and Lyapunov stability, we prove convergence forgeneral strongly sign-preserving and odd nonlinear models.This generalized 1st-order solution can be adopted for prac-tical resource allocation by considering physical constraintson the agents over sparse dynamic networks. To our bestknowledge, the literature provides no such general solutionto serve various nonlinearities.

Outline: Section II formulates the problem. Section III in-troduces the related definitions and lemmas. Our continuous-time solution is proposed in Section IV with convergenceanalysis in Section V. Simulations are given in Sections VIand VII. Finally, Section VIII concludes the paper.

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II. PROBLEM STATEMENT

The network resource allocation problem is in the form1,

minX

F (X, t) =

n∑i=1

fi(xi, t), s.t. Xa = b (1)

with xi ∈ Rd, X = [x1, . . . ,xn] ∈ Rd×n, column vectorsa = [a1; . . . ; an] ∈ Rn, and b = [b1; . . . ; bd] ∈ Rd. Also,fi(xi, t) : Rd+1 → R represents the local time-varying costfunction at agent i in the form fi(xi, t) = fi(xi) + fi(t)where, in general, fi(t) 6= 0 represents the time-varying partof the objective. In some applications, the states are subjectto the so-called box constraints, m 4 xi 4 m, where ”4”is an element-wise comparison operator implying that forevery entry p ∈ {1, . . . , d} we have mp ≤ xi,p ≤ mp

(xi,p denotes the pth entry of xi). Using exact penaltyfunctions [36], one can incorporate these constraints into thelocal objective functions, i.e., to modify the objectives asf εi (xi, t) = fi(xi, t) + εhε(xi − m) + εhε(m − xi) withhε(u) = max{u,0}. The smooth equivalents 1

µ log(1 +

exp(µu)) [14] or quadratic penalty (max{u,0})2 [37] canbe used as well, where the gap between the smooth and exactpenalties inversely scales with ε [38]. Note that problem (1)differs from unconstrained distributed optimization [12], [13]as it further needs the constraint Xa = b to be feasible.

Assumption 1: The (time-independent part of) local func-tions, fi(xi) : Rd → R, are strictly convex and differentiable.This assumption ensures a unique optimizer (see Lemma 2)and existence of the function gradient. This paper aimsto design a general nonlinear dynamic to solve (1) in adistributed way over a multi-agent network, where eachagent’s dynamics is based on local information on its ownobjective function and the data received from its directneighbors. The proposed dynamics captures many possiblenonlinearities on the agents’ model and can be extended todifferent nonlinear protocols for various purposes, some ofwhich are discussed in Section IV.

III. DEFINITIONS AND AUXILIARY RESULTS

A. Preliminaries on Graph Theory

The multi-agent network is modeled by an undirectedgraph G(t) = {V, E(t)} with (possibly time-varying) set oflinks E(t) and (time-invariant) set of nodes V = {1, . . . , n}.A link (i, j) ∈ E(t) represents the connection from agent i toj, and the set Ni(t) = {j|(j, i) ∈ E(t)} represents the directneighbors of agent i over G(t). Every link (i, j) ∈ E(t) isassigned with a positive weight Wij > 0, in the associatedweight matrix W (t) = [Wij(t)] ∈ Rn×n≥0 of G(t). In G(t)define a spanning tree as a subset of links in which thereis only one path between every two nodes (covering all nnodes).

Assumption 2: The following assumptions hold on G(t):

1Note the subtle abuse of notation where the overall state X is representedin matrix form to simplify the notation in proof analysis throughout thepaper.

• The network G(t) is undirected. This implies a sym-metric associated weight matrix W (t), i.e., Wij(t) =Wji(t) ≥ 0 for i, j ∈ {1, . . . , n} at all time t ≥ 0.

• There exist a sequence of non-overlapping finite time-intervals [tk, tk + lk] in which

⋃tk+lkt=tk

G(t) includes anundirected spanning tree (uniform-connectivity).

Unlike many works, we do not require W to be row, column,or doubly stochastic. The link weights only need to bepositive.

B. Preliminary Results on Convex Optimization

Following the Karush-Kuhn-Tucker (KKT) condition andLagrange multipliers method [36], optimal solution to prob-lem (1) satisfies the feasibility condition as described below.

Definition 1: (Feasibility Condition) Define Sb = {X ∈Rd×n|Xa = b} as the feasible set and X ∈ Sb as a feasiblevalue for X.

Lemma 1: Problem (1) under Assumption 1 has a uniqueoptimal feasible solution X∗ ∈ Sb as ∇F (X∗) = ϕϕϕ∗ ⊗a>, with ϕϕϕ∗ ∈ Rd, F (X) =

∑ni=1 fi(xi), ∇F (X∗) =

[∇f1(x∗1), . . . ,∇fn(x∗n)] as the gradient (with respect to X)of the function F at X∗, and ⊗ as the Kronecker product.

Proof: The proof follows [39] by using KKT method-ology with respect to a as the gradient of the constraint.

In the following, we analyze the solution for every feasibleset. First, recall the concept of level sets. Given a functionh(X) : Rd×n → R, the level set Lγ(h) for a given γ ∈ R isthe set Lγ(h) = {X ∈ Rd×n|h(X) ≤ γ}. It can be shownthat for a strictly convex function h(X), all its level setsLγ(h) are strictly convex for all scalars γ [36]. Further, fortwo distinct points X and Y with h(X) > h(Y) on twolevel sets Lγ1(h) and Lγ2(h) with γ1 = h(X), γ2 = h(Y)we have [40],

e>p ∇h(X)(Y −X)>ep > 0. (2)

with ep as the unit vector of the p’s coordinate (p ∈{1, . . . , d}). This is used in the proof of the following lemma.

Lemma 2: For every feasible set Sb there is only oneunique point X∗ ∈ Sb (under Assumption 1) such that∇F (X∗) = Λ⊗ a> with Λ ∈ Rd.

Proof: From strict convexity of F (X) (Assumption 1),only one of its strict convex level sets, say Lγ(F ), touches (isadjacent) the constraint facet Sb, where the touching happensonly at a single point, say X∗. Clearly, the gradient ∇F (X∗)

is orthogonal to Sb, and ∇fi(x∗i )

ai=∇fj(x∗j )

aj= Λ for all i.

Now, by contradiction consider two points X∗1,X∗2 ∈ Sbfor which ∇F (X∗1) = Λ1 ⊗ a> and ∇F (X∗2) = Λ2 ⊗ a>

(two possible optimum), implying that either (i) one levelset Lγ(F ), γ = F (X∗1) = F (X∗2) is adjacent to the affineconstraint Sb at both X∗1,X∗2, or (ii) there are two levelsets Lγ1(F ), γ1 = F (X∗1) and Lγ2(F ), γ2 = F (X∗2),each touching the affine set Sb at X∗1 and X∗2 respectively.Since Sb forms a linear facet, the former case contradictsthe strict convexity of the level sets. In the latter case,

e>p ∇F (X∗2)(X∗1 −X∗2)>ep = 0,∀p (3)

Assume F (X∗2) > F (X∗1) for the two level sets. From (2)e>p ∇F (X∗2)(X∗1 − X∗2)>ep > 0, which contradicts (3).

This proof analysis is further recalled in the next sections.

IV. THE PROPOSED 1ST-ORDER NONLINEAR DYNAMICS

In many applications (e.g., generator coordination) theagents’ states (e.g., the generated power) evolves continu-ously in time [16]. We propose a 1st-order continuous-timeprotocol coupling the agents’ dynamics to solve problem(1), while addressing model nonlinearities and satisfyingfeasibility condition at all times,

xi = − 1

ai

∑j∈Ni

Wijg(∇fi(xi)

ai− ∇fj(xj)

aj

), (4)

with Wij as the weight of the link between agents i andj and ∇fi(xi) as the gradient of (time-invariant part of)the local objective fi with respect to xi. Recall that thetime-varying and time-invariant parts of the local objectivesare decoupled. Dynamics (4) represents a 1st-order weightedgradient tracking, with no use of the Hessian matrix, Thus,function fi(·) is not needed to be twice-differentiable (incontrast to 2nd-order dynamics, e.g., in [29]). In case of com-munication network among agents, periodic communicationwith sufficiently small period τ is considered, see [41] fordetails. The state of every agent i evolves under influenceof its direct neighbors j ∈ Ni weighted by Wji, e.g,via information sharing networks [41] where every agent ishares its local gradients ∇fi(xi) along with the weight Wji.Therefore, the proposed resource allocation dynamics (4) isonly based on local information-update, and is distributedover the multi-agent network.

Assumption 3: (Strongly sign-preserving nonlinearity)In dynamics (4), g : Rd → Rd is a nonlinear odd mappingsuch that g(x) = −g(−x), g(x) � 0 for x � 0, g(0) = 0,and g(x) ≺ 0 for x ≺ 0. Further, ∇g(0) 6= 0.

Some causes of such nonlinearities in practical applica-tions, e.g., physics-based nonlinearities, are given next.

Application 1: The nonlinear function g(·) in (4) canbe adopted from finite-time and fixed-time literature [1],[2], [42]. These protocols are based on the odd functionsgnµ(x) = x‖x‖µ−1, where ‖ · ‖ denotes the Euclideannorm and µ ≥ 0. Recall that system dynamics in the fromxi = −

∑nj=1Wij(sgnµ1(xi − xj) + sgnµ2(xi − xj)) are

known to converge in finite/fixed-time [2]. This motivatesthe following fast-convergent allocation dynamics [42], [43],

xi =−∑j∈Ni

Wij(sgnµ1(z) + sgnµ2(z)), (5)

with z = ∇fi(xi)ai

− ∇fj(xj)aj, 0 < µ1 < 1, and 0 < µ2 < 1

(finite-time case) or 1 < µ2 (fixed-time case).Application 2: Nearly all digital signal processing appli-

cations are involved with quantization, which is the processof representing a signal in digital form (known as quantized

value) [3], [4], [5]. This motivates quantized resource allo-cation via choosing the function g(·) in (4) as,

gl(z) = sgn(z) exp(gu(log(|z|))), (6)

where gu(z) = δ[zδ

]represents the uniform quantizer with

[·] as rounding operation to the nearest integer. Functionsgn(·) follows sgnµ(·) with µ = 0, and δ is the quantizationlevel. The function gl represents the logarithmic quantizer.

Application 3: The other application is in consideringrobustified sign-preserving nonlinearities [6], [7] to makethe dynamics (4) robust to impulsive noise (i.e., noise ofgenerally low nominal-value with high intensity impulse-likeoutliers). In this case, the optimal choice for g(·) depends onthe noise density p as gp(z) = −d(log p(z))dz . For example, incase the noise density p follows from approximately uniformclass P1 or Laplace class P2, the function gp(·) is [7],

p ∈ P1 : gp(z) =

{1−εεd sgn(z) |z| > d

0 |z| ≤ d(7)

p ∈ P2 : gp(z) = 2εsgn(z), (8)

with 0 < ε < 1, d > 0. The first function represents a relaywith dead-zone and the latter is a weighted sign function.

Application 4: Saturation nolinearities [8], [9] (alsoknown as clipping) are typically due to, e.g., data rangeconstraints and limited range of analog/digital signal trans-formation on the sensors/actuators (e.g., in the presence ofovershoots/undershoots). It is known that the saturation levelaffects the stability, convergence, and general behavior of thedynamical systems. For a given saturation level κ > 0, thesaturated version of the dynamics (4) is associated with thefollowing function,

gκ(z) =

{κsgn(z) |z| > κ

z |z| ≤ κ(9)

V. ANALYSIS OF CONVERGENCE

In this section, combining convex analysis from Lemma 1-2 with Lyapunov theory, we prove the convergence of thegeneral protocol (4) to the optimal value of problem (1)subject to the constraint on the weighted-sum of resources.The proof is, in general, irrespective of the nonlinearity types,i.e., holds for any nonlinearity satisfying Assumption 3,including (5)-(9).

Lemma 3: (Anytime Feasibility) Suppose Assumption 3holds. The states of the agents under dynamics (4) remainfeasible, i.e., if X(0) ∈ Sb, then X(t) ∈ Sb for t > 0.

Proof: Having X(0) ∈ Sb implies that X(0)a = b.For the general state dynamics (4),

d

dt(Xa) =

n∑i=1

xiai = −n∑

i=1

∑j∈Ni

Wijg(∇fi(xi)

ai− ∇fj(xj)

aj

).

(10)

From Assumptions 2 and 3, Wij = Wji and g(−x) =−g(x). Therefore, the summation in (10) is equal to zero,ddt (Xa) = 0, and Xa is time-invariant under dynamics(4). Thus, having feasible initial states X(0)a = b, then

X(t)a = b remains feasible over time, i.e. X(t) ∈ Sb fort > 0.

The above lemma proves anytime feasibility, i.e., thesolutions under dynamic (4) remain feasible at all times.

Theorem 1: (Equilibrium-Uniqueness) Under Assump-tions 2 and 3, the equilibrium point X∗ of the solutiondynamics (4) is only in the form ∇F (X∗) = Λ ⊗ a> withΛ ∈ Rd, and coincides with the unique optimal point of (1).

Proof: From dynamics (4), x∗i = 0,∀i for X∗ satisfying∇F (X∗) = Λ ⊗ a>, and such point X∗ is clearly anequilibrium of (4). We prove that there is no other equi-librium with ∇F (X∗) 6= Λ⊗ a> by contradiction. AssumeX as the equilibrium of (4) such that ∇fi(xi)ai

6= ∇fj(xj)aj

for at least two agents i, j. Let ∇F (X) = (Λ1, . . . , Λn).Consider two agents α = argmaxq∈{1,...,n} Λq,p and β =

argminq∈{1,...,n} Λq,p for any entry p ∈ {1, . . . , d}. Fol-lowing the Assumption 2, the existence of an (undirected)spanning tree in the union network

⋃tk+lkt=tk

G(t) implies thatthere is a mutual path between nodes (agents) α and β.In this path, there exists at least two agents α and β forwhich Λα,p ≥ ΛNα,p, Λβ,p ≤ ΛNβ ,p with Nα and Nβ asthe neighbors of α and β, respectively. The strict inequalityholds for at least one neighboring node in Nα and Nβ . FromAssumption 2 and 3, in a sub-domain of [tk, tk+lk], we have˙xα,p < 0 and ˙xβ,p > 0. Therefore, ˙

X 6= 0 which contradictsthe assumption that X is the equilibrium of (4). Recall that,from Lemma 2, this point coincides with the optimal solutionof (1), as for every feasible initialization in Sb there is onlyone such point X∗ satisfying ∇F (X∗) = Λ ⊗ a>. Thiscompletes the proof.

The above lemma paves the way for convergence analysisvia Lyapunov stability theorem, as it shows that the dynamics(4) has a unique equilibrium for any feasible initial condi-tion.

Lemma 4: [42, Lemma 3] Consider nonlinearity g(·) andmatrix W satisfying Assumptions 2 and 3. Then, for ψψψ ∈ Rdwe have,

n∑i=1

ψψψ>i

n∑j=1

Wijg(ψψψj −ψψψi) =

n∑i,j=1

Wij

2(ψψψj −ψψψi)

>g(ψψψj −ψψψi).

Following the convex analysis in Lemmas 2-3, and Theo-rem 1 along with Lemma 4, we provide our main theoremnext.

Theorem 2: (Convergence) Suppose Assumptions 1-3hold. Then, initializing by X(0) ∈ Sb, the proposed dy-namics (4) solves the network resource allocation problem(1).

Proof: Following Lemmas 2, 3, and Theorem 1 andinitializing from a feasible state X(0) ∈ Sb, there is a uniquefeasible equilibrium X∗ for solution dynamics (4) in the form∇F (X∗) = ϕϕϕ∗ ⊗ a>. Define the Lyapunov function,

F (X) = F (X, t)−F (X∗, t) =

n∑i=1

(fi(xi)+fi(t))−(fi(x∗i )+fi(t)).

Clearly, F (X) =∑ni=1(fi(xi) − fi(x

∗i )) > 0 is purely a

function of X, with X∗ is the unique equilibrium of F (X)

Fig. 1. (Left) This figure shows the time-variation of the multi-agentnetwork. None of the 4 networks contains a spanning tree (not connected),while their union (Right) is connected.

and F (X∗) = 0. We have,

F (X) =

n∑i=1

∇fi(xi)>xi

=

n∑i=1

−∇fi(xi)

ai

> ∑j∈Ni

Wijg(∇fi(xi)

ai− ∇fj(xj)

aj

)).

Following Lemma 4,

F (X) = −n∑

i,j=1

Wij

2

(∇fi(xi)ai

−∇fj(xj)aj

)>g(∇fi(xi)

ai−∇fj(xj)aj

).

From Assumption 3, g(x) is odd and strongly sign-preserving, i.e., x>g(x) ≥ 0. Therefore, F (X) ≤ 0 where,

F (X∗) = 0 ⇐⇒ ∇fi(x∗i )ai

=∇fj(x∗j )

aj= ϕϕϕ∗, i, j ∈ {1, ..., n}

Recall that the invariance set I = {X∗} includes a uniqueequilibrium point X∗ for every feasible set Sb (Theorem 1)and F is negative-definite ∀X /∈ I. Thus, X∗ is globallyasymptotically stable and agents’ states under dynamics (4)converge to X∗.

VI. SIMULATION OVER SPARSE NETWORKS:QUANTIZED ACTUATION AND SATURATED ACTUATION

In this section, we simulate protocol (4) for (i) quan-tized and (ii) saturated resource allocation over weakly-connected multi-agent networks. We consider Erdos-Renyi(ER) networks of n = 100 agents as shown in Fig. 1 withrandom symmetric weights. Every 0.1 second the networkswitches between the 4 graphs Gs with switching commands : d10t − 4b2.5tce. The Fiedler-values of all 4 graphs are0, implying dis-connectivity, while the Fiedler-value of theirunion

⋃tk+0.4t=tk

G(t) (Fig. 1-(Right)) is 6.167 > 0, implyingthat Assumption 2 holds. We consider strictly convex localcost function at agent i as [39],

fi(xi) =

4∑j=1

ai,j(xi,j − ci,j)2

+ log(1 + exp(bi,j(xi,j − di,j))),fi(t) =

∑4j=1 ei,j sin(αi,jt+ φi,j)

(11)

with parameters chosen randomly. The resource allocationconstraint is Xa = b = [10; 10; 10; 10] with ai randomlychosen in [0.1, 1]. To solve distributed resource allocation (1)

0 5 10 15 2060

80

100 Optimal valueCost function

0 5 10 15 200

20

40

Fig. 2. The time-evolution of (Top) the cost function versus the time-varying optimal value and (Bottom) the associated Lyapunov function forquantized resource allocation over switching networks in Fig. 1.

0 5 10 15 2040

60

80

100

120Optimal valueCost function

0 5 10 15 200

20

40

Fig. 3. The time-evolution of (Top) the cost function versus the time-varying optimal value and (Bottom) the associated Lyapunov function forsaturated resource allocation over the same switching network topology.

we consider two dynamics in the form (4) to accommodate(i) quantized actuation via the logarithmic quantizer (6) withδ = 1, and (ii) saturated actuation (9) with κ = 1. Thetime-evolution of the cost function (11) and its associatedLyapunov function F (X) = F (X, t) − F ∗(t) are shown inFig. 2 and Fig. 3, respectively, for case (i) and (ii) withrandom cost parameters in (11) for each case. As it is clear,the cost functions converge to the optimal (time-varying)values, while the Lyapunov functions (residuals) decreaseover time.

VII. APPLICATION TO ECONOMIC DISPATCH PROBLEM

The smart grid commonly consists of many power gen-erators with continuous-time dynamics. The EDP is an op-timization problem to find an output combination of thesepower generators to reach minimum operating cost, whilesatisfying the load demand constraint [16], [30], [44], [45],[46], [47], [48], [17]. Parameter xi represents the amountof power assigned to be produced by the generator i. Thetotal operating cost is the sum of local costs in the following

0 20 40 60 80 100 120 140 1606.6

6.8

7

7.2

7.4

7.6

7.8

8103

linearsign-basedaccelerated linearinitialization-free

finite-timeThis work: fixed-timeThis work: robustifiedOptimal cost

5 10 15 20 25 30

7300

7320

7340

7360

0 20 40 60 80 100 120 140 16040

50

60

70

80

90

100

110

Fig. 4. (Top) This figure shows the performance of the dynamics (5)(solid black) and the robustified dynamics (4) via (8) (dashed blue) to solvethe distributed economic dispatch problem in comparison with some recentliterature. (Bottom) The time-variation of the states x under dynamics (5)are shown, which represent the amount of power in MW allocated to thegenerators over time. The average of generated powers at all generators isfixed over time (shown by dashed black line), implying that the feasibilitycondition is met at all times.

quadratic form [17]:

F (x) =

n∑i=1

γix2i + βixi + αi, s.t.

n∑i=1

xi = D (12)

and the total load demand constraint is equal to the totalpower produced at the n generators represented by thefixed quantity D. Following the problem formulation (1),we have 40 ≤ xi ∈ R ≤ 110 (d = 1), ai = 1, and b = D.Assume n = 10 power generators under the supply demandconstraint D = 800 MW . Initially, the allocated power atall the generators is equal to D

n = 80 MW . The parametersof the quadratic cost function (12) depend on the type ofpower generators (coal-fired, oil-fired, etc.) [45] and arechosen randomly such that αi ∈ [100, 600], βi ∈ [2, 5],and γi ∈ [0.02, 0.05]. We apply (i) the dynamics (5) withµ1 = 0.5, µ2 = 1.5 and (ii) the robustified version of thedynamics (4) via gp(·) in (8) with ε = 0.4 to optimallyallocate power to these generators. The coordination networkamong the generators (agents) is a cycle with random linkweights in (0, 1]. We compare the results with linear [39],sign-based consensus [24], accelerated linear [23], finite-time[46], and initialization-free [48] protocols in Fig. 4. The time-evolution of the allocated power to the generators under the

proposed dynamics (5) is shown in Fig. 4-(Bottom). Clearly,from the figure, all the power generators have reached stablestates (while satisfying the feasibility condition) for whichthe operating cost is optimal according to Fig. 4-(Top).

VIII. CONCLUSION

This paper proposes general nonlinear-constrained solu-tions for resource allocation over uniformly-connected net-works. The dynamical nonlinearity generalizes many scenar-ios, including (i) fixed/finite-time dynamics, (ii) quantizedactuation, (iii) robustified (to noise) dynamics, and (iv)saturated actuation. The proposed solution can solve the allo-cation problem subject to a composition of the nonlinearities(i)-(iv) (as their compositions are also odd and strongly sign-preserving mappings) and any other nonlinearity satisfyingAssumption 3.

As a direction of future research, one can extend the 1st-order nonlinear actuation constraint in this work to nonlinearcommunication and/or node-based constraints (see [3]) or to2nd-order nonlinear dynamics. Note that for the 2nd-ordercase, the objective function needs to be twice-differentiable.Further, Assumption 1 can be relaxed to the convex case,which requires more complex analysis on the uniqueness ofthe solution. Possible applications in (i) distributed dynamicoffloading control for edge computing, and (ii) networkcongestion control are also of interest.

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