Distributed Interval Optimization Over Time-Varying Networks: A Numerical Programming Perspective

In this study, we investigate a distributed interval optimization problem involving agents linked by a time-varying network, optimizing interval objective functions under global convex constraints. Through scalarization, we first reformulate the distributed interval optimization problem as a distributed constrained optimization problem. The optimal Pareto solutions to the reformulated problem are then illustrated. We establish a distributed subgradient-free algorithm for the distributed constrained optimization problems by generating random differences of reformulated optimal objective functions, and the optimal solutions of the distributed constrained optimization problem are equivalent to Pareto optimal solutions of the distributed interval optimization problem. Moreover, we demonstrate that a Pareto optimal solution can be reached over the time-varying network using the proposed algorithm almost surely. FInally, we conclude with a numerical simulation to demonstrate the effectiveness of the proposed algorithm.


I. INTRODUCTION
Recently, distributed optimization and control in a network environment have attracted a growing amount of interest, as they are more effective than centralized designs for many large-scale problems when agents only have access to local information and exchange data with their neighbours over the network.In fact, distributed algorithms for a variety of (constrained) optimization problems have been extensively studied, with potential applications to sensor networks, smart grids, and equation solutions (see [1], [2], [3], [4], [5], [6], [7], [8]).Note that connectivity is an important aspect of distributed design.Although fixed topologies are still required for distributed optimization designs in certain The associate editor coordinating the review of this manuscript and approving it for publication was Xiwang Dong.circumstances, time-varying jointly connected networks have been considered in a number of algorithms, including [1], [3], [7], and [9].
Nonetheless, objective functions and constraints of some practical optimization problems may not be described precisely or explicitly.For instance, some conditions in power systems may be time-varying or uncertain, and data mining can produce inaccurate results (see [10], [11], and [12]).Motivated by the present setting, interval optimization is investigated in [13], [14], and [15], which provides a framework for capturing the uncertainty in optimization.In fact, interval optimization problems (IOP), which were first proposed by [13] and further studied in [14] and [15] and references thereto, have been extensively studied in numerous research fields, including economic systems [16] and smart grids [17].Objective functions in an interval optimization problem are intervalvalued, i.e., they are described by intervals rather than real integers, via interval-valued maps.The well-defined partial orderings and convexity of interval-valued maps [16], [18], [19] guarantee the existence of solutions for maximization and minimization of interval optimization problems.Literature (referring to [20], [21], [22], [23]) has provided various programming methods, particularly based on Wolfe's method or Lamke's algorithm (Lingo software provides different algorithm boxes to solve linear programming problems), to address centralized interval optimization problems.
With this background, it is nature for us to consider the construction of effective algorithms for distributed interval optimization problems over multi-agent networks.However, the distributed interval optimization problems are still under investigation.It may be because it is so simple to distribute the ideas of Wolfe or Lamke's algorithms, and very few papers with related theoretical results have been published on the subject of [24].A further reason is that the partial order resulting from the interval makes gradient-based methods challenging, particularly when only local information is available in a distributed design.
In systems and control, randomization and stochastic methods have proven to be effective instruments.In contrast to traditional robustness methods [25], stochastic gradients are advantageous for controlling uncertain systems, for instance.When developing distributed algorithms, for instance [26], randomization and stochastic methods can enhance the system's overall performance as they are quite natural for the study of network dynamics and have a close relationship with real-world systems.
The purpose of this paper is to propose a distributed algorithm for interval optimization problems, based on recent results on distributed subgradient-free algorithms that overcome the challenge of obtaining subgradient information of local interval-valued functions.Zeroth-order/subgradientfree algorithms have been extensively researched in [27], [28], [29], [30], and [31] and references listed therein due to their applications in situations where obtaining gradient/subgradient information is computationally expensive or even impractical.Therefore, we propose a subgradient-free stochastic algorithm for a class of interval optimization problems based on numerical programming.The contributions of this paper are summarized as follows: (a) In keeping with the rapid evolution of data science and engineering systems, we extend the centralized interval optimization problem [14], [15] to a distributed setting.Through well-defined partial orderings and convexity [16], [18], [19] of the distributed interval optimization problem, we transform it into a solvable distributed optimization problem with convex global constraints.The optimal solutions of the distributed constrained optimization problem are equivalent to Pareto optimal solutions of the distributed interval optimization problem in this reformulation.Wolfe's and Lamke's methods cannot be easily extended to distributed versions; therefore, we employ zeroth-order or subgradient-free ideas in the distributed design.(b) We develop a new distributed subgradient-free algorithm with random differences for solving the reformulated distributed constrained non-smooth optimization problem because it is challenging to acquire the subgradient of the interval optimization problem.The algorithm employs random differences to approximate subgradients of locally reformulated objective functions, which is distinct from many existing distributed gradient/subgradient-free algorithms see, [32], [33], [34], and [35] though consistent with them when the local objective function is smooth.(c) The proposed algorithm, which is related to the distributed stochastic optimization algorithm, is subjected to both theoretical and numerical analysis.We establish the mean-square convergence rate of O( 1

√ k
). after establishing the consensus of estimates and accomplishment of global minimization with probability one with the proposed algorithm.With decreasing step-size, the convergence results match the best of the first-order stochastic algorithms [4], [36], [37] The main contributions of this article could also be outlined in fig. 1.The remaining sections are organized as follows.In Section II, preliminary information regarding the analysis and design of distributed interval optimization is provided.The distributed interval optimization problem is then formulated and the corresponding distributed algorithm is presented in Section III, while Section IV analyses the proposed algorithm.The following numerical example is provided in Section V.The section VI concludes with some concluding remarks.
122542 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Definition 1 ([38] Sub-Gradient): Let f (x) : R p → R be a non-smooth convex function.Vector-valued function ▽f (x) ∈ ∂f (x) ⊂ R p is called the subgradient of f (x) if for any x, y ∈ dom(f ), the following inequality holds: Then, we summarize the Euclidean norm inequalities [3], [39] that will be used in this paper.
Lemma 2 [4]: Let x 1 , x 2 , . . ., x n be vectors in R p .Then Denote the projection of x onto set X by P X (x), i.e., P X (x) = arg min y∈X x − y , where X is a closed bounded convex set in R p .The following results are on the projection operators in Euclidean norm: Lemma 3 [3], [38]: Let X be a closed convex set in R p .Then for any x ∈ R p , it holds that (a) x − P X (x), y − P X (x) ⩽ 0, for all y ∈ X (b) P X (x) − P X (y)∥ ⩽ x − y , for all x, y ∈ R m .(c) x − y, P X (y) − P X (x) ⩽ − P X (x) − P X (y) 2 , for all 2 , for any y ∈ X .

B. PROBABILITY THEORY
Denote ( , F, P) as the probability space, where is the whole event space, F is the σ -algebra on , and P is the probability measure on ( , F).Then, definitions of convergence in ( , F, P) and convergence of super-martingales theorem is given.
Definition 3 [16], [18]: For any A, B ∈ P(R).denote: Definition 4 [16], [18]: For any A, B ∈ P(R), denote: Let G : R p ⇒ R be any interval-valued map, where G(•) is an interval with respect to x.We give the definition of Lipschitz continuity and convexity of the interval-valued map G : R p ⇒ R as follows: Definition 5 ( [42] Lipschitz Continuity): Let G : R p ⇒ R be any interval-valued map.G is locally Lipschitz at x if there exist K > 0 and a neighborhood W of x such that We say that G is locally Lipschitz at x if there exist a neighbourhood W of x and a constant K ⩾ 0, such that We denote by B(A, ϱ) = {y|d(y, A) ⩽ ϱ}, the ball of radius ϱ around subset A, where y is chosen from a metric space.

Then according to Definitions 3 and 4, L(•), R(•) : R p
→ R are convex functions with respect to x ∈ R p .Namely, for any x 1 , x 2 ∈ R p and any t ∈ [0, 1], the following inequalities hold:

2) INTERVAL OPTIMIZATION PROBLEMS WITH INTERVAL-VALUED MAP
Let G : R p ⇒ R be any interval-valued map.Now we consider the following interval optimization problem:

where C k v stands for kdimensinal column vecter whose elements are vectors. For the given interval vector
In light of definitions of L(x) and R(x) of the example, we see that we can not get the explicit expressions of L(x) and R(x), and this (IOP) problem can be solved through set-valued optimization rather than vector valued optimization.
Based on quasi orderings of compact intervals in C(R) given in Definitions 3 and 4, we define the Parato optimal solution to (IOP): Definition 7 [44]: A point x * ∈ is said to be a Pareto optimal solution (PO) to (IOP Remark 2: There is no solution to interval optimization problem given in fig. 2. However, [x 1 , x 2 ] are Pareto optimal solutions to this given problem.Associated with (IOP), consider the following interval optimazation problem with its scalarization: (SIOP) min The following lemma holds according to [44].We gave its proof here just for self-reminder.∈ is an optimal solution to (SIOP).Proof: (a) Given a real number λ ∈ (0, 1) and let x * ∈ be an optimal solution to (SIOP).Suppose that there exists a point x ∈ , such that G(x) ≦ G(x * ), which implies L(x) ⩽ L(x * ) and R(x) ⩽ R(x * ).Therefore, which contradicts that x * is an optimal solution to (SIOP).(b) Let x * ∈ be a Pareto optimal solution to (IOP).Since G is compact-valued and convex with respect to x, according to Remark 1, L(x) and U (x) are convex functions.Following Definition 7, there exists a vector which proves that there exists a real number λ ∈ [0, 1] such that x * ∈ is an optimal solution to (SIOP).

III. DISTRIBUTED INTERVAL OPTIMIZATION
Consider the following distributed interval optimization problem over an n-agent network: where and G i : R p ⇒ R is a compact and convex interval-valued function.In this setting, the state of an agent i is the estimate of solution to problem (DIOP).Each agent i knows local functions G i and global constraint X .We first make the following assumption on the local functions and constraints for the distributed interval optimal problem (DIOP): Consider solving (DIOP) over a time-varying multigenerator network.Define a directed network G(k) = N , E(k), W (k) as the communication topology among generators, where N = {1, 2, . . .n} is the agent set, the edge 122544 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
. Assumption 2 reveals that each agent i can periodically collect data from all its neighbours.It is also a common connectivity condition for time-varying distributed network designs (see [1], [3]).
Define the function f : where Remark 3: Note that both L(x) and R(x) are separable, that is We can convert the distributed interval optimization problem into a standard distributed optimization problem with scalar values.Let λ = λ 0 1 n with λ 0 ∈ (0, 1).To solve problem (1), the following distributed optimization problem is solved: where agent i is acquainted with the information of f i , x i , λ i ∈ (0, 1), and its vicinity.4) degenerates to a typical distributed constrained optimization problem [4] if each agent i chooses a common parameter λ or λ i s don't not need to be common.Due to the need for λ i = λ j via iterations, the reformulated problem is more challenging than typical distributed problems.Remark 5: According to Definitions 5-6 and Assumption 1, we have: (a) Each f i x, λ is convex with respect to x, i. e. for any x 1 , x 2 : e. for all x 1 , x 2 and λ: (d) Each f i x, λ is Lipschitz continuous with respect to λ i. e. for all λ 1 , λ 2 and x: Proof of (e): Suppose there exists a vector x, such that we can choose a subgradient ▽f i x (x, λ) ∈ ∂f i x (x, λ), where ∥▽f i x (x, λ)∥ > L. Suppose y = x + ▽f i x (x, λ), according to the definition of subgradient given in Definition 1, we have which contradicts the Lipschitz continuity of f i x, λ with respect to x.By an analogous proof, ∂f i λ (x, λ) ⩽ K .
, is an optimal solution to problem (4), then x * is a Pareto solution to problem (1).
Due to the fact that the differentiability of f (x, λ) with respect to x does not hold in general, we propose a Distributed (sub)gradient-free inteval-valued algorithm 1 for solving reformulated problem (4).
Calculation of local measurement d i (k) Descent Step: Projection Step: Average of local observations λ i (k): 6: end for where d i (k) is used as an estimate for ∂f i ϵ i (k) ξ i (k), λ i (k) .
, . . ., , where q i (k) k⩾0 , q = 1, 2, . . ., p, k = 1, 2, . . . is a sequence of mutually independent and identically distributed random variables with zero mean.The measurements y + i (k) and y − i (k) are given by is the σ -algebra created by the whole history of Distributed (sub)gradient-free inteval-valued algorithm (Algorithm 1) up to moment k (referring to [4]).We further make the following hypotheses on the dither signal i (k): Assumption 3: (a) For any fixed (i, q), let q i (k) k⩾0 be a sequence of independent and identically distributed (i.i. d.) random variables.For all k ⩾ 0 and for (b) For i ̸ = j or q ̸ = r, q i (k) k⩾0 and r j (k) k⩾0 are mutually independent of each other.Then, we introduce conditions on the step-size ι(k) of Algorithm 1 and c(k) used in the randomized differences (6): The chosen of unite parameter ι(k) c(k) satisfies the stochastic approximation stepsize condition in [27] and [36].

IV. PROPERTIES OF DISTRIBUTED (SUB)GRADIENT-FREE STOCHASTIC ALGORITHM
In this section, we first analyse that the estimate (x i (k), λ i (k)) converges to a consensus optimal point (x * , λ * ) almost surely of Algorithm 1.Then, the mean-square convergence rate of Algorithm 1 is provided.
Denote the transition matrix of The following lemma about (k, s) holds ture, given in Proposition 1 of [1].
First, we present a theorem regarding the proposed algorithm's convergence analysis.
Theorem 1: With Assumptions 1-4, (a) all the sequences {λ i (k)}, i ∈ N reach consensus (which is depended by initail parameters λ i (0) ′ s) almost surely by Algorithm 1.(b) all the sequences {x i (k)}, i ∈ N by Algorithm 1 converge to the same optimal point x * in consensus and almost surely.The proof of Theorem 1 relies on Lemmas 8-12.Lemma 8 provides an upper bound for the Euclidean norm of d i (k) in expection; Lemma 9 investigates the consensus in the L 1 norm of estimates x i (k) in Algorithm r1; Lemma 10 investigates the lower bound of the cross term of d i (k) and (ξ i (k) − x * )in expection and in conditional expection with respect to F(k), where x * is the optimal solution of (4) for fixed common point λ * ; Lemma 11 analyses that {x i (k)}, i ∈ N converge to the same random variable almost surely and Lemma 12 analyses that {x i (k)}, i ∈ N converge to x * in L 2 .The proofs for these lemmas can be found in the Appendix.
Lemma 8: Let Assumption 1 and 3 hold.Then the first order moments and second moments of d i (k) are bounded by where L is the Lipschitz constant with respect to x in Remark 5.
122546 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
(b) in expection as follows: where L is the Lipschitz constant with respect to x, K is the Lipschitz constant with respect to λ given in Remark 5, B is a positive constant.Lemma 11: With Assumptions 1-4, for fixed common point λ * , all the sequences {x i (k)}, i ∈ N converge to the same random variable consensusly and almost surely by Algorithm 1.
Proof of Theorem 1: (a) We prove that for i, j = 1, 2, . . ., n, According to the definition of transition matrix (k, s) and the definition of λ i (k + 1) in ( 9), we have: . According to Assumption 1 and by an analoglous induction, the following equality holds: Therefore, ∀i ∈ N , Plugging in the estimate of (k, s) in Lemma 7, we have = 0, which means that {x i (k)}, i ∈ N generated from Algorithm 1 converge to the optimal solution x * in L 2 .Therefore, Then we give the following convergence rate result of the proposed algorithm.
Proof: According to Lemma 12, we have Following (13) in the proof of Theorem 1, we obtain The proof is completed.Remark 6: It directly follows from Theorem 2 that the optimal values for ϵ and δ are ϵ = 1 2 and δ = 1 2 , respectively, , and Not only does the rate match the best rate for centralized stochastic approximation algorithms, see [36] and references therein, but it also matches the best rate given for first-order stochastic subgradient algorithms [37] with diminishing step-size..
Then, let us investigate the convergence performance of the distributed (sub)gradient-free interval-valued algorithm.The simulation results are based on a 5-agent time-varying network whose communication topology is depicted in Fig. 2. Both Fig. 4 and Fig. 5 illustrate the convergence performance of the proposed algorithm.For 500 iterations, we can attain a (0.500, 0.996) pareto solution.

VI. CONCLUSION
This paper investigated the problem of distributed interval optimization subject to local convex constraints.The objective functions for the distributed design are compact interval-valued functions, and the network is time-varying.A distributed subgradient-free methodology for finding a Pareto-optimal solution to a distributed interval optimization problem was developed by constructing random differences.In addition, we showed that a Pareto optimal solution can be achieved with probability one over a time-varying network and provided a numerical illustration of the algorithm's efficacy.

APPENDIX A PROOF OF LEMMA 8
According to the definition of d i (k) in (6), we have 122548 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where i (k) according to Remark 5. Due to Assumption 3(a), we have and

APPENDIX B PROOF OF LEMMA 9
For all i ∈ N , k ⩾ 0, define as the error between x i (k + 1) and ξ i (k).According to Lemma 5(b) and the fact that X is a closed, convex set, we get Rewrite ( 9) compactly in terms of (k, s) and the definition of p i (k + 1) as follows: (k, 0) ij x j (0) + p i (k + 1) for k ⩾ s.Define x(k + 1) = 1 n n i=1 x i (k + 1).Moreover, with Assumption 1(b) and by an analoglous induction, the following equality holds: Therefore, ∀i ∈ N , x j (0) Taking the expectation of ( 25), we get Plugging in the estimate of (k, s) in Lemma 7 and the estimate of p i (k + 1) in ( 22), we have From Theorem 1, we have E d i (k) ⩽ L. Therefore, According to Lemma 1: where θ i ∈ [−1, 1] is a constant.Therefore, there exists By taking conditional expectation of d i (k), ξ i (k) − x * with respect to F(k) and noticing (32), we obtain the following inequality: where We further formulate D i (k) as follows: By Definition 1 and Remark 5, we obtain and where B is a positive constant.Combining ( 35), (28) with (33) gives (b) By Definition 1 and Remark 5, we obtain Similar to the proof of part (a), we can get The proof of second part of Lemma 10 can be given by taking expection to both side of (39).

APPENDIX D PROOF OF LEMMA 11
(a) We prove that for i, j = 1, 2, . . ., n, which yields E lim inf 2 according to Lemma 2 and According to Assumption 2(b), Thus, taking the conditional expectation of both side of (41) yields According to Assumption 4 and Lemma 8(b) ⩽ ξi (k) − x * 2 according to the properties of Euclidean norm in Lemma 3.Then, Taking conditional expection of both sides of (44), we obtain for all k = 0, 1, 2, . .., By the double stochasticity of matrix W (k) in Assumption 2(b), Then, with probability 1, for i ∈ N , it holds where According to Assumption 4 and Lemma 8,

APPENDIX E PROOF OF LEMMA 12
By taking expectation to both sides of (44), we obtain Noticing that ι(s) = 1 s 1+ϵ , c(s) = 1 s δ , and 1 2 + ϵ > δ > 0. By Lemma 8, for the first term on the right hand side of (52), we have Since X is bounded in R m , for x ∈ X , there exists a constant M x such that x ⩽ M x .Still, λ ∈ [0, 1].For the terms on the right hand side of (52), we have where

FIGURE 1 .
FIGURE 1. Outline.Notations: Let R p be the p-dimensional Euclidean space.Denote R p + as its non-negative orthants.∥ • ∥ denotes the Euclidean norm.Denote the sets of all non-empty compact intervals of R by C(R).
and L(y) ⩽ L(x 2 ) according to Definition 7. Therefore, [x 1 , x 2 ] are Pareto optimal solutions to this given problem.

FIGURE 2 .
FIGURE 2. L(x) and R(x) for vector x.

Lemma 5 :
We assume that G is compact-valued and convex with respect to x: (a) If there exists a real number λ ∈ (0, 1) such that x * ∈ is an optimal solution to (SIOP), then x * ∈ is a Pareto optimization to (IOP).(b) A point x * ∈ is a Pareto optimization to (IOP), then there exists a real number λ ∈ [0, 1] such that x * set E(k) ⊂ N × N represents information communication at time k and W (k) = w ij (k) ij represents adjacency matrix at time k.Each agent interacts with its neighbors in G(k) = (N , E(k), W (k)) at time k.We make the following assumption about communication topology G(k) = N , E(k), W (k) .Assumption 2: The graph G(k) = N , E(k), W (k) satisfies: (a) There exists a constant η with 0 < η < 1 such that,

Remark 4 :
In Problem(4), We need λ i = λ j through the design of algorithms for the following reasons: (a) The aforementioned articles on interval optimization are centralized.This paper provides a framework for solving distributedly centralized interval optimization problems.(b) The distributed interval optimization problem may also be applied to the resolution of stochastic problems involving distributed stripe disturbances.The iteratively generated λ reflects the intrinsic stripe properties of such distributed stochastic problems.(c) The problem of distributed interval optimization could also be applied to privacy protection problems.In current distributed privacy protection settings, agents offer stochastic function information with noises to other agents.If each agent chooses to provide others with a confidence region (interval-valued function) without revealing λ i , these methods can better protect their data.(d) Reformulated problem (

Lemma 9 :
With Assumptions 1-4, for given common point λ * , the consensus of estimate x i (k) in L 1 is achieved by Algorithm 1, that is, for i, j = 1, 2, . . ., n,lim k→∞ E x i (k) − x j (k) = 0.Lemma 10: With Assumptions 1 and 3, the cross term of d i (k) and ξ i (k) − ξ * is lower bounded (a) in conditional expection with respect to F(k) as follows: ) (b) According to Lemma 11, lim k→∞ n i=1 x i (k) − x * 2 converges to a non-negative random variable almost surely.According to Lemma 12, we have lim k→∞ E x i (k) − x * 2

FIGURE 5 .
FIGURE 5. x i (k) for agent i .