UAV-Assisted Data Offloading for Smart Container in Offshore Maritime Communications

Yanpeng Dai,Bin Lin,2,*,Yudi Che,Ling Lyu,3

1 School of Information Science and Technology,Dalian Maritime University,Dalian 116026,China

2 Network Communication Research Centre,Peng Cheng Laboratory,Shenzhen 518052,China

3 State Key Laboratory of Integrated Services Networks,Xidian University,Shaanxi 710071,China

Abstract: Smart containers have been extensively applied in the maritime industry by embracing the Internet of Things to realize container status monitoring and data offloading without human intervention.However, the offloading rate and delay in the offshore region are limited by the coverage of the onshore base station (BS).In this paper, we investigate the unmanned aerial vehicle(UAV)-assisted data offloading for smart containers in offshore maritime communications where the UAV is as a relay node between smart containers and onshore BS.We first consider the mobility of container vessel in the offshore region and establish a UAV-assisted data offloading model.Based on this model, a data offloading algorithm is proposed to reduce the average offloading delay under data-size requirements and available energy constraints of smart containers.Specifically, the convex-concave procedure is used to update time-slot assignment,offloading approach selection, and power allocation in an iterative manner.Simulation results show that the proposed algorithm can efficiently reduce average offloading delay and increase offloading success ratio.Moreover, it is shown that the UAV relay cannot always bring the performance gain on offloading delay especially in the close-to-shore area, which could give an insight on the deployment of UAV relay in offshore communications.

Keywords: data offloading; smart container;unmanned aerial vehicle; maritime Internet of Things

I.INTRODUCTION

1.1 Background and Motivation

With the rapid development of digital shipping,the maritime industry is experiencing deep integration with advanced wireless communications technology[1, 2].The smart container is one of the representatives [3] which embraces the Internet of Things (IoT) techniques to monitor container status and deliver collected data to various sectors of the supply chain to reduce human intervention and improve logistics safety and efficiency.Latest satellite communications systems have supported the data offloading service during ocean shipping at low transmission rate, e.g.the Inmarsat-4 system with a peak rate of 492 kbps [4].In the offshore region, the onshore base station (BS) can extend the coverage of terrestrial wireless networks to realize high-rate data transmission such as time-division long-term evolution (TD-LTE) networks of Qingdao port with a peak rate of 7 Mbps covering the area up to 30km offshore[4].To improve offloading rate and link robustness, the unmanned aerial vehicle (UAV)is recently introduced to offshore communications as the relay node between onshore BS and smart containers [5-7].The UAV relay can be deployed dynamically according to the mobility of container vessel to provide on-demand service on the sea.

To enable logistics-chain excellence and sustainability,smart containers are required to offload real-time data accurately and timely to onshore BS.However, limited time-spectrum resources would not support low-latency data offloading for all smart containers especially when the number of which is large.Thus, time-spectrum resources should be carefully assigned according to data sizes of all smart containers to guarantee the successful execution of data offloading.Furthermore, the channel conditions among smart containers, UAV relay, and onshore BS are varying during the container vessel sails.In each time slot, each smart container needs to select the suitable offloading approach between direct offloading and UAV relay-assisted offloading to reduce the offloading delay.Therefore, it is necessary to design the UAV-assisted data offloading scheme for smart containers in offshore maritime communications.

1.2 Related Works

Smart containers have attracted the attention of many international organizations and research institutions.The United Nations Centre for Trade Facilitation and Electronic Business (UN/CEFACT) launched the Smart Container Project to develop the data exchange standards, aiming to promote and simplify the deployment of smart container solutions [3].The Digital Container Shipping Association (DCSA) is established in 2019 which proposed the DCSA IoT standards to enable mass deployment of interoperable smart container solutions[8].

There have been many research works that investigate the data exchange and communications of smart containers [9-11, 6, 12, 13].P.Ruckebuschet al.in [9] investigated the wireless sensor networks for smart container monitoring and proposed a tailored sensor communication solution to reduce transmission delay and extend battery lifetime.E.H.Fortet al.in [10] proposed a low-power sensor network for tracking and monitoring of smart containers and goods.K.Salahet al.in [11] presented an IoT-enabled monitoring system for smart container which provides continuous information readings and offloading related to temperature, humidity, location,open/close conditions and so on.Z.I.Bellet al.in [12] and G.Zhu and J.Du in [13] further investigated the tracking control methods for the container shipping.In above-mentioned works, the satellite communications are mainly used to offload the collected data on the sea.H.Seoet al.in [14],R.Camposet al.in [15], and Y.Huoet al.in [16]further investigated the broadband communications on offshore sea via the terrestrial networks such as long-term evaluation(LTE)system and wireless local area network (WLAN) to improve the transmission rate and delay.However, the coverage capability of onshore BS and ocean waves restricts the data offloading performance in offshore region.

The UAV relay is a promising way to extend the coverage of terrestrial BSs and overcome harsh environment of sea surface[17-19].X.Chenet al.in[17] investigated the channel modeling and analyzed the performance of UAV relay systems.Y.Zenget al.in [18] and S.Zenget al.in [19] proposed the power allocation algorithms for UAV-enabled mobile relaying systems to maximize the throughput and minimize energy consumption,respectively.J.Zhanget al.in [20] investigated the UAV relay-assisted maritime communications and proposed a UAV relay placement algorithm to improve average transmission rate of users on the sea.S.Kavuriet al.in[6] analyzed the performance gain brought by the UAV relay for data offloading of smart containers in offshore region.It is shown that the UAV relay can efficiently reduce average transmission delay and message loss probability compared with direct access to onshore BS.

In the existing literature, it has been well realized that UAVs can enhance the performance of offshore communications and data offloading [21,22].However, there are challenges that hinder the application of UAVs in data offloading for smart containers.The collected data of each smart container should be offloaded accurately and timely to onshore BS to realize real-time cargo tracking and monitoring.However, time-frequency resources are finite which may not satisfy the offloading for all smart containers.Furthermore, the selection of offloading approaches between direct offloading and UAV relay-assisted offloading for smart containers is important for reducing offloading delay.For instance, the smart containers with large data sizes should be selected to perform UAV relay-assisted data offloading and others with small data sizes should be selected to offload data directly to onshore BS, which would be better for reducing offloading delay.Therefore, it is necessary to design the efficient data offloading scheme for releasing the potential of UAV relay in offshore communications.

1.3 Outcomes and Contributions

In this paper, we investigate the UAV-assisted data offloading for smart containers in offshore maritime communications to reduce average offloading delay and increase offloading success ratio.Specifically,we consider the mobility of container vessel in offshore region and establish a data offloading model where the UAV relay works in the decode-and-forward mode.Then, an average offloading delay minimization for all smart containers on the vessel is formulated to jointly optimize the time slot assignment, offloading approaches selection, and power allocation while satisfying data-size requirements and available energy constraints of all smart containers.We show that besides the binary constraints,the objective functions and constraints of original minimization problem can be transformed into be convex via variable substitution and quasi-convex approximation.Finally, a data offloading algorithm is proposed based on convexconcave procedure (CCP) and its convergence is analyzed.The main contributions of our paper are summarized as follows.

• We construct a data offloading model in the offshore maritime communications which considers the mobility of container vessel, the UAV movements, and wireless propagation on the sea.This model can be used to evaluate the impact of key system parameters such as the offshore distance,the heights of UAV and onshore BS, and data sizes of smart containers on the performance of data offloading.

• We propose a maritime UAV-assisted data offloading (MarUADO) algorithm for smart containers which can achieve a sub-optimal solution of the original minimization problem in an iterative manner.The results show that the proposed algorithm can efficiently reduce average offloading delay and improve offloading success ratio.

• We find in the results that the UAV relay cannot always bring the performance gain for data offloading especially when the offshore distance of the container vessel is short.It exactly demonstrates the importance of conducting offloading approach selection and UAV deployment planning in offshore region.

The reminder of this paper is organized as follows.Section II presents the system model and problem formulation.The problem solution and algorithm design are presented in Section III.We show the simulation results in Section IV.

II.SYSTEM MODEL AND PROBLEM FORMULATION

We consider a UAV-assisted data offloading system in offshore maritime scenario which includes an onshore BS, a UAV relay, and a container vessel loaded withMsmart containers, as shown in Figure 1.Smart containers are equipped with monitors and sensors to collect the data of container internal environment and the status of products and have the communication capability to offload collected data to the onshore BS[9,23].LetC={1,2,...,M}denote the set of smart containers.Dmdenotes the size of collected data of smart containerm,∀m ∈C.

Figure 1. An illustration of UAV-assisted data offloading in offshore maritime communications.

There are two data offloading approaches, which are the direct offloading and UAV relay-assisted offloading.One smart container directly offloads its data to onshore BS if it performs the direct offloading.If one smart container performs UAV relay-assisted offloading, it first transmits its data to UAV relay and then UAV relay forwards the received data to onshore BS [17, 6].Tdenotes the time period for all smart containers on the vessel to offload the data, which is divided intoNtime slots with identical lengthTs=T/N.LetT={1,2,...,N}denote the set of time slots.

2.1 Mobility and Channel Models

It is assumed that the container vessel sails into the port along with a straight line and towards onshore BS.The initial sailing velocity of container vessel is denoted byand its accelerated velocity is denoted byaC(≤0).Hence,the distance between onshore BS and container vessel in time slotnis expressed as

Proof.See Appendix A.

Generally speaking, the line-of-sight (LOS) path and the sea-surface reflection path are the two most dominant components of maritime wireless propagation due to the sparse distribution of obstacles on the sea surface [24].However, the shadowing fading is non-negligible for the signal transmission of smart containers.Because all smart containers are stacked and crowded on the deck of the vessel, the signal transmission of one smart container may be blocked by others.Therefore, we adopt the two-ray channel model with shadowing effect to characterize wireless propagation between onshore BS and smart containers [25].The channel power gain between smart containermand onshore BS in time slotnis given by

2.2 Transmission Model

Each smart container is allowed to perform only one approach between direct offloading and UAV relay-assisted offloading in each time slot.In order to restrain interference between different links,two offloading approaches are allocated different and exclusive spectrum bandwidths.WDdenotes the bandwidth of the spectrum channel for direct offloading.WUdenotes the bandwidth of the spectrum channel for UAV relay-assisted offloading.If smart containermperforms direct offloading in time slotn,its transmission rate is expressed as

wherepm[n]is the transmit power for smart containermin time slotn.N0is the additive noise power.

The UAV relay adopts the decode-and-forward mode[20,28,19].In the first half of one time slot,the UAV relay receives the data from the smart container.After decoding the signal, the UAV relay forwards the received data to onshore BS in the second half of one time slot.Hence, if smart containermperforms UAV relay-assisted offloading,its transmission rate is expressed as

whereandare transmission rates in the first half and the second half of time slotn,respectively.pUis the transmit power of UAV relay.

Letαm,ndenote the indicator of UAV relayassisted offloading.αm,n= 1 if smart containermperforms UAV relay-assisted offloading in time slotn.Otherwise,αm,n= 0.Letβm,ndenote the indicator of direct offloading.βm,n= 1 if smart containermperforms direct offloading in time slotn.Otherwise,βm,n= 0.The data bits offloaded by smart containermin time periodTis expressed as

The energy consumption of smart containermin time periodTis expressed as

2.3 Problem Formulation

Let A ={αm,n,m ∈C,n ∈T}and B ={βm,n, m ∈C,n ∈T}denote indicator variables for UAV relay-assisted offloading and direct offloading,respectively.P ={pm[m],m ∈C,n ∈T}denotes the transmit power variables for smart containers.The average offloading delay for all smart containers minimization problem can be formulated as

whereEmis the total available energy for smart containerm.is the maximum transmit power for smart containers.C1-C3 constrain that each offloading approach can be used by at most one smart container in one time slot,and each smart container can perform at most one offloading approach in one time slot.C4 restricts that the energy consumption of smart containermcannot exceedEm.C5 guarantees that the data of each smart container is entirely offloaded inT.C6 restricts the maximum transmit power for all smart containers.C7 and C8 are binary constraints for eachαm,nandβm,n.

Due to the binary variables, i.e., A and B, and the non-convex constraints, i.e., C4 and C5, Problem P0 is a mixed-integer nonlinear programming problem,which cannot be solved directly[29].Hence, we aim to design an efficient data offloading algorithm with moderate complexity for Problem P0.Furthermore,the channel conditions among smart containers,offshore BS, and UAV relay are varying in different time slots because of the mobility of container vessel.We should allocate offloading approaches and time slots to smart containers according to the data sizes and channel condition in each time slot.

III.PROBLEM SOLUTION AND ALGORITHM DESIGN

In this section, we propose a UAV-assisted data offloading algorithm which assigns the offloading approach and transmit powers for smart containers in each time slot.First, we transform Problem P0 into a convex problem by variable substitution and relaxation.Then, the penalty convex-concave procedure (CCP) method is used to propose an algorithm to solve Problem P0.

3.1 Problem Transformation

Sinceis a minimum function ofandwhich is not continuous in entire feasible region.To replace the minimum function in[n],we introduce new variable[n]and new constraints C9 and C10 as follows.

Furthermore,constraint C5 is rewritten as

Problem P0 can be transformed into Problem P1 as follows.

Then,the following proposition is given.

Proposition 1.If ProblemP0is feasible,the optimal solution of ProblemP0can be obtained by solving ProblemP1.

Furthermore, constraint C6 can be replaced by the following two constraints:

Based on constraint C13 and C14, we further rewrite constraint C11 as

Then, Problem P1 can be further transformed into Problem P2 as follows.

Then,we can give the following proposition.

Proposition 2.The optimal value of ProblemP1can be obtained by solving ProblemP2.

We can see that besides integer constraints C7 and C8, the objective function and all other constraints in Problem P2 are convex.The branch-and-bound method[29,30]is a classic method to solve the integer programming problem which however is with high computational complexity.We give an alternative method based on penalty CCP method [31] with moderate complexity in this paper.

3.2 Algorithm Design

According to the feature of Problem P2, we utilize the penalty CCP method to design the MarUADO algorithm.Specifically, we first relax the integer constraints C7 and C8 as follows.

where constraints C16 and C17 are equivalent to C7.Constraints C18 and C19 are equivalent to C8.Although constraints C16 and C18 are concave,the first-order Taylor approximation can be used to respectively transform them as:∀m ∈C,∀n ∈T,

Algorithm 1. Maritime UAV-assisted data offloading(MarUADO)algorithm.1: Initialize:Set iteration i=0 and O[0]=+∞.Give initial points A[0],B[0]and τ[0].2: repeat 3: Solve Problem P3 based on A[i] and B[i] to obtain A∗,B∗,P∗,and O∗.4: Set i=i+1 and τ[i]=µτ[i-1].5: Update A[i] = A∗, B[i] = B∗, and O[i] =O∗.6: ε=|O[i]-O[i-1]|.7: until ε ≤εmin and i ≥imax.8: if τ[i]images/BZ_170_1480_989_1528_1034.png m∈C images/BZ_170_1610_989_1658_1034.png n∈T sm,n ≥10-2 then 9: The problem is infeasible.10: end if 11: return A∗, B∗, P∗and O∗if the problem is feasible.

whereαm,n[i]andβm,n[i]are given points at thei-th iteration.Because the given point at the first iteration is not guaranteed to satisfy all the constraints, the slack variable S ={sm,n,∀m ∈C,∀n ∈T}and the penalty variableτ[i] are introduced.Then, we can formulate the Problem P3 as follows.

whereτ[i] =µτ[i-1]increases with the growth of the number of iterations due toµ≥1.According to Problem P3, we can construct and solve a series of convex problems to obtain a sub-optimal solution of Problem P2.The detailed procedure of proposed algorithm is concluded in Algorithm 1.

It can be seen from Algorithm 1 that step 1 is to give initial points, i.e., A[0] and B[0].Steps 2-7 are to solve Problem P3 in an iteration manner untilthe condition of convergence is met or the maximum number of iterationsimaxis achieved.The condition of convergence is that the difference of the objective function values between two adjacent iterations is smaller thanεmin.Steps 8-10 is to check the feasibility of considered problem.If the problem is feasible,the obtained solution and objective function value are returned.Furthermore, the following corollary is given.

Corollary 1.The convergence of Algorithm 1 can be guaranteed.

Proof.See Appendix B.

The complexity of solving Problem P3 isif the interior point method [29]is applied.Therefore, the complexity of Algorithm 1 is

IV.SIMULATION RESULTS

In this section, simulation results are presented to evaluate the performance of our proposed MarUADO algorithm.vU[1] andaUof UAV relay are set as the half ofvC[1] andaC.The default setting ofaCis 0 m/s2.Tsis set as 10 s.The data size of each smart container follows the uniform distribution in[5,dmax]Mbits, wheredmaxis the maximum data size.The available energy of each smart container follows the uniform distribution in[8,Emax]J,whereEmaxis the maximum energy.The height of smart containers is normalized to that of container vessel which is usually 15～30m.The default simulation parameters are summarized in Table 1.The path-loss model between smart containers and onshore BS is set asPLC[dB]=-20 lg(hT)-20 lg(hR)+37.6 lg(l[km])[32]wherehTandhRare the heights of transmitter and receiver,respectively.The path-loss model between smart containers and UAV relay and that between offshore BS and UAV relay are set asPLU[dB] = 78 +20.3 lg(l[km]) [27].Furthermore, two benchmark algorithms are used for the performance comparison,which are listed as follows.

Table 1. Default simulation parameters.

• Only direct offloading (OnlyDirect) algorithm:This algorithm is only allowed to use the direct offloading and without the assist of UAV relay.Other procedures are similar with our proposed MarUADO algorithm.

• Greedy offloading(GreedyChG)algorithm: This algorithm selects the pair of offloading approach and smart container with the best channel gain at each time slot and allocates them with each other.Moreover,the transmit power always uses

4.1 Impact of the Number of Smart Containers on Offloading Performance

Figure 2 shows the average offloading delay versusMof different algorithms.It can be seen that the average offloading delay increases with the growth ofM.This is because the increase ofMaggravates the resource competition.WhenMis small, most of smart containers can be allocated with UAV relayassisted offloading and consume a few time slots to offload their stored data.WhenMis large,more smart containers cannot obtain the assist of UAV relay and just use direct offloading which leads to consume more time slots to finishing data offloading.Furthermore,we can see that our proposed MarUADO algorithm can achieve lower average offloading delay than other benchmark algorithms.This is because that different with OnlyDirect algorithm,MarUADO algorithm can utilize UAV relay to assist smart containers to offload data to reduce the offloading delay.Compared with GreedyChG algorithm, MarUADO algorithm allocates time slots and offloading approach according to channel condition,data size,residual energy instead of only channel gain.

Figure 2. Average offloading delay versus M of different algorithms(dmax =10 Mbits).

Figure 3. Offloading success ratio versus M of different algorithms(dmax =10 Mbits).

Figure 3 shows the offloading success ratio versusMof different algorithms.The offloading success ratio is the probability of all smart containers successfully offloading all their stored data.It can be seen that the offloading success ratio decreases with the growth ofM, and our proposed MarUADO algorithm outperforms other benchmark algorithms.This is because whenMis sufficient large,the limited number of time slots cannot satisfy the offloading requirement of all smart containers.Furthermore,our proposed MarUADO algorithm can efficiently allocate offloading approaches and time slots for smart containers to increase the offloading success ratio compared with other two benchmark algorithms.

4.2 Impact of the Data Size on Offloading Performance

Figure 4 shows the average offloading delay versusdmaxof different algorithms.It can be seen that the average offloading delay increases with the growth ofdmax.This is because the smart container needs more time slots to entirely offload its stored data when the data size increases.Furthermore, we can see that even whendmaxis large, our proposed MarUADO algorithm can achieve lower average offloading delay than other two benchmark algorithms.The reasons are twofold.On the one hand, the UAV relay can efficiently improve the achievable transmission rate of data offloading in offshore region.The smart container can offload more data in one time slot with the assist of UAV relay.On the other hand,our proposed MarUADO algorithm can allocate the offloading approaches and time slots according to the data size of each smart container which improves the efficiency of offshore data offloading.

Figure 4. Average offloading delay versus dmax of different algorithms(M =4).

Figure 5 shows the offloading success ratio versusdmaxof different algorithms.It can be seen that the offloading success ratio decreases with the growth ofdmax, and our proposed MarUADO algorithm can improve the offloading success ratio compared with other two benchmark algorithms.This is because when the data size is sufficiently large, it is difficult to offload total data of all smart containers inNtime slots which reduces the offloading success ratio.

Figure 5. Offloading success ratio versus dmax of different algorithms(M =4).

4.3 Impact of Offshore Distance on Offloading Performance

Figure 6 shows the average offloading delay versuslC,BS[1] under differentdmax, which is obtained by performing our proposed MarUADO algorithm.It can be seen that the average offloading delay increases with the growth oflC,BS[1].Furthermore,we can see that higherdmaxcan lead to larger average offloading delay under the samelC,BS[1].Long offshore distance can result in large path loss resulting in that the offloading performance obviously decreases.It demonstrates that the deployment of UAV relay is necessary for improving the offloading performance in offshore region and enhancing coverage capability of offshore communications.Our proposed MarUADO algorithm can release the potential of UAV relay on data offloading in offshore region.

Figure 6. Average offloading delay versus lC,BS[1] under different dmax (M =4).

Figures 7 and 8 show the usage ratios of UAV relay and direct offloading versuslC,BS[1] under differentdmax, respectively.The usage ratio of one offloading approach is the ratio of the usage count of this approach to the total number of offloading transmissions.It can be seen from Figure 7 that the usage ratio of UAV relay is first increasing and then decreasing with the growth oflC,BS[1].Meanwhile,we can see from Figure 8 that the usage of ratio of direct offloading is first decreasing and then increasing with the growth oflC,BS[1].It demonstrates that if the offshore distance is very short, the direct offloading is more efficient than that of UAV relayassisted offloading.This is because although the UAV relay can improve the channel condition, offshore BS only has half time slot to receive the data of smart container via UAV relay and however direct offloading can allow offshore BS to receive the data on entire time slot.The direct offloading can deliver more data size than UAV relay-assisted offloading in short offshore distance.With the growth of offshore distance, the channel condition between offshore BS and smart container degrades.There are more smart containers to select the UAV relay to offload their stored data.In other words, the advantage of UAV relay-assisted offloading becomes increasing with the growth of offshore distance which greatly increases the usage of UAV relay.When the offshore distance is sufficiently long, the UAV relay cannot satisfy the offloading requirement of all smart containers which makes partial smart containers offload the data via direct offloading.The results reveal that the performance gain brought by the UAV relay is closely dependent on its offshore distance and deployment.

Figure 7. Usage ratio of UAV relay versus lC,BS[1] with different dmax (M =4).

Figure 8. Usage ratio of direct offloading versus lC,BS[1]with different dmax (M =4).

V.CONCLUSION

This paper has investigated the UAV-assisted data offloading algorithm from smart containers, aiming to reduce the average offloading delay in offshore maritime communications.First, we have considered the mobility of container vessel and established a data offloading model in offshore region where the UAV relay works in the decode-and-forward mode.Then, the MarUADO algorithm has been proposed based on the penalty CCP method to jointly optimize the offloading approach, time-slot assignment and power allocation.Simulation results show that the deployment of UAV relay can effectively improve the performance of data offloading in offshore region,and our proposed algorithm can efficiently reduce the average offloading delay and offloading success ratio.Furthermore,it is shown from simulation results that the performance gain brought by UAV relay diminishes when the offshore distance of container vessel is short.In future work, we will investigate the location deployment and trajectory planning of UAV relay for data offloading in offshore maritime communications.

ACKNOWLEDGEMENT

This work was supported in part by National Key Research and Development Program of China under Grant 2019YFE0111600, in part by National Natural Science Foundation of China under Grants 62101089, 62002042, 61971083, and 51939001, in part by China Postdoctoral Science Foundation under Grants 2021M700655 and 2021M690022, in part by Cooperative Scientific Research Project, Chunhui Program of Ministry of Education, P.R.China,in part by LiaoNing Revitalization Talents Program under Grant XLYC2002078, in part by Dalian Science and Technology Innovation Fund under Grant 2019J11CY015, and in part by the Fundamental Research Funds for Central Universities under Grants 3132021237 and 3132021223.

APPENDIX

A.Proof of Lemma 1

The distance between UAV relay and onshore BS is expressed as

The distance between container vessel and onshore BS is expressed as

In order to ensure that the location of UAV relay is always in the middle of container vessel and onshore BS, it should ensure that the flight distance of UAV relay is always half of the sailing distance of container vessel.Hence,we can get the following formula

Now,the proof is finished.

B.Proof of Corollary 1

We prove the convergence of Algorithm 1 from two cases,i.e.,τ=τmaxandτ≤τmax.

First, we consider that the case ofτ=τmax.LetO[i]denote the optimal value of thei-th iteration.We know thatO[i] is obtained according to Problem P3 with given points A[i-1] and B[i-1].Because Problem P3 minimizes the value of the objective function, it must be obtained thatO[i] ≥O[i+1].At least, A[i] and B[i] can be chosen as the optimal solution of thei+ 1-th iteration such thatO[i] =O[i+1].Otherwise,O[i]> O[i+1].Due to the existence of minimum value of Problem P3,Algorithm 1 can converge whenτ=τmax.

Second, we consider that the case ofτ < τmax.τis increasing with the process of the iterations.If the feasible solution of Problem P2 is not found in current iteration, the sum of slack variables, i.e.,is greater than zero.Hence,Algorithm 1 is not a descent algorithm in this case such that its objective function value is not decreasing with the process of the iterations.Once the feasible solution is found,all slack variables are approximately equal to to zeros such that0.Then,we can use the way of the first case to prove the convergence.

Now,the proof of Corollary 1 is finished.