Mechanism Design for Wireless Powered Spatial Crowdsourcing Networks

03/27/2020 ∙ by Yutao Jiao, et al. ∙ IEEE 0

Wireless power transfer (WPT) is a promising technology to prolong the lifetime of the sensors and communication devices, i.e., workers, in completing crowdsourcing tasks by providing continuous and cost-effective energy supplies. In this paper, we propose a wireless powered spatial crowdsourcing framework which consists of two mutually dependent phases: task allocation phase and data crowdsourcing phase. In the task allocation phase, we propose a Stackelberg game based mechanism for the spatial crowdsourcing platform to efficiently allocate spatial tasks and wireless charging power to each worker. In the data crowdsourcing phase, the workers may have an incentive to misreport its real working location to improve its utility, which causes adverse effects to the spatial crowdsourcing platform. To address this issue, we present three strategyproof deployment mechanisms for the spatial crowdsourcing platform to place a mobile base station, e.g., vehicle or robot, which is responsible for transferring the wireless power and collecting the crowdsourced data. As the benchmark, we first apply the classical median mechanism and evaluate its worst-case performance. Then, we design a conventional strategyproof deployment mechanism to improve the expected utility of the spatial crowdsourcing platform under the condition that the workers' locations follow a known geographical distribution. For a more general case with only the historical location data available, we propose a deep learning based strategyproof deployment mechanism to maximize the spatial crowdsourcing platform's utility. Extensive experimental results based on synthetic and real-world datasets reveal the effectiveness of the proposed framework in allocating tasks and charging power to workers while avoiding the dishonest worker's manipulation.

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I Introduction

Crowdsourcing is becoming a popular paradigm which efficiently completes tasks and solves problems by aggregating information and intelligence from crowds. Integrated with advanced sensing and communication techniques, mobile devices can help complete diverse location-aware tasks, such as the large-scale data acquisition and analysis in real-time traffic monitoring111An example is the crowdsourcing-based traffic and navigation app “Waze” (https://www.waze.com). or weather monitoring and forecasting [20] at different places. By focusing on the geospatial data, a new paradigm called spatial crowdsourcing (SC) [17] has received increasing attention in the last few years [5, 33, 10]. Typically, there are three entities in the SC system, including an online SC platform, requesters and workers. As a core component of the SC ecosystem, the SC platform is a broker which allows requesters to post tasks and recruits workers to complete them. Each employed worker then stays at or travels to its target task area to collect and transmit the requested data back. Since the relationship between the SC platform and the workers are incentive-driven, we study the interactions between them to understand and develop an effective mechanism to enable sustainable and efficient operations of the SC systems.

Most existing work assumes that there is always reliable communication infrastructure and enough energy available for workers to complete the data transmission. However, this assumption may not be realistic, especially when the workers have to perform tasks in remote areas without a wireless base station. Moreover, workers can be battery-powered wireless mobile devices. Their energy constraint limits the working time and ultimately affects the task completion. Fortunately, some studies [22, 34, 18] in wireless powered sensor networks have illustrated the feasibility of using wireless power transfer (WPT) [2] in sensing data collection to prolong the lifetime of sensors. Given this, we consider a paradigm called wireless powered spatial crowdsourcing where the SC platform deploys a mobile base station (BS), e.g., robots, drones or vehicles, to assist the data collection. The mobile BS serves as the infrastructure for communication and wireless power transfer. A typical scenario suitable for applying this paradigm is the information collection in an emergency rescue mission. The requester can be the relief headquarter which needs the SC platform to organize workers to continually transmit the live video or environmental monitoring data from the target task area, e.g., seismic site. These data and data analytics results will significantly help to increase the efficiency of succour. Meanwhile, those workers with battery-powered devices will need wireless charging due to the possible power outage.

To ensure successful and stable operations of the crowdsourcing system, designing an incentive mechanism that stimulates workers’ participation and efficiently allocates tasks is essential. A number of studies have proposed mechanisms satisfying various requirements, such as profitability, strategyproofness, i.e., truthfulness, and individual rationality [32, 28]. Nevertheless, in wireless powered spatial crowdsourcing networks, the reward offered by the SC platform to workers can be the wireless power supply, which is location-dependent. The major difference from those existing mechanisms, the incentive of which is based on the monetary reward222The monetary reward can be tokens, virtual money, reputation, etc.. The difference introduces a few major issues for incentive mechanism design in wireless powered crowdsourcing networks, and the following questions have to be answered. First, what is the optimal total charging power supply for the SC platform to configure for maximizing its utility? The SC platform can encourage workers to transmit sensed data at a higher transmission rate, i.e., more collected data per unit time, but it is at the cost of a higher power supply. Second, how to allocate the tasks and charging power to workers which are spatially distributed in the target task area? The allocation is based on not only each worker’s sensing cost but also the working location, which affects the communication cost and transferred power. Note that the workers’ sensing cost and working location can be private information and unknown to the SC platform. Lastly, how to deploy the mobile BS taking the workers’ strategic behaviours into account? Since the workers’ working locations are private, workers need to report their locations before the mobile BS chooses the best location to deploy.

Under the assumption of rationality, a worker may dishonestly misreport its location to increase its utility while reducing the SC platform’s utility. Figure 1 shows such an example. In the task area, there are one dishonest worker at location and two honest workers respectively at locations and . The SC platform would place the mobile BS at for optimal utility if all the workers report true locations and . However, the dishonest worker has the incentive to report a fake location , so that according to the reported locations and , the mobile BS will be deployed at . In this case, the dishonest worker at can be closer to the mobile BS and then enjoy more transferred power from the mobile BS while consuming less power to transmit its sensed data. This inevitably increases other workers’ and SC platform’s energy consumption and damages their utility. Most current studies on incentive mechanisms for the crowdsourcing system have not addressed such issue yet.

Fig. 1: An example where a dishonest worker misreports its true location.

In this paper, we propose a strategyproof and energy-efficient SC framework which jointly solves the problems of task and wireless charging power allocation as well as the truthful working location reporting. In the framework, there are two phases: task allocation phase and data crowdsourcing phase. In the task allocation phase, the SC platform determines and announces a fixed total charging power supply. Each worker interested in participating needs to choose and submits the preferred crowdsourcing plan, i.e., its data transmission rate to the SC platform. In return, they can obtain the corresponding portion of the supplied charging power from the SC platform. We use the Stackelberg game to model the interactions between workers and the SC platform, in which each worker’s transmission rate and allocated power can be determined. In the data crowdsourcing phase, the mobile BS requests for workers’ working locations. Based on the Moulin’s generalization median rule [19], we present three strategyproof mobile BS deployment mechanisms for the mobile BS to determine its service location. The first one is the classical median mechanism. The other two mechanisms are designed from the Bayesian viewpoint. One is a conventional mechanism which assumes that each worker’s working location follows a priori known distribution. For more general scenarios with only historical working location data available, we resort to the advanced deep learning technique to develop another mechanism for higher robustness and larger utility.

Fig. 2: Wireless powered spatial crowdsourcing system with two phases.

The major contributions of this paper can be summarized as follows:

  • We propose a strategyproof and energy-efficient framework for implementing the wireless powered spatial crowdsourcing. The task allocation phase and the data crowdsourcing phase jointly coordinate the task/power allocation and the mobile BS deployment to maximize the SC platform’s utility.

  • We propose an incentive mechanism for the task and wireless power transfer allocation based on the Stackelberg game model in the task allocation phase. We prove that a unique Nash equilibrium exists among workers’ strategies, i.e., the data transmission rates, and the Stackelberg equilibrium can be efficiently calculated to optimize the SC platform’s utility.

  • In the data crowdsourcing phase, we first present two strategyproof mobile BS deployment mechanisms to prevent the dishonest worker’s manipulation while maximizing the SC platform’s utility under different scenarios respectively with 1) no prior information 2) prior location distribution. Moreover, for the complex scenario with only historical working location data available, we utilize the deep learning technique and construct a novel deep neural network to design a strategyproof deployment mechanism.

  • Based on synthetic and real-world datasets, the experimental results illustrate the effectiveness of the proposed incentive mechanisms in assisting the SC platform in allocating the task and charging power efficiently. In particular, the deep learning based mechanism shows significant improvement in performance and stability compared with the conventional mechanism.

To the best of our knowledge, this is the first work that investigates the incentive mechanism design in wireless powered spatial crowdsourcing and, for the first time, the deep learning method to address the problem of potential working location misreporting in spatial crowdsourcing systems.

The rest of the paper is organized as follows. In Section II, we discuss the related work and motivations in detail. In Section III, we describe the system model of wireless powered spatial crowdsourcing. Section IV proposes the task and charging power allocation mechanism. In Section V, we present three mechanisms for strategyproof mobile BS deployment in the data crowdsourcing phase. In Section VI, we provide the experimental results. Finally, we conclude the paper in Section VII.

Ii Related Work and Motivations

There have already been studies about the incentive mechanisms in crowdsourcing systems [28, 31]. The authors in [28] proposed platform-centric and user-centric incentive mechanisms, respectively based on the Stackelberg game and the reverse auction. Each worker is free to determine its own strategy, i.e., working time or cost, for a reward. Some desirable economic properties, such as strategyproofness333We use “strategyproofness” and “truthfulness” interchangeably in this paper. and individual rationality, are guaranteed in the auction. In [31], the authors designed a reputation-based incentive mechanism and used the repeated gift-giving game in analyzing the interaction between task requesters and workers. The authors in [4] made use of the historical data about the workers’ visiting records to the task locations to investigate workers’ skills. For crowdsourcing in wireless-powered task-oriented networks, a game-based distributive incentive mechanism was proposed in [29] for reducing energy consumption while ensuring task completion. Particularly, in [29], the authors also used the energy as the reward and introduced an energy bank as the trusted medium of the energy service exchange to avoid using the unreliable and unspecific monetary reward among the workers. However, to the best of our knowledge, most of the existing studies on the crowdsourcing rely on the monetary transfer, i.e., payment, to guarantee the property of truthfulness in reporting private valuations. Moreover, none of the existing work has addressed the issue that a dishonest worker could possibly misreport its working location and manipulate the crowdsourcing system in the data crowdsourcing phase, which cannot be solved using monetary transfer. The study of approximate mechanism design without money was initialized in [23], where the authors discussed the strategyproof single facility deployment mechanism in one-dimensional space. The authors in [9] designed two neural network structures, including MoulinNet and RegretNet, to solve the strategyproof multiple facilities location problem in one-dimensional space. Inspired by these works, we propose mobile BS deployment mechanisms for the SC system, which can achieve high utility while guaranteeing the strategyproofness without any money or reward transfer.

Fig. 3: Data transmission and power transfer in the data crowdsourcing phase.

Iii System Model: Wireless Powered Spatial Crowdsourcing Market

Figure 2 depicts the wireless powered spatial crowdsourcing system model where there are three entities, including the requesters, the SC platform residing in the cloud and the workers with mobile sensing devices. The workers can be human, unmanned vehicles or robots. Initially, the requesters publish spatial tasks with requirements, such as the target task area, the task duration, and the sensed data type. Then, the SC platform advertises the task information to workers on behalf of the requesters and collects the crowdsourced or sensed data. As shown in Fig. 3, we denote by the set of workers and denote by the task area on a Cartesian coordinate plane. The worker ’s working location is described by a 2-tuple, i.e., . We use to represent the deployed mobile BS’s service location projected on the XY-plane and use to denote its height. We assume that each worker knows its preferred area to work, i.e., working area, such as the area near to its commuting route or around home [26]. In the task area, worker  has its own working area and its working location falls in this area, i.e., . In this section, we first model the power cost of communication and sensing for the mobile BS and workers in the data crowdsourcing phase. Then, we elaborate on both the task allocation phase and the data crowdsourcing phase and present the problem formulations.

Iii-a Power cost model

Iii-A1 Worker’s power cost

We consider a frequency division duplexing (FDD) system where sufficient channels are available to ensure interference-free transmission. Note that with this assumption, we can better focus on the incentive mechanism design between the SC platform and workers. Furthermore, we assume that the communication channels are dominated by line-of-sight (LoS) links. Given the mobile BS’s service location , we can write the worker ’s transmission rate according to Shannon’s formula as follows:

(1)

where is the channel gain to noise ratio (CNR), represents the corresponding channel power gain at the reference distance of meter, is the noise power at the receiver mobile BS, is the channel bandwidth, is worker ’s data transmission power, and is the path-loss exponent. In addition, we define

(2)

as the Euclidean distance between the worker and the mobile BS. Again, is the height of the mobile BS. Hereby, we can derive the worker ’s transmission power as

(3)

Besides the power used to transmit data, for the worker , we have the power cost function of data sensing where is the energy cost per bit. Here, the power cost of data sensing is linear to the sampling rate [6], i.e., the transmission rate. Therefore, the worker ’s total power cost can be expressed as follows:

(4)

Iii-A2 Power cost of the mobile base station

The mobile BS consumes energy mainly for WPT to workers. If the charging power transferred to the worker is , the mobile BS at the service location has to consume power as follows [35]:

(5)

where , denotes the receiver energy conversion efficiency, denotes the combined antenna gain at the reference distance of meter.

Iii-B Utility function in the wireless powered spatial crowdsourcingsystem

We define the utility of the crowdsourced data based on the transmission rate, which combines two common metrics, i.e., the data size and timeliness. For example, the requesters may perform the data analysis and prediction based on the real-time crowdsourced data. Higher data transmission rate means that the requesters can process more data during a unit time and yield more accurate prediction results. The utility of the crowdsourced data is equivalent to the utility of the SC task completion. The utility of data collected from the SC task completion is calculated by

(6)

where

is the transmission rate vector reported by workers,

and are parameters. The inner logarithmic function reflects the SC platform’s diminishing return of the worker ’s contribution, and the outer logarithmic function reflects the diminishing return of all participating workers’ contributions [14, 27]. In this paper, the mobile BS serves as a dedicated power transmitter which applies the directional beamforming technique [12]. Taking the power cost of WPT (5) into consideration, the SC platform’s utility function can be expressed as [12]

(7)

Similarly, we obtain the worker ’s utility function as

(8)

Iii-C The procedure of wireless powered spatial crowdsourcing

Note that we aim to maximize the SC platform’s utility. Recalling the utility functions in (7) and (8), how to determine each worker’s transmission rate and charging power as the reward and where to deploy the mobile BS are two critical issues for utility maximization.

Iii-C1 Task allocation phase

Before the mobile BS departs to collect data and workers execute the assigned tasks, the SC platform announces a total charging power supply () to assist workers in the data crowdsourcing. The charging power transferred to worker is proportional to its contribution (the data transmission rate), i.e., . Based on the sensing tasks and the other workers’ responses, each worker reports the preferred data rate to maximize its own utility. In practice, the SC platform may serve as a relay to receive and broadcast the workers’ responses. As workers have not determined the suitable working place and perform the allocated task, they are exposed to uncertainty of working location and the mobile BS’s service location which are only known in the next data crowdsourcing phase. We assume that the workers are risk-averse, which means that they choose to minimize the uncertainty and avoid any possible loss in the future. This concept can be found in the well-known prospect theory [16]. A common example is that a majority of people prefer to deposit money at the bank for safekeeping and low return instead of buying financial products with a high risk of loss. Note that given the power supply and other workers’ transmission rates, the worker ’s utility function in (8) is monotonically decreasing with . Since the worker knows its working area and the task area , it can obtain the maximum value of , i.e., . Therefore, if the worker plans the transmission rate for the worst case where is its distance from the mobile BS, the worker will achieve the utility which is not lower than the worst case in the data crowdsourcing phase. In addition, we use to denote the reported transmission rate vector for all workers except the worker . Hereby, the worker ’s utility function in the task allocation phase can be expressed as

(9)

The SC platform’s utility in (7) is rewritten as

(10)

Iii-C2 Data crowdsourcing phase

In the task allocation phase, the total charging power supply , each worker’s allocated charging power and transmission rate have been determined. Each worker decides the working location according to the task and its available working area. For example, if the task requires collecting data about road traffic condition, workers may choose the roadside or crossing. As our paper mainly focuses on establishing a spatial crowdsourcing market with wireless energy transfer and designing relevant trading mechanisms, how to choose a good working location is beyond our scope. Once working locations are decided, they will travel to the working locations and the SC platform sends out the mobile BS to serve the workers. However, the mobile BS has to know each worker’s working location. Then, it can determine the service location for maximizing the SC platform’s utility. The worker ’s and the SC platform’s utility functions in the data crowdsourcing phase can be respectively expressed as

(11)

and

(12)

To make workers reveal their private working location , the mobile BS organizes the following voting process on the spot.

  1. The mobile BS first broadcasts its deployment mechanism, i.e, the mechanism or rule to place the mobile BS according to the locations reported by workers, to the task area.

  2. Once receiving the notification about the deployment mechanism, each worker sends its working location to the mobile BS.

  3. Based on the collected locations and the deployment mechanism, the service location is calculated for the mobile BS to deploy.

Let denote the applied deployment mechanism which takes the workers’ reported working location vector as input and outputs the mobile BS’s service location , i.e., . During the above voting process, a worker may have an incentive to improve its own utility in (11) by misreporting its true working location . To make the location voting process robust and implementable, our designed mobile BS deployment mechanism should have the property of strategyproofness (truthfulness), which is defined as follows:

Definition 1.

(Strategyproofness) Regardless of other workers’ reported locations, a worker cannot increase the utility by misreporting its working location . Formally, given a deployment mechanism and a misreported location , we have

(13)

where is the vector containing all workers’ working locations except the worker ’s.

Iii-C3 Mutual Dependence

The task allocation phase and the data crowdsourcing phase are mutually dependent. On the one hand, each worker’s transmission rate in data crowdsourcing is determined from the task allocation phase. On the other hand, a prerequisite of the successful charging power allocation is to guarantee that the data crowdsourcing phase cannot be strategically manipulated. The untruthful or dishonest worker may overestimate its risk preference, i.e., the maximum distance , due to its deliberate manipulation. Both the two phases affect the efficient use of the power as well as all the participants’ utilities.

Iv Task and Wireless Transferred Power Allocation Mechanism

We utilize the Stackelberg game approach [8] to analyze the model introduced in the task allocation phase (Section III-C1). There are two levels in the Stackelberg game. In the first (upper) level, the SC platform acts as the leader which strategizes and announces the total charging power supply . In the second (lower) level, each worker is the follower which determines the strategy, i.e., the preferred transmission rate , to maximize its utility. Mathematically, the SC platform chooses the strategy by solving the following optimization problem:

Meanwhile, the worker makes the decision on its reported to solve the following problem:

The objective of the Stackelberg game is to find the Stackelberg Equilibrium (SE). We next introduce the concept of the SE for our proposed model.

Definition 2.

(Stackelberg Equilibrium) Let be a solution for Problem and be a solution for Problem of the workers. Then, a point is the SE for the proposed Stackelberg game if it satisfies the following conditions:

(14)
(15)

for any with and .

In general, the first step to obtain the SE is to find the perfect Nash Equilibrium (NE) [8] for the non-cooperative transmission Rate Determination Game (RDG) in the lower level. Then, we can optimize the strategy of the SC platform at the upper level. Given a fixed , the NE is defined as a set of strategies that no worker can improve utility by unilaterally changing its own strategy while other workers’ strategies are kept unchanged. Since workers are rational and not willing to provide service for a negative utility, they shall set if . To analyze the NE, we introduce the concept of the concave game and the theorem about the existence and uniqueness of NE in the concave game.

Definition 3.

(Concave game [24]) A game is called concave if each worker chooses a strategy to maximize utility , where is concave in .

Theorem 1.

([24]) Concave games have (possibly multiple) Nash Equilibrium. Define matrix function in which ,. Let denote the transpose of . If is strictly negative definite, then the Nash equilibrium is unique.

Hereby, we calculate the first-order and second-order derivatives of the worker ’s utility function with respect to as follows:

(16)
(17)

Since , is a strictly concave function with respect to . Then, the non-cooperative RDG is a concave game and the NE exists when . Otherwise the worker ’s best strategy does not exist. Given any and any strategy profile , the worker ’s best response strategy exists and is unique. To prove the uniqueness of the NE, we also calculate the second-order mixed partial derivative of for with respect to as follows:

where and if . Then, we have the specific expression of the matrix function defined in Theorem 1. Furthermore, the matrix function can be decomposed into a sum of several matrix functions: , where , and Let denote the sum of over , i.e., . Since and , if , we can find that is strictly negative definite, and and are negative semi-definite. Thus, is proved to be strictly negative definite which shows the NE in the RDG is unique. In other words, once the SC platform decides a strategy , the workers’ strategies, i.e., the transmission rates, will be uniquely determined. We then can use the iterative best response [11] to find the SE point in the first level, i.e., the optimal strategy of .

V Mobile BS Deployment Mechanisms in Data Crowdsourcing Phase

Given the SE points () calculated from the task allocation phase, we use (break ties randomly) to represent the set of employed workers whose transmission rate . Hence, the specific problems for the SC platform in the data crowdsourcing phase is

(18)

Based on workers’ reported working locations, the SC platform decides the mobile BS’s location to maximize its utility. For simplicity, we write the equivalent problems as follows:

(19)

where is the crowdsourcing cost of SC platform. Minimizing the SC platform’s crowdsourcing cost is equivalent to maximizing its utility. Similarly, the worker ’s utility and crowdsourcing cost can be respectively expressed as

(20)
(21)

To address the mobile BS’s location problem introduced in Section III-C2, we first present the classical median mechanism and analyze its worst-case performance. Then, we propose a conventional mechanism to improve the utility of the SC platform in expectation. For more general scenarios and achieving better performance, we also propose a deep learning based strategyproof mechanism. The design rationale of the deep neural network is the Moulin’s generalized median mechanism.

V-a Conventional strategyproof mechanism under Bayesian settings

We first introduce an important concept of -dimensional single-peaked preference for the discussed problem.

Definition 4.

(-dimensional single-peaked preference [1]) Let be the set of possible mobile BS’s service locations output by the deployment mechanism on the XY-plane where and are respectively a one-dimensional axis. The worker ’s preference for the mobile BS’s location is -dimensional single-peaked with respect to if 1) there is a single most-preferred location outcome , and 2) for any two outcomes , whenever or for , i.e., both and axes.

In the above definition, means that is preferred by worker to . “” is a strict ordering by worker on the dimension . An explanation of this condition is that is preferred by worker to as long as is nearer to its most-preferred location on each dimension.

Proposition 1.

In the data crowdsourcing phase, the worker’s preference for the mobile BS’s service location is -dimensional single-peaked.

Proof:

We first expand the worker ’s crowdsourcing cost function given in (21) as . We can then show that is convex with respect to and there is a unique optimal solution to minimizing the cost. In other words, the worker ’s most preferred mobile BS’s service location is its working location, i.e., , which satisfies the first condition in Definition 4. In the task area , we randomly choose two locations , . Note that the convexity of guarantees the convexity on one dimension if fixing the variable on the other dimension is fixed. implies that for any on axis and then . We can have the similar implication from . If and are both satisfied, we can have and thus . Therefore, the worker prefers to , i.e., , which proves the condition 2 in Definition 4 and completes the proof. ∎

Theorem 2.

(Moulin’s one-dimensional generalized median mechanism [19]) A mechanism for single-peaked preferences in a one-dimensional space is strategyproof and anonymous if and only if there exist constants such that:

(22)

where is the set of workers’ most-preferred mobile BS’s locations and is the median function. An outcome rule is anonymous, if for any permutation of , we have for all .

Theorem 3.

(Multi-dimensional generalized median mechanism [1]) A mechanism for multi-dimensional single-peaked preferences in a multi-dimensional space is strategyproof and anonymous if and only if it is an -dimensional generalized median mechanism, which straightforwardly applies the one-dimensional generalized median mechanism on each of the dimensions.

1:Workers’ reported locations .
2:Mobile BS’s service location .
3:begin
4:       Repectively sort the x coordinates and y coordinates of workers’ locations in ascending order.
5:       if 

is odd 

then
6:             
7:       else
8:             
9:       end if
10:end
Algorithm 1 MED mechanism

A straightforward benchmark mechanism is the median mechanism [19, 1], as shown in Algorithm 1. We simply name it as MED mechanism, i.e., . This algorithm directly computes the median of workers’ reported locations as the mobile BS’s service location. Apparently, it is a special case of the multi-dimensional generalized median mechanism, so it is strategyproof. We next analyze its performance by comparing it with the optimal mechanism which achieves the maximum utility of the SC platform without considering incentive constraints. Let and respectively denote the maximum and the minimum transmission rate among workers, i.e., .

Proposition 2.

The benchmark MED mechanism has an approximation ratio , which means its worst-case performance for minimizing the SC platform’s crowdsourcing cost can guarantee .

Proof:

We expand the SC platform’s utility function in (18) as follows:

(23)

Let and respectively denote the median and mean of and . Also, we use to denote the optimal solution to maximizing the utility function in (23), i.e., . We also note that the optimal solution to minimizing the is where and . As , we have

(24)

According to [7, Theorem 4.3], we have and . Then, we can verify that

(25)
(26)

Since , we can prove that

(27)

Hence, based on Theorem in [13] and the fact that and , we can obtain