## I Introduction

As predicted in [2], the number of unmanned aerial vehicles (UAVs), also known as drones, will continue to surge over the next several years, with the global annual unit shipment increasing more than tenfold from 6.4 million in 2015 to 67.7 million by 2021. Such a dramatic increase is due to the steadily decreasing cost of UAVs and their fast-growing demand in applications such as surveillance and monitoring, aerial camera and radar, cargo delivery, communication platforms, etc. In June 2017, the International Olympic Committee (IOC) and Intel announced to use UAVs to enhance the future Olympic Games experience, such as the drone light show [3]. Just in the same month, the “Safe DRONE Act of 2017” was proposed to the United States Congress for accelerating the development of the UAV technology, which requested 14 million dollars for funding a host of research projects on UAVs [4]. In fact, many prominent companies such as Google, Qualcomm, Amazon, and Nokia, have already launched their respective programs to advance the UAV research and conduct UAV field tests. In particular, the recent trial results released by Qualcomm have shown that the current fourth-generation (4G) cellular network can provide reliable communications for UAVs at an altitude up to 400 feet [5], which paves the way for realizing cellular-enabled UAV communications in future. Besides being “drone clients” of wireless networks, UAVs can also be employed as various aerial communication platforms, to help improve the performance of existing wireless communication systems such as cellular networks [6].

Compared to the traditional terrestrial wireless communications, UAV-enabled communications mainly have the following three advantages. First, they are significantly less affected by channel impairments such as shadowing and fading, and in general possess more reliable air-to-ground channels due to the much higher possibility of having line-of-sight (LoS) links with ground users. Second, UAVs can be deployed more flexibly and moved more freely in the three-dimensional (3D) space. Third, the high mobility of UAVs can be fully controlled to enhance desired communication links and/or avoid undesired interference via proper trajectory design. These features bring both opportunities and challenges in designing UAV-enabled wireless communications, which were not explored before in conventional terrestrial systems with fixed ground base stations (BSs) [6]. As UAVs are suitable to serve as airborne communication hubs such as aerial BSs and/or relays, they are especially useful for the practical scenarios that require on-demand deployment in temporary events or emergency situations (such as natural disaster), when the ground infrastructures are insufficient or even unavailable. However, there are also open challenges that need to be tackled before realizing the full potential of UAV-enabled communications, such as UAV deployment and trajectory design, communications resource allocation and multiple access, etc.

Recently, UAV deployment problems have been extensively studied in the literature [7, 8, 9, 10, 11, 12]. The main design objective is to optimize the UAVs’ altitude, horizontal positions, and/or spatial density to achieve the maximum communication coverage in a given area. However, the high mobility or trajectory design of UAVs is not considered in these works. An energy-efficient relaying scheme is proposed in [13] where multiple UAVs cooperatively relay data packets from ground sensors to a remote BS based on time division multiple access (TDMA). Although the UAV mobility is considered in [13], the UAVs’ trajectories are assumed to be pre-determined and not optimized, which simplifies the design to a UAV-packet matching problem. In [14], both the static and mobile UAV-enabled wireless networks are studied which are underlaid with a device-to-device (D2D) communication network. Yet, the mobile UAV is only allowed to communicate at a set of stop points. As a result, the UAV trajectory is highly restricted and does not fully exploit the UAV’s high mobility for performance optimization. As such, a joint UAV trajectory and adaptive communication design is more promising. Motivated by this, a general trajectory and communication joint optimization framework is proposed in [15] for a UAV-enabled mobile relaying system, which is also extended to the energy efficiency maximization in a point-to-point UAV-ground communication system [16]. The UAV trajectory designs in [15] and [16] can be considered as a generalization of the UAV deployment problem, subject to practical constraints on the UAV’s mobility, such as its initial/final locations, maximum speed and acceleration, etc. Thus, it is intuitive that a mobile UAV with optimized trajectory in general can achieve higher throughput than a static UAV, even with optimal deployment/placement. For UAV-enabled multiuser communication networks, a novel cyclical multiple access scheme is proposed in [17] where the UAV periodically serves each of the ground users along its cyclical trajectory via TDMA. In [18], a joint user scheduling, power control, and trajectory optimization problem is investigated for a multi-UAV enabled multiuser system. It is shown that with joint trajectory design and power control, the UAVs can cooperatively serve the ground users in a periodic manner with strong LoS links and yet avoiding severe interference. However, this is achieved by scheduling each user to communicate only when its associated UAV is sufficiently close to it in each UAV flight period, which implies that the user has to wait for the next UAV flight period to communicate again. As a result, the throughput gain brought by the UAV’s mobility does not come for free but in fact at the cost of user communication delay.

Future wireless networks are expected to provide different quality-of-service (QoS) guarantees for a wide range of applications with diversified requirements [19, 20]. In fact, the delay requirements of wireless multimedia services may vary dramatically in a large scale from milliseconds such as for video conferencing and online gaming, to several seconds for file downloading/sharing and data backup. For example, when a user is downloading files which are delay-tolerant in general, some other users in the same area may be watching high-definition (HD) movies on YouTube that require minimum rates at any time. In light of this, wireless networks with such mixed services should be optimized to not only maximize the system total throughput but also meet the heterogeneous delay requirements of different applications. Although the UAV trajectory design has been shown to significantly enhance the throughput of various wireless communication systems such as for mobile relaying channel [15, 13], multiple access channel (MAC) and broadcast channel (BC) [21, 22, 17], interference channel (IFC)[18, 14], and wiretap channel [23, 14], all these works consider only delay-tolerant applications where the users’ instantaneous rates in general cannot be guaranteed. Therefore, it remains unknown whether mobile UAVs are still able to provide throughput gains over static UAVs/ground BSs, when the users’ delay or minimum-rate requirements are considered.

Motivated by such an open question, we study in this paper a UAV-enabled orthogonal frequency division multiple access (OFDMA) system, where a UAV is dispatched as a mobile BS to serve a group of users on the ground during a given finite period, as shown in Fig. 1. Besides having been standardized as the downlink multiple access scheme in the current 4G networks, OFDMA is also deemed as a promising candidate for the forthcoming fifth-generation (5G) wireless networks [24, 25, 26]. This is essentially attributed to its various advantages, e.g., flexible bandwidth and power allocation over users, which fits particularly well in our considered UAV-enabled network with heterogeneous user delay requirements. Without loss of generality, we consider two types of data traffic for each user: delay-constrained traffic where a minimum rate should be supported at any time during the UAV service period versus delay-tolerant traffic where no minimum rate is required and the transmission rate can be elastically allocated over the period. To this end, we introduce a minimum-rate ratio (MRR) for each user which is defined as the minimum instantaneous rate required over the achievable average throughput of the whole period. As such, the MRR can be flexibly adjusted by each user to specify the percentage of the delay-constrained traffic required versus that of the delay-tolerant traffic, depending on real-time applications. Our goal is to maximize the minimum average throughput of all ground users while meeting a given set of constraints on the users’ MRRs, by jointly optimizing the UAV trajectory and OFDMA resource allocation. Compared to prior works [13, 15, 16, 27, 17, 18, 21, 14, 28], such a joint UAV trajectory and resource allocation design is more general and practically useful since the user communication delay requirements are taken into account. This thus leads to a more fair throughput comparison with the terrestrial BSs or static UAVs and also helps to reveal a better understanding of the fundamental throughput gain achievable by exploiting the UAV’s mobility control subject to delay constraints. Intuitively, when the UAV flies towards some users to capture better channels with them, it gets farther away spontaneously from other users that are not in its heading direction and thus experiences degraded channels. As a result, more bandwidth and transmit power need to be allocated to those users so that their minimum rates can be achieved. This in turn would limit the potential rate increase of the users in the UAV’s heading direction. Thus, there exists a new and non-trivial tradeoff in the UAV trajectory design for throughput maximization when minimum-rate or delay constraints are considered.

The main contributions of this paper are summarized as follows. First, we formulate a joint UAV trajectory and OFDMA resource allocation optimization problem to maximize the minimum throughput of ground users while guaranteeing their specified MRR constraints. Next, we show that the system max-min throughput is non-increasing with respect to the users’ MRRs, which implies that the throughput gain of mobile UAVs over static UAVs reduces as the delay constraints become more stringent. Although the formulated problem is non-convex and challenging to solve, we propose an efficient iterative block coordinate descent algorithm to solve the bandwidth and power allocation problem and the UAV trajectory optimization problem alternately, which is guaranteed to converge. Specifically, in each iteration, the bandwidth and power allocation problem is solved optimally by applying the Lagrange duality method with given fixed UAV trajectory. While for the UAV trajectory optimization problem with fixed OFDMA resource allocation, the successive convex optimization technique is applied to tackle its non-convexity. However, due to the MRR constraints, it is shown that the conventional block coordinate descent method which directly iterates between the OFDMA resource allocation and UAV trajectory design will very likely get stuck at the initial point, which leads to an ineffective update of the UAV trajectory. To overcome this issue, we propose a new parameter-assisted block coordinate descent method where each parameter is a temporary MRR set to be larger than the target MRR for a corresponding user. Then at each iteration, we gradually decrease the temporary MRRs before solving the UAV trajectory optimization problem, until they reach the target MRRs for all users. It is shown that this new method can effectively update the UAV trajectory in each iteration, thus resolving the issue of conventional block coordinate descent method. Furthermore, we propose a systematic and low-complexity UAV trajectory initialization scheme based on the simple circular trajectory. Finally, numerical results are provided to verify the fundamental tradeoff between the system max-min throughput and the users’ delay/MRR constraints and demonstrate the effectiveness of our proposed designs.

The rest of this paper is organized as follows. Section II introduces the system model and the problem formulation for a UAV-enabled OFDMA network. In Section III, we propose an efficient iterative algorithm as well as a general UAV trajectory initialization scheme for the considered problem. Section VI presents the numerical results to demonstrate the performance of the proposed designs. Finally, we conclude the paper in Section VI.

*Notations:*

In this paper, scalars are denoted by italic letters, while vectors and matrices are respectively denoted by bold-face lower-case and upper-case letters.

denotes the space of -dimensional real-valued vector. For a vector , represents its Euclidean norm, denotes its transpose, and indicates that is element-wise larger than or equal to 0. For a time-dependent function , denotes its derivative with respect to time . For a set , denotes its cardinality.## Ii System Model and Problem Formulation

### Ii-a System Model

As shown in Fig. 1, we consider a UAV-enabled OFDMA system where the UAV is employed as an aerial BS to serve a group of users on the ground. The user set is denoted by with . In practice, the users that need to be served by the UAV can be either the terrestrial BSs that have no ground backhaul links or the ground mobile terminals in a geographical area that is poorly or not even covered by existing terrestrial BSs. At any time during the UAV flight period, denoted by , the UAV can communicate with multiple ground users simultaneously by employing OFDMA, i.e., assigning each user a fraction of the total bandwidth/transmit power. In general, from the perspective of throughput maximization, it is intuitive that larger is desirable since it will provide the UAV more time to fly closer to each ground user, leading to better air-to-ground links. The effect of on the system performance will be investigated in detail in Section IV. Since we target for a centralized design that is implemented off-line, the proposed algorithm can be performed at a central controller (e.g., installed on the UAV) who is able to collect all the users’ information such as their locations and MRRs. Then the obtained solutions can be programmed into the control and communication units of the UAV.

We assume that the horizontal coordinate of each ground user is known in advance and fixed at , . The UAV is assumed to fly at a fixed altitude above ground and the time-varying horizontal coordinate of the UAV at time instant is denoted by , with . To serve ground users periodically, we assume that the UAV needs to return to its initial location by the end of each period , i.e., . In addition, the UAV trajectory is also subject to the maximum speed constraints in practice, i.e, , where denotes the maximum UAV speed in meter/second (m/s). However, the continuous variable essentially implies an infinite number of UAV speed constraints which are difficult to tackle in general. To facilitate our analysis and algorithm design, we apply the discrete linear state-space approximation technique, which results in a finite number of constraints. Specifically, we discretize the UAV flight period into equally-spaced time slots with step size , i.e., . Note that for the given maximum UAV speed and altitude , the number of time slots can be chosen sufficiently large such that the UAV’s location change within each time slot can be assumed to be negligible, compared to the link distances from the UAV to all ground users. However, a larger value of inevitably increases the complexity of the proposed design. Thus, the number of time slots can be properly chosen in practice to balance between the accuracy and algorithm complexity. More discussions on the choice of as well as its impact on the system performance can be found in [18]. Based on such a discretization, the UAV trajectory can be characterized by a sequence of UAV locations , . As a result, the above constraints can be equivalently modeled as

(1) | ||||

(2) |

where is the maximum horizontal distance that the UAV can travel within one time slot. Furthermore, the distance from the UAV to user in time slot is assumed to be a constant that can be expressed as

(3) |

The measurement results in [29, 30, 31] have shown that the air-to-ground communication channels are mainly dominated by the LoS links even when the UAV is at a moderate altitude. For example, for the UAV at an altitude of

m, the LoS probability of air-to-ground links in rural environment exceeds 95% for a horizontal ground distance of 4 kilometers (Km)

[29]. Thus, we assume that the channel quality depends mainly on the UAV-user distance for simplicity. In addition, the Doppler effect induced by the UAV mobility is assumed to be perfectly compensated at the receivers. Following the free-space path loss model, the channel power gain from the UAV to user in time slot can thus be expressed as(4) |

where denotes the channel power gain at the reference distance m. Denote the total available system bandwidth by in Hertz (Hz). The fraction of bandwidth assigned to user in time slot is denoted by . In a practical OFDMA system, is in general a discrete value between and , which increases linearly with the number of subcarriers assigned to user in time slot . It is known that when the number of subcarriers is sufficiently large, can be approximated to a continuous value between and . Thus, the bandwidth allocation constraint can be expressed as

(5) | ||||

(6) |

Denote the transmit power allocated to user in time slot by . Then the total transmit power constraint of the UAV can be expressed as

(7) |

where is the maximum allowed transmit power of the UAV in each time slot. Accordingly, the instantaneous achievable rate of user in time slot , denoted by in bits/second/Hz (bps/Hz), can be expressed as

(8) |

where , with denoting the power spectral density of the additive white Gaussian noise (AWGN) at the receivers. As a result, the average achievable throughput of user over time slots in bps/Hz, is given by

(9) |

Motivated by the diversified user applications and heterogeneous delay requirements in the forthcoming 5G wireless networks, we consider both delay-constrained and delay-tolerant services for users in the system. Specifically, a parameter , , is introduced to denote the MRR of user , which means that at any time slot, fraction of its average throughput over slots is delay-constrained and the remaining fraction is delay-tolerant. In particular, and indicate that all the services of user are delay-tolerant and delay-constrained, respectively. As such, the MRR constraint of user in time slot can be expressed as

(10) |

which implies that in any of the time slots, at least fraction of the average throughput needs to be satisfied for each user . Therefore, a system where some users’ services are all delay-constrained while those of the others are all delay-tolerant, i.e., , is a special case of our general setup.

### Ii-B Problem Formulation

Let , , and . By taking into account the MRR constraints of all users, we aim to maximize the minimum average throughput among them via jointly optimizing the bandwidth and power allocation (i.e., and ) as well as the UAV trajectory (i.e., ). Define . The optimization problem is formulated as

(11a) | |||

(11b) | |||

(11c) | |||

(11d) | |||

(11e) | |||

(11f) | |||

(11g) | |||

(11h) | |||

(11i) |

Note that the challenges of solving problem (11) lie in the following three aspects. First, and in constraints (11b) and (11c) are not jointly concave with respect to the optimization variables , , and . Second, for fixed UAV trajectory , although and are jointly concave with respect to and , (11c) is non-convex due to the presence of in its left-hand-side (LHS). Third, for fixed bandwidth and power allocation and , and are neither convex nor concave with respect to . Consequently, problem (11) is a non-convex optimization problem and in general, there is no standard method for solving such a problem efficiently. To tackle the above challenges, we first transform problem (11) into a more tractable form as follows,

(12a) | |||

(12b) | |||

(12c) |

Comparing (12b) with (11c), it follows that the feasible set of problem (11) is a subset of that of problem (12) in general. However, the equivalence of problems (11) and (12) holds if all users achieve the equal average throughput in the optimal solution to problem (11). This can be easily verified since otherwise the objective value of (11) can be further improved by allocating more transmit power and/or bandwidth to the user with a lower average throughput without violating the total transmit power and bandwidth allocation constraints (11d) and (11f). With such a transformation, we only need to focus on solving problem (12) in the rest of the paper. Although it is still a non-convex optimization problem, problem (12) facilitates the development of an efficient algorithm. Before proceeding to solve problem (12), we first show the effect of the users’ MRRs on the maximum objective value of problem (12).

###### Theorem 1.

The maximum objective value of problem (12) is an element-wise non-increasing function with respect to .

###### Proof.

Denote the optimal solutions of problem (12) with and by and , respectively. To prove Theorem 1, we only need to show that holds when , where indicates element-wisely less than or equal to, i.e., , . Note that in problem (12), the MRRs are only involved in constraint (12b). Thus, we have the following inequalities

(13) |

which implies that is also a feasible solution of problem (12) with . Since is the maximum objective value of problem (12) with , it follows that , which thus completes the proof of Theorem 1. ∎

Theorem 1 sheds light on the fundamental tradeoff between the max-min average throughput and the user communication delay requirement: as the required MRR increases for any user

, the max-min average throughput of the system decreases in general. This is because imposing more stringent minimum-rate requirements on users fundamentally limit the UAV’s mobility to fly closer to any user to achieve a better channel since at the same time it needs to meet the minimum-rate requirements of other users that are not in its heading direction and thus will have degraded channels with it. As a result, the degree of freedom for exploiting the UAV’s mobility via its trajectory design is restricted by such delay/MRR constraints, thus leading to lower max-min average throughput.

## Iii Proposed Solution

In this section, we propose an efficient parameter-assisted block coordinate descent algorithm for solving problem (12). Specifically, we first optimize the bandwidth and power allocation for given UAV trajectory and then optimize the UAV trajectory for given bandwidth and power allocation. These two optimization problems are solved alternately until convergence is achieved.

### Iii-a Joint Bandwidth and Power Allocation

Besides being a subproblem of problem (12), the bandwidth and power allocation optimization problem for given UAV trajectory may also correspond to a practical scenario when the UAV trajectory is pre-specified due to other tasks such as aerial imaging, rather than being optimized for achieving best communication performance. Specifically, for any given UAV trajectory , the bandwidth and power allocation can be optimized by solving the following problem,

(14a) | |||

(14b) | |||

(14c) | |||

(14d) | |||

(14e) | |||

(14f) | |||

(14g) |

where . Define when , , such that the LHSs of both (14b) and (14c) are continuous with respect to over the whole domain . As such, problem (14) is a convex optimization problem since in (14b) and (14c) is jointly concave with respect to and and (14d)-(14g) are all affine constraints. Furthermore, it can be verified that Slater’s constraint qualification is satisfied for problem (14) [32]. Therefore, strong duality holds and the duality gap between problem (14) and its dual problem is thus zero, which means that the optimal solution can be obtained efficiently by applying the Lagrange duality. The partial Lagrange function of problem (14) can be expressed as

(15) |

where , , , and are the non-negative Lagrange multipliers associated with constraints (14b), (14c), (14d), and (14f), respectively. The boundary constraints (14e) and (14g) will be absorbed into the optimal solution in the following. Accordingly, the dual function is given by

(16a) | ||||

(16b) | ||||

(16c) | ||||

for which the following lemma holds. |

###### Lemma 1.

To make bounded from the above in (16), i.e., , it follows that must hold.

###### Proof.

This is shown by contradiction. If or , it follows that by setting or . Thus, neither of the above two inequalities can be true and the lemma is proved. ∎

Lemma 1 imposes additional constraints for dual variables and . As such, the dual problem of problem (14) is given by

(17a) | |||

(17b) | |||

(17c) |

Next, we show how to obtain the primal optimal solution by applying the Lagrange duality.

#### Iii-A1 Obtaining by Solving Problem (16)

With the given dual variables, problem (16) can be decomposed into subproblems that can be solved independently in parallel. Specifically, one subproblem is for optimizing and the other subproblems are for optimizing and , i.e.,

(18) |

(19a) | |||

(19b) | |||

(19c) |

where each subproblem in (19) is for user in time slot . For problem (18), since holds as in Lemma 1, the objective value is zero which is independent of the value of . This implies that we can choose any arbitrary real number as the optimal solution, denoted by . Without loss of generality, we simply set for the purpose of obtaining the dual function and updating the dual variables^{1}^{1}1We note that cannot be the optimal primal solution to problem (14). How to obtain the optimal primal solution for this problem will be discussed later in Section III-A-3). . Next, we consider problem (19). Since problem (19) is jointly concave with respect to and , the solution that satisfies the Karush-Kuhn-Tucker (KKT) conditions is also the optimal solution. By taking the derivative of the objective function of (19) with respect to , the optimal power allocation, denoted by , can be obtained as

(20) |

where . Let , which can be regarded as the power spectrum density of user in time slot . Note that in (20), the power allocation follows a multi-level water-filling structure. Substituting the obtained into problem (19) yields

(21a) | |||

(21b) |

It is evident that problem (21

) is a linear program (LP) with only one optimization variable,

. Thus, the optimal bandwidth allocation, denoted by , can be obtained as(22) |

where can be any arbitrary real number between 0 and 1 since the objective value of problem (21) is not affected in this case. For simplicity, we set as for the case of . In general, (22) cannot provide the optimal primal solution for problem (14) even with optimal dual variables. Nevertheless, with the above proposed solutions to problems (18) and (19), the dual function is obtained.

#### Iii-A2 Obtaining Optimal Dual Solution to Problem (17)

After obtaining for given , and , we next solve the dual problem (17) to find the optimal dual variables that maximize . Note that although the dual function is always convex by definition, it is non-differentiable in general. As a result, the commonly used subgradient based method such as the ellipsoid method, can be used to solve problem (17). In each iteration, the dual variables and are updated based on the subgradients of both the objective function and the constraint functions in problem (17). Specifically, the subgradient of the objective function is denoted by where , and are vectors with the elements respectively given by

(23) | ||||

(24) | ||||

(25) | ||||

(26) |

Furthermore, the equality constraint in (17b) is equivalent to two inequality constraints: and . Thus, the subgradient of the former constraint function is denoted by where the corresponding elements are given by . In addition, the subgradient of the latter constraint function is given by . With the above subgradients, the dual variables can be updated by the constrained ellipsoid method toward the optimal solution with global convergence [33].

#### Iii-A3 Constructing Optimal Primal Solution to Problem (14)

Based on the obtained dual optimal solution , , , and , it remains to obtain the optimal primal solution to problem (14). It is worth pointing out that for a convex optimization problem, the optimal solution that maximizes the Lagrangian function is the optimal primal solution if and only if such a solution is unique and primal feasible [32]. However, in our case, the optimal solutions and that maximize are not unique given that and as in (18) and (22). Therefore, additional steps are needed in order to construct the optimal primal solution. The key observation is that with given , , and , the optimal power spectrum density (the ratio of the optimal power allocation to the optimal bandwidth allocation), i.e., , can be uniquely obtained from (20). By substituting into the primal problem (14), we have

(27a) | |||

(27b) | |||

(27c) | |||

(27d) | |||

(27e) | |||

(27f) |

Given , it is easy to observe that problem (27) is an LP with respect to and , which thus can be efficiently solved by using standard convex optimization solvers such as CVX [34]. After obtaining the optimal bandwidth allocation , the corresponding power allocation can be obtained as , . The details of the procedures for obtaining the optimal solution to problem (14) are summarized in Algorithm 1. The computational complexity of Algorithm 1 consists of three parts. The first part is for solving problems (18) and (19), the second part is for updating the dual variables by the ellipsoid method, and the third part is for solving linear program problem (27). In step 3) of Algorithm 1, the complexity of solving problem (18) is and that of solving (19) is . The complexities of step 4) and 5) are and , respectively. Since the ellipsoid method takes to converge, the total complexity for step 2) to 6) is [33]. The complexity of solving (27) is . Therefore, the total complexity of Algorithm 1 is .

### Iii-B UAV Trajectory Optimization

Given any feasible bandwidth and power allocation , problem (12) is simplified into the following problem for optimizing the UAV trajectory only, i.e.,

(28a) | |||

(28b) | |||

(28c) | |||

(28d) | |||

(28e) |

where . Note that problem (28) is not a convex optimization problem since the LHSs of constraints (28b) and (28c) are not concave with respect to . In general, there is no efficient method to obtain the optimal solution for such a non-convex problem. However, we observe that the LHSs of both (28b) and (28c) are convex with respect to

. Note that for a convex function, its first-order Taylor expansion is the global under-estimator at any point

[32]. This thus motivates us to leverage the successive convex optimization technique to tackle the non-convex problem (28) by an iterative algorithm, where in each iteration, the LHSs of both (28b) and (28c) are replaced by more tractable functions derived from the Taylor expansion at a given local point. Specifically, with given local point , we have the following inequality(29) |

where

(30) | ||||

(31) |

For any given local point , define the function . With the lower bounds ,

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