Branch and Bound Algorithm for Globally Solving Minimax Linear Fractional Programming

In this paper,we study the minimax linear fractional programming problem on a non-empty bounded set,called problem(MLFP),and we design a branch and bound algorithm to find a globally optimal solution of(MLFP).Firstly,we convert the problem(MLFP)to a problem(EP2)that is equivalent to it.Secondly,by applying the convex relaxation technique to problem(EP2),a convex quadratic relaxation problem(CQRP)is obtained.Then,the overall framework of the algorithm is given and its convergence is proved,the worst-case iteration number is also estimated.Finally,experimental data are listed to illustrate the effectiveness of the algorithm.

Sufficiency and Duality for Nondif ferentiable Multiobjective Fractional Programming Problems with (Φ,ρ,α)-V-Invexity

A new concept of(Φ,ρ,α)-V-invexity for differentiable vector-valued functions is introduced,which is a generalization of differentiable scalar-valued(Φ,ρ)-invexity.Based upon the(Φ,ρ,α)-V-invex functions,sufficient optimality conditions and MondWeir type dual theorems are derived for a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function and each constraint function contain a term involving the support function of a compact convex set.

Optimality conditions and duality for a class of nondifferentiable multiobjective generalized fractional programming problems

This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of （F, α, ρ, d）-V-convexity. Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.

The Optimal Conditions of the Linear Fractional Programming Problem with Constraint

In this article,the authors discuss the optimal conditions of the linear fractionalprogramming problem and prove that a locally optional solution is a globally optional solution and the locally optimal solution can be attained at a basic feasible solution withconstraint condition.

Optimality and Duality on Fractional Multi-objective Programming Under Semilocal E-convexity

In this paper, some necessary and sufficient optimality conditions are obtained for a fractional multiple objective programming involving semilocal E-convex and related functions. Also, some dual results are established under this kind of generalized convex functions. Our results generalize the ones obtained by Preda[J Math Anal Appl, 288（2003） 365-382].

DUALITY FOR MULTIOBJECTIVE FRACTIONAL PROGRAMMING INVOLVING n-SET FUNCTIONS

In this paper, two duality results are established under generalized ρ-convexity conditions for a class of multiobjective fractional programmign involvign differentiable n-sten functions.

THE SUFFICIENT EFFICIENCY CONDITIONS IN SEMIINFINITE MULTIOBJECTIVE FRACTIONAL PROGRAMMING UNDER HIGHER ORDER EXPONENTIAL TYPE HYBRID TYPE INVEXITIES

First, a class of higher order exponential type hybrid （α,β, γ, η, p, h（.,.）, κ（., .）, w（.,., .）, ω（.,.,.）, θ）-invexities is introduced, second, some parametrically sufficient efficiency conditions based on the higher order exponential type hybrid invexities are established, and finally some parametrically sufficient efficiency results under the higher order exponential type hybrid （a,β, γ, ρ, h（.,.）, k（.,-）, w（-,., .）, w（.,., .）, 0）-invexities are investigated to the context of solving semiinfinite multiobjective fractional programming problems. The notions of the higher order exponential type hybrid （a, β, γ η, p, h（., .）, n（., .）, w（-,.,-）, ω（.,.,.）, 0）-invexities encompass most of the generalized invexities in the literature. To the best of our knowledge, the results on semiinfinite multiobjective fractional programming problems established in this communication are new and application-oriented toward multitime multi- objectve problems as well as multiobiective control problems.

Joint Active and Passive Beamforming Design for Hybrid RIS-Aided Integrated Sensing and Communication

Integrated sensing and communication(ISAC) is considered an effective technique to solve spectrum congestion in the future. In this paper, we consider a hybrid reconfigurable intelligent surface(RIS)-assisted downlink ISAC system that simultaneously serves multiple single-antenna communication users and senses multiple targets. Hybrid RIS differs from fully passive RIS in that it is composed of both active and passive elements, with the active elements having the effect of amplifying the signal in addition to phase-shifting. We maximize the achievable sum rate of communication users by collaboratively improving the beamforming matrix at the dual function base station(DFBS) and the phase-shifting matrix of the hybrid RIS, subject to the transmit power constraint at the DFBS, the signal-to-interference-plus-noise-ratio(SINR) constraint of the radar echo signal and the RIS constraint are satisfied at the same time. The builtin RIS-assisted ISAC design problem model is significantly non-convex due to the fractional objective function of this optimization problem and the coupling of the optimization variables in the objective function and constraints. As a result, we provide an effective alternating optimization approach based on fractional programming(FP) with block coordinate descent(BCD)to solve the optimization variables. Results from simulations show that the hybrid RIS-assisted ISAC system outperforms the other benchmark solutions.

Joint Beamforming Design for Dual-Functional Radar-Communication Systems Under Beampattern Gain Constraints

The joint beamforming design challenge for dual-functional radar-communication systems is addressed in this paper.The base station in these systems is tasked with simultaneously sending shared signals for both multi-user communication and target sensing.The primary objective is to maximize the sum rate of multi-user communication,while also ensuring sufficient beampattern gain at particular angles that are of interest for sensing,all within the constraints of the transmit power budget.To tackle this complex non-convex problem,an effective algorithm that iteratively optimizes the joint beamformers is developed.This algorithm leverages the techniques of fractional programming and semidefinite relaxation to achieve its goals.The numerical results confirm the effectiveness of the proposed algorithm.

CoMP-transmission-based energy-efficient scheme selection algorithm for LTE-A systems

In order to achieve higher system energy efficiency （EE）,a new coordinated multipoint （CoMP）-transmission-based scheme selection energy saving （CTSES）algorithm is proposed for downlink homogeneous cellular networks.The problem is formulated as an optimization of maximizing system EE,under the constraints of the data rate requirement and the maximum transmit power.The problem is decomposed into power allocation and alternative scheme selection problems.Optimal power allocation is calculated for CoMP-JT （joint transmission）and CoMP-CS （coordinated scheduling） transmissions,and the scheme with higher EE is chosen. Since the optimal problem is a nonlinear fractional optimization problem for both CoMP transmission schemes, the problem is transformed into an equivalent problem using the parametric method. The optimal transmit power and optimal EE are obtained by an iteration algorithm in CoMP-JT and CoMP-CS schemes.Simulation results show that the proposed algorithm offers obvious energy-saving potential and outperforms the fixed CoMP transmission scheme.Under the condition of the same maximum transmit power limit,the empirical regularity of user distribution for scheme choice is presented, and using this regularity, the computational complexity can be reduced.

Joint Beamforming Optimization for Reconfigurable Intelligent Surface-Enabled MISO-OFDM Systems

In this paper,we investigate the reconfigurable intelligent surface(RIS)-enabled multiple-input-single-output orthogonal frequency division multiplexing(MISO-OFDM)system under frequency-selective channels,and propose a low-complexity alternating optimization(AO)based joint beamforming and RIS phase shifts optimization algorithm to maximize the achievable rate.First,with fixed RIS phase shifts,we devise the optimal closedform transmit beamforming vectors corresponding to different subcarriers.Then,with given active beamforming vectors,near-optimal RIS reflection coefficients can be determined efficiently leveraging fractional programming(FP)combined with manifold optimization(MO)or majorization-minimization(MM)framework.Additionally,we also propose a heuristic RIS phase shifts design approach based on the sum of subcarrier gain maximization(SSGM)criterion requiring lower complexity.Numerical results indicate that the proposed MO/MM algorithm can achieve almost the same rate as the upper bound achieved by the semidefinite relaxation(SDR)algorithm,and the proposed SSGM based scheme is only slightly inferior to the upper bound while has much lower complexity.These results demonstrate the effectiveness of the proposed algorithms.

Proportional Fairness-Based Energy-Efficient Power Allocation in Downlink MIMO-NOMA Systems with Statistical CSI

In this paper, proportional fairness(PF)-based energy-efficient power allocation is studied for multiple-input multiple-output(MIMO) non-orthogonal multiple access(NOMA) systems. In our schemes, statistical channel state information(CSI) is utilized for perfect CSI is impossible to achieve in practice. PF is used to balance the transmission efficiency and user fairness. Energy efficiency(EE) is formulated under basic data rate requirements and maximum transmitting power constraints. Due to the non-convex nature of EE, a two-step algorithm is proposed to obtain sub-optimal solution with a low complexity. Firstly, power allocation is determined by golden section search for fixed power. Secondly total transmitting power is determined by fractional programming method in the feasible regions. Compared to the performance of MIMO-NOMA without PF constraint, fairness is obtained at expense of decreasing of EE.

Delay-Throughput Tradeoff in Satellite Data Relay Networks with Prioritized User Satellites

The efficient antenna scheduling strategy for data relay satellites(DRSs)is essential to optimize the throughput or delay of the satellite data relay network.However,these two objectives conflict with each other since the user satellites(USs)with higher priorities take up more transmission time of DRSs’antennas for greater throughput but the USs storing more packets cause a severer waiting delay to the whole network.To balance the conflicting metrics for meeting the delay-throughput integrated requirements,we formulate the antenna scheduling as a stochastic non-convex fractional programming,which is challenging to be solved.For the tractability,we equivalently transform the fractional programming to a parametric problem and implement the Lyapunov drift to guarantee the constraint of mean rate stability.By proposing a delay and throughput tradeoff based antenna scheduling algorithm,we further transform the parametric problem to a solvable weight matching problem.Simulation results reveal the feasible region of the preference control parameter for integrated QoS cases and its variation relationship with network delay and throughput.

User scheduling and power allocation for downlink multi-cell multi-carrier NOMA systems

In Non-Orthogonal Multiple Access(NOMA),the best way to fully exploit the benefits of the system is the efficient resource allocation.For the NOMA power domain,the allocation of power and spectrum require solving the mixed-integer nonlinear programming NP-hard problem.In this paper,we investigate user scheduling and power allocation in Multi-Cell Multi-Carrier NOMA(MCMC-NOMA)networks.To achieve that,we consider Weighted Sum Rate Maximization(WSRM)and Weighted Sum Energy Efficiency Maximization(WSEEM)problems.First,we tackle the problem of user scheduling for fixed power using Fractional Programming(FP),the Lagrange dual method,and the decomposition method.Then,we consider Successive Pseudo-Convex Approximation(SPCA)to deal with the WSRM problem.Finally,for the WSEEM problem,SPCA is utilized to convert the problem into separable scalar problems,which can be parallelly solved.Thus,the Dinkelbach algorithm and constraints relaxation are used to characterize the closed-form solution for power allocation.Extensive simulations have been implemented to show the efficiency of the proposed framework and its superiority over other existing schemes.

Energy efficient resource allocation for D2D multicast communications

The resource allocation for device-to-device(D2D)multicast communications is investigated.To achieve fair energy efficiency(EE)among different multicast groups,the max-min fairness criterion is used as the optimization criterion and the EE of D2D multicast groups are taken as the optimization objective function.The aim is to maximize the minimum EE for different D2D multicast groups under the constraints of the maximum transmit power and minimum transmit rate,which is modeled as a non-convex and mixed-integer fractional programming problem.Here,suboptimal resource allocation algorithms are proposed to solve this problem.First,channel assignment scheme is performed to assign channel to D2D multicast groups.Second,for a given channel assignment,iterative power allocation schemes with and without loss of cellular users’rate are completed,respectively.Simulation results corroborate the convergence performance of the proposed algorithms.In addition,compared with the traditional throughput maximization algorithm,the proposed algorithms can improve the energy efficiency of the system and the fairness achieved among different multicast groups.

Energy-Efficient Target Channel Sequence Design for Spectrum Handoffs in Cognitive Radio Networks

Cognitive radio is considered as an efficient way to improve the spectrum efficiency. As one of its key technologies,spectrum handoff can guarantee the transmission continuity of secondary users(SUs). In this paper,we address a new and more generalized spectrum handoff problem in cognitive radio networks(CRNs),by considering simultaneously energy efficiency,multiple spectrum handoffs and multiple channels. Furthermore,effects of the primary users'(PUs')arrival and service rate on the target channel sequence selection are also considered. In order to obtain the energy-efficient target channel sequence,we firstly analyze the energy consumption and the number of delivered bits per hertz in the spectrum handoff process,and formulate a ratio-type energy efficiency optimization problem,which can be transformed into a parametric problem by utilizing fractional programming. Then,we propose an algorithm combining dynamic programming with bisection(DPB)algorithm to solve the energy efficiency optimization problem. Our simulation results verify that the designed target channel sequence has better performance than the existing algorithms in terms of energy efficiency.

Energy-Efficient Sensing and Transmission for Multi-Hop Relay Cognitive Radio Sensor Networks

How to achieve transmissions in an energy-efficient way in multi-hop decode and forward(DF) relay cognitive radio sensor networks(CRSNs) is important since sensor nodes in CRSNs are usually battery powered. This paper aims to maximize energy efficiency(EE) by joint optimizing sensing time and power allocation in multi-channels & multihops DF relay CRSNs under constraints on outage probability and sensing performance. First, we design a channel selection scheme for sensing according to the available probabilities of multi channels. Second, we analyze the expected throughput and energy consumption and formulate the EE problem as a concave/concave fractional program. Third, coordinate ascent and Charnes-Cooper Transformation(CCT) methods are used to transform the nonlinear fractional problem into an equivalent concave problem. Subsequently, the closed form of outage probability is derived and the convergence rate of the iterative algorithm is analyzed. Finally, simulation results show that the proposed scheme can achieve effective EE.

Optimization of energy efficiency for the full-duplex massive MIMO systems

The energy efficiency（EE） for the full-duplex massive multi-input multi-output（MIMO） system is investigated. Given the transmit powers of both the uplink and the downlink, the closed-form solutions of the optimal number of antennas and the maximum EE are achieved in the high regime of the signal-to-noise ratio（SNR）. It is shown that the optimal number of antennas and the maximum EE gets larger with the increase in user numbers. To further improve the EE, an optimization algorithm with low complexity is proposed to jointly determine the number of antennas and the transmit powers of both the uplink and the downlink. It is shown that, the proposed algorithm can achieve the system performance very close to the exhaustive search.

Optimal scenario balance of reduction in costs and greenhouse gas emissions for municipal solid waste management

To reduce carbon intensity, an improved management method balancing the reduction in costs and greenhouse gas(GHG)emissions is required for Tianjin's waste management system. Firstly, six objective functions, namely, cost minimization, GHG minimization, eco-efficiency minimization, cost maximization, GHG maximization and eco-efficiency maximization, are built and subjected to the same constraints with each objective function corresponding to one scenario. Secondly, GHG emissions and costs are derived from the waste flow of each scenario. Thirdly, the range of GHG emissions and costs of other potential scenarios are obtained and plotted through adjusting waste flow with infinitely possible step sizes according to the correlation among the above six scenarios. And the optimal scenario is determined based on this range. The results suggest the following conclusions. 1) The scenarios located on the border between scenario cost minimization and GHG minimization create an optimum curve, and scenario GHG minimization has the smallest eco-efficiency on the curve; 2) Simple pursuit of eco-efficiency minimization using fractional programming may be unreasonable; 3) Balancing GHG emissions from incineration and landfills benefits Tianjin's waste management system as it reduces GHG emissions and costs.

Energy Efficient Resource Allocation in Timesharing Multiuser Systems with Hybrid Energy Harvesting Transmitter

An energy effi cient resource allocation scheme in timesharing multiuser system with a hybrid energy harvesting transmitter is studied in this paper. Specially, the operation energy of system is supplied by constant energy and energy harvesting, which harvests energy from external environment. Our goal is to maximize the energy effi ciency of timesharing multiuser systems by considering jointly allocation of transmission time and power control in an off-line manner. The original nonconvex objective function is transformed into convex optimization problem via the fractional programming approach. Then, we solve the convex problem by Lagrange dual decomposition method. Simulation results show that the proposed energy efficient resource allocation scheme has a better performance than the scheme which decomposes optimization problem into two parts(power allocation, time allocation) to solve iteratively.