Charnes-Cooper Transformation Research Articles

A one-dimensional branching rule based branch-and-bound algorithm for minimax linear fractional programming

This paper investigates a type of minimax linear fractional program (MLFP) that often occurs in practical problems such as design of electronic circuits, finance and investment. We first transform the MLFP problem into an equivalent problem (EP) by using the Charnes–Cooper transformation and introducing an auxiliary variable. A linear relaxation strategy should simplify the nonconvex parts of the constraints in (EP). For globally solving MLFP, a branch-and-bound algorithm is then developed. It integrates the presented relaxation with the one-dimensional branching. The convergence of the algorithm is demonstrated and the number of worst-case iterations is estimated. Finally, preliminary numerical experiments verify that the proposed algorithm in this article is robust and efficient for solving the tested instances.

Investing in Wind Energy Using Bi-Level Linear Fractional Programming

Investing in wind energy is a tool to reduce greenhouse gas emissions without negatively impacting the environment to accelerate progress towards global net zero. The objective of this study is to present a methodology for efficiently solving the wind energy investment problem, which aims to identify an optimal wind farm placement and capacity based on fractional programming (FP). This study adopts a bi-level approach whereby a private price-taker investor seeks to maximize its profit at the upper level. Given the optimal placement and capacity of the wind farm, the lower level aims to optimize a fractional objective function defined as the ratio of total generation cost to total wind power output. To solve this problem, the Charnes-Cooper transformation is applied to reformulate the initial bi-level problem with a fractional objective function in the lower-level problem as a bi-level problem with a fractional objective function in the upper-level problem. Afterward, using the primal-dual formulation, a single-level linear FP model is created, which can be solved via a sequence of mixed-integer linear programming (MILP). The presented technique is implemented on the IEEE 118-bus power system, where the results show the model can achieve the best performance in terms of wind power output.

Joint Active and Passive Beamforming Design for Reconfigurable Intelligent Surface Enabled Integrated Sensing and Communication

To exploit the potential of the reconfigurable intelligent surface (RIS) in supporting integrated sensing and communication (ISAC), this paper proposes a novel joint active and passive beamforming design for RIS-enabled ISAC system in consideration of the target size. First, the detection probability for target sensing is derived in closed-form based on the illumination power on an approximated scattering surface area of the target, and a new concept of ultimate detection resolution (UDR) is defined for the first time to measure the target detection capability. Then, an optimization problem is formulated to maximize the signal-to-noise ratio (SNR) at the user-equipment (UE) under a minimum detection probability constraint. To solve this non-convex problem, a novel alternative optimization approach is developed. In this approach, the solutions of the communication and sensing beamformers are obtained by our proposed bisection-search based method. The optimal receive combining vector is derived from an equivalent Rayleigh-quotient problem. To optimize the RIS phase shifts, the Charnes-Cooper transformation is conducted to cope with the fractional objective, and a novel convexification process is proposed to convexify the detection probability constraint with matrix operations and a real-valued first-order Taylor expansion. After the convexification, a successive convex approximation (SCA) based algorithm is designed to yield a suboptimal phase-shift solution. Finally, the overall optimization algorithm is built, followed by detailed analyses on its computational complexity, convergence behavior and problem feasibility condition. Extensive simulations are carried out to testify the analytical properties of the proposed beamforming design, and to reveal two important trade-offs, namely, communication vs. sensing trade-off and UDR vs. sensing-duration trade-off. In comparison with several existing benchmarks, our proposed approach is validated to be superior when detecting targets with practical sizes.

An Asynchronous Collision-Tolerant ACRDA Scheme Based on Satellite-Selection Collaboration-Beamforming for LEO Satellite IoT Networks

In this paper, an asynchronous collision-tolerant ACRDA scheme based on satellite-selection collaboration-beamforming (SC-ACRDA) is proposed to solve the avalanche effect caused by packet collision under random access (RA) high load in the low earth orbit (LEO) satellite Internet of Things (IoT) networks. A non-convex optimization problem is formulated to realize the satellite selection problem in multi-satellite collaboration-beamforming. To solve this problem, we employ the Charnes-Cooper transformation to transform a convex optimization problem. In addition, an iterative binary search algorithm is also designed to obtain the optimization parameter. Furthermore, we present a signal processing flow combined with ACRDA protocol and serial interference cancellation (SIC) to solve the packet collision problem effectively in the gateway station. Simulation results show that the proposed SC-ACRDA scheme can effectively solve the avalanche effect and improve the performance of the RA protocol in LEO satellite IoT networks compared with benchmark problems.

Joint Reconfigurable Intelligent Surface Location and Passive Beamforming Optimization for Maximizing the Secrecy-Rate

The physical layer security (PLS) is investigated for reconfigurable intelligent surface (RIS) assisted wireless networks, where a source transmits its confidential information to a legitimate destination with the aid of a single small RIS in the presence of a malicious eavesdropper. A new joint RIS location and passive beamforming (J-LPB) optimization scheme is proposed for the sake of maximizing the secrecy rate under the RIS location constraint and the constraint that the modulus of the reflecting coefficient at each RIS's unit is not larger than 1. Specifically, we analyze the optimal location of the RIS, and conclude that the product involving the source-RIS distance and the RIS-destination distance should be minimized. Since the product minimization problem is nonconvex, we then propose a two-tier optimization algorithm for solving it. Based on the near-optimal RIS 3D location obtained, we further formulate the passive beamforming optimization problem, and then propose to apply the Charnes-Cooper transformation along with the sequential rank-one constraint relaxation (SROCR) algorithm to solve it. Our numerical results show that the secrecy rate of the proposed J-LPB optimization scheme is higher than that of the benchmarks. Explicitly, we use the following benchmarks: the near-source-based RIS location and passive beamforming (NSB-LPB) optimization scheme, the near-destination-based RIS location and passive beamforming (NDB-LPB) optimization scheme, and the random RIS location and passive beamforming (R-LPB) optimization scheme. Finally, the benefits of our J-LPB scheme are further increased with the number of RIS units.

Radio resource allocation for energy efficiency maximization in satellite–terrestrial integrated networks

Satellite–Terrestrial Integrated Networks (STIN), where a satellite access network liaising with terrestrial networks, is useful not only in proffering seamless coverage but also in improving the backhaul capacity for heavy traffic/dense network scenarios. Therefore, an energy-efficient STIN is envisioned to be a valued gap filler both in public safety networks and in the provision of high-speed data services with ubiquitous coverage in remote areas. STIN necessitate admission control, user association, optimal power distribution and spectrum resource allocation to attain the desired quality-of-service standards. This paper investigates joint admission control, user association and power distribution for ensuring fairness while associating users in STIN and fairness in the allocation of spectrum resources to associated users in STIN with an overall objective to maximize the energy efficiency of STIN. The reviewed problem is a concave fractional programming problem which by utilizing Charnes–Cooper transformation is converted into a concave optimization problem. Subsequently, the concave optimization problem is resolved via the proposed outer approximation algorithm. The performance of the ϵ-optimum solution is extensively evaluated via the execution of different system parameters including number of users, user association, user fairness and resource block fairness.

Penerapan algoritma Dinkelbach dan transformasi Charnes Cooper pada pemrograman fraksional linear di UD Bintang Furniture

Linear fractional programming is a special case of non-linear programming with an objective function consisting of the ratio of two linear functions. The problem can be solved using the Dinkelbach algorithm and the Charnes Cooper transformation. The essence of these two methods is to convert the problem of linear fractional programming into a linear programming problem which then provides an optimal value of each variable in its objective function. In this study, we will solve the problem of linear fractional programming at UD Bintang Furniture, that is to determine the optimal value of the comparison between profits and production costs of the company. The results show that the Dinkelbach algorithm method requires more iterations than Charnes Cooper's transformation. Despite this, both methods produce the same optimal value. 
 Keywords: Linear fractional programming, Dinkelbach algorithm, Charnes Cooper transformation. MSC2020: 90C32

Radar Target Detection Based on Waveform Design under Atmospheric Attenuation

Under the condition of atmospheric attenuation, the mismatch of radar echo signals leads to the decrease of signal-to-noise ratio (SNR) and detection probability. A joint optimization algorithm for transmitting signal and receiving filter of MIMO radar with atmospheric system response is presented in this paper. Considering the energy constraint, similarity constraint and spectral coexistence constraint of the transmitting signal, and introducing the atmospheric channel response matrix, the echo model under the atmospheric response condition is constructed. By optimizing the transmitting signal and the receiving filter, the echo signal SNR and radar detection probability can be improved. On this basis, the ARMA stochastic model is used to simulate the atmospheric system response. Solve non-convex problems by matrix optimization, positive semi-definite relaxation, Charnes-Cooper transformation, and loop optimization methods. Finally, the rank-one matrix decomposition method is used to extract the optimal signal from the signal autocorrelation matrix. And we analyze the convergence of the algorithm, the degree of SNR improvement and the detection probability. Through numerical experiments, it is proved that after the atmospheric channel response is introduced and matched correctly, the SNR of the echo signal can be improved and the radar detection probability can be greatly increased.

UAV-Assisted Physical Layer Security in Multi-Beam Satellite-Enabled Vehicle Communications

In this paper, we investigate unmanned aerial vehicle (UAV) assisted physical layer security in multi-beam satellite enabled vehicle communications. Particularly, the UAV is exploited as a relay to improve the secure satellite-to-vehicle link, and simultaneously serves as a jammer by deliberately generating artificial noise (AN) to confuse Eve. The satellite beamforming (BF) and UAV power allocation (PA) are jointly optimized to maximize the secrecy rate of the legitimate user within a target beam while guaranteeing the quality of service (QoS) of users within other beams. Since the problem is nonconvex, we first convert it into an equivalent two-stage problem. Then, the outer-stage problem is solved by using one-dimensional search, and the inner-stage problem is transformed to a bi-convex problem by using the semi-definite relaxation (SDR) and Charnes Cooper transformation. To solve the inner-stage bi-convex problem, we propose an iterative alternating optimization algorithm, where the optimal BF is obtained by semi-definite programming (SDP), and the optimal UAV PA is subsequently obtained by solving the reformulated fractional programming problem with an iterative Dinkelbach method. The tightness of SDR and the complexity of our proposed approach are analyzed, and extensive simulations are carried out to evaluate the effectiveness of our proposed approach.

Joint user association and energy efficiency maximization in beyond 5G heterogeneous networks

SummaryHeterogeneous networks (HetNets) seem to be the future of data networks supported by macrocells, small cells, relays, and D2D communication. The main purpose of HetNets is to facilitate maximum users while keeping energy efficiency (EE) to its peak. In this research work, we have formulated joint users association and EE maximization problem for HetNets, where our main goal is to increase the number of users associated with HetNets while keeping EE to its maximum. Formulated problem is a concave fractional problem in nature. We have used Charnes–Cooper Transformation to convert it to the concave optimization problem. We have used the (MADS) algorithm to solve the formulated optimization problem. Results have been analyzed after extensive simulations. Performance of MADS algorithm have been shown with respect to different system parameters, that is, the number of users associated, the minimum required data rate, and joint maximization of users associated and EE.

Mixed-integer linear-fractional programming model and it’s linear analogue for reducing inconsistency of pairwise comparison matrices☆

Pairwise comparison matrices are often used in multicriteria decision making (MCDM). The most critical part of this technique is the inconsistency, which emerges for logical reasons but can cause significant problems during the decision making process. Hence it is necessary to keep inconsistency below an acceptable threshold. In order to support the decision maker (DM) in making a rational decision, we must keep in mind the following: Our suggestion should be as close to the DM’s original result as possible, moreover it should have as low inconsistency as possible. We have studied various linear programming (LP) models that are used for reducing the inconsistency of pairwise comparison matrices (PCM) (Bozóki et al., 2011, 2015). These models, however aim at fulfilling only one of the previously mentioned two goals at a time. Therefore, the optimal solutions given may differ from each in the other respect, which is not taken as an objective but as a constraint in the model. So, they cannot be considered as equally good optimal solutions from a wider perspective. Based on our experiences concerning these models, we have developed a mixed-integer linear-fractional programming (MILFP) model that takes both mentioned goals as objectives by combining them into a linear-fractional objective function. We also provide the linear analogue (LA) of our MILFP model using an appropriate adaptation and combination of the Charnes-Cooper transformation and Glover’s linearization scheme.

Optimal Cooperative Transmission for Overlay Cognitive Satellite Terrestrial Networks

In this letter, we investigate an overlay cognitive satellite terrestrial network, where the secondary terrestrial network cooperates with the primary satellite network for spectrum access in a time splitting manner. To maximize the spectrum efficiency of such a cooperative cognitive network, we propose an optimal cooperative transmission scheme by optimizing the time splitting factors. Specifically, we firstly analyze the feasible range of time splitting factors to guarantee both transmission requirements of the satellite and terrestrial networks. On this basis, we decompose the non-concave optimization problem into several categories and employ the Charnes-Cooper transformation to determine the optimal time splitting factors.

Energy Efficient Resource Allocation for H-NOMA Assisted B5G HetNets

The resource allocation solution offered based on Non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) schemes are sub-optimal to address the challenging quality of service (QoS) and higher data rate viz-a-viz energy efficiency (EE) requirements in 5th generation (5G) cellular networks. In this work, we maximize the EE using user equipment (UE) clustering (UE-C) with downlink hybrid NOMA (H-NOMA) assisted beyond 5G (B5G) HetNets. We formulate an optimization problem incorporating UE admission in a cluster, UE association with a base station (BS), and power allocation assisted by H-NOMA, i.e., OMA and NOMA schemes in the macro base station (MBS) only and heterogeneous networks (HetNets) environments. The problem formulated is a type of non-linear concave fractional programming (CFP) problem. The Charnes-Cooper transformation (CCT) is applied to the formulated non-linear CFP problem to convert it into a concave optimization, i.e., mixed-integer non-linear programming (MINLP) problem. A two-phase ϵ-optimal outer approximation algorithm (OAA) is used to solve the MINLP problem. The simulation results show that H-NOMA with HetNets outperforms H-NOMA with MBS only in terms of UE admission, UE association, throughput, and EE.

A Novel Pilot Spoofing Scheme via Intelligent Reflecting Surface Based on Statistical CSI

Pilot spoofing attack brings new challenges to physical layer secure transmission. However, this method will not work without any knowledge about the pilot sequence and active eavesdropping can be detected by constructing random pilot sequence. Intelligent reflecting surface (IRS), with the real-time programmable features for wireless channels, provides new possibilities for effective pilot spoofing. In this paper, the IRS is deployed near the legitimate users and the legitimate signal can always be passively reflected. Then the control strategy is embedded into the communication process under time-division duplex mode to assist eavesdroppers to conduct pilot spoofing. By setting different phase shifts at the IRS during the uplink phase and downlink phase, the channel reciprocity disappears, and thus secure beamforming vector is biased towards the eavesdropper. Furthermore, in order to overhear more information, the average secrecy rate minimization problem based on statistical channel state information is established by carefully designing the phase shifts, which is non-trivial to solve. With alternating optimization and Charnes-Cooper transformation technique, the original problem is transformed into convex form and a near optimal solution is achieved. Finally, simulation results show that our proposed scheme can pose serious secure threat without any energy footprint. What's more, if the IRS is not utilized by the internal users properly, it will bring more threat.

Interference Suppression for Railway Wireless Communication Systems: A Reconfigurable Intelligent Surface Approach

Due to transmission of train control and dispatch commands, the ultra high reliability is required for railway wireless communication systems (RWCS). However, the quality of service (QoS) of RWCS is affected by external interference and jamming seriously in railway transportation. The external interference and jamming generated from intentional and/or unintentional sources can disturb the transmission of RWCS and be extremely dangerous for the trains’ safety. In this paper, a reconfigurable intelligent surface (RIS) approach is proposed to suppress the interference and jamming in RWCS. By taking the signal-to-interference-plus-noise (SINR) as the QoS metrics, the interference suppression problem with RIS is transformed into a maximizing receiving SINR problem. Two optimum solutions, which one is based on Charnes-Cooper transformation and the other is based on single linear programming, are provided. In order to deploy the RIS in RWCS practically, two low-complexity sub-optimum solutions, which one is to bypass the channel estimation for interference links and the other is to maximize the receiving desired power, are also discussed. The simulations results show that the RIS can be employed in RWCS with considerable anti-interference gain.

The strict complementarity in linear fractional optimization

As an important duality result in linear optimization, the Goldman–Tucker theorem establishes strict complementarity between a pair of primal and dual linear programs. Our study extends this result into the framework of linear fractional optimization. Associated with a linear fractional program, a dual program can be defined as the dual of the equivalent linear program obtained from applying the Charnes–Cooper transformation to the given program. Based on this definition, we propose new criteria for primal and dual optimality by showing that the primal and dual optimal sets can be equivalently modeled as the optimal sets of a pair of primal and dual linear programs. Then, we define the concept of strict complementarity and establish the existence of at least one, called strict complementary, pair of primal and dual optimal solutions such that in every pair of comple mentary variables, exactly one variable is positive and the other is zero. We geometrically interpret the strict complementarity in terms of the relative interiors of two sets that represent the primal and dual optimal setsin higher dimensions. Finally, using this interpretation, we develop two approaches for finding a strict complementary solution in linear fractional optimization. We illustrate our results with two numerical examples.

Maximizing the minimum secrecy rate of two-way relay networks

This paper concerns maximizing the minimum achievable secrecy rate in a two-way multi-antenna relay network in which two single-antenna nodes aim at exchanging their confidential messages with the help of some multi-antenna relays incorporating the Amplify and Forward (AF) strategy, while a single-antenna passive eavesdropper attempts to overhear the exchanged information. The communication is carried out in two hops. During the first hop, two source nodes transmit their information, hence, the relays get a combination of transmitted signals. Then, throughout the second hop, each relay applies a beamforming matrix to its received signal vector and broadcasts the resulting signal vector to the transmitting ends. Also, the eavesdropper overhears the exchanged information of two hops. The main goal persuaded in the current work is to devise proper beamforming strategies at the relays to improve the minimum secrecy rate. The problem is cast as an optimization problem, where it is shown that the underlying problem is not convex in general. Hence, two suboptimal approaches, called null-space (NS) and interference-leakage alignment (ILA) are devised, where using the so-called semi-definite relaxation (SDR) and Charnes–Cooper transformation techniques, the underlying problems are formulated as semi-definite programming (SDP) problems which can be numerically solved.

Resource allocation for energy efficiency optimization in uplink–downlink decoupled 5G heterogeneous networks

SummaryHeterogeneous networks (HetNets) are a practical solution for traffic offloading from a high powered base station (HBS) to a low powered base station (LBS). In the HetNets, uplink (UL)–downlink (DL) decoupled access (UDDa) strategy is an optimal solution that ensures cell association, independently, in the UL and DL. This strategy offloads HBS cell edge mobile device (MD) to the nearby LBS in the UL. However, energy‐efficient cell association for traffic offloading from HBS to LBS, relay, or device to device (D2D) in the UL employing UDDa strategy has not been explored in the past work. We formulate mathematical models which ensure energy‐efficient cell association, power allocation, and traffic offloading employing traditional UL‐DL coupled access (UDCa) and UDDa strategies in the HetNets. The formulated problems are concave fractional programming (CFP) problems. The CFP is changed to a concave optimization problem using Charnes–Cooper transformation (CCT). The transformed problems are solved using an outer approximation algorithm to obtain optimal solution. Simulation results show the effectiveness of UDDa strategy over UDCa strategy in terms of MDs association, traffic offloading in the UL and DL, interference mitigation in the UL, data rate, and energy efficiency in the HetNets.

Solving data envelopment analysis models with sum-of-fractional objectives: a global optimal approach based on the multiparametric disaggregation technique

The majority of data envelopment analysis (DEA) models can be linearized via the classical Charnes–Cooper transformation. Nevertheless, this transformation does not apply to sum-of-fractional DEA efficiencies models, such as the secondary goal I (SG-I) cross efficiency model and the arithmetic mean two-stage network DEA model. To solve a sum-of-fractional DEA efficiencies model, we convert it into bilinear programming. Then, the obtained bilinear programming is relaxed to mixed-integer linear programming (MILP) by using a multiparametric disaggregation technique. We reveal the hidden mathematical structures of sum-of-fractional DEA efficiencies models, and propose corresponding discretization strategies to make the models more easily to be solved. Discretization of the multipliers of inputs or the DEA efficiencies in the objective function depends on the number of multipliers and decision-making units. The obtained MILP provides an upper bound for the solution and can be tightened as desired by adding binary variables. Finally, an algorithm based on MILP is developed to search for the global optimal solution. The effectiveness of the proposed method is verified by using it to solve the SG-I cross efficiency model and the arithmetic mean two-stage network DEA model. Results of the numerical applications show that the proposed approach can solve the SG-I cross efficiency model with 100 decision-making units, 3 inputs, and 3 outputs in 329.6 s. Moreover, the proposed approach obtains more accurate solutions in less time than the heuristic search procedure when solving the arithmetic mean two-stage network DEA model.

Algorithm for Solving Linear Fractional Programming

To solve the Linear Fractional Programming system, the Charnes-Cooper Transformation is used to split the original objective function to an equivalent, simplified linear programming system. To prove the congruence of two systems, the feasibility and optimality of the linear fractional program maximizes or minimizes a ratio of affine functions over a polyhedral set is proved. In the last part of the paper, a MATLAB function is used to design an algorithm which accepts the input of parameters of a fractional polyhedral set, and outputs the optimal solution.