Geometric Programming Problem Research Articles

Design of a compact MOSFET model suitable for geometric programming

ABSTRACT The design of analog integrated circuits is a demanding task that involves many constraints and objectives. Also, the transistor models employed must consider high-order physics effects to achieve accurate solutions. Geometric Programming (GP) allows finding the global minimum in optimisation problems, but requires design equations to be in monomial or posynomial form. This work proposes a simplified model inspired on the Advanced Compact MOSFET model tailored to be used in GP problems. The proposed model considers short-channel effects and it is valid in the saturation region with all inversion levels. The equations are presented, first showing the transistor current-voltage characteristic and then the associated capacitances. The validity of the equations is proven by their capacity to fit to the simulated data, achieving a mean error of 0.4% for the main NMOS characteristic. After that, the optimal design of two basic amplifier stages is performed to demonstrate the model usability with GP. The predicted voltage gain and bandwidth of the designed amplifiers are compared with simulations, presenting mean errors of 24.7% and 31.7%, respectively. These errors are low when viewed from a logarithmic perspective and considering the wide design space covered. As GP optimisation guarantees the global minimum location and does not require the usage of a circuit simulator in the loop, the proposed GP-friendly compact model can enable fast and accurate optimisation of analog integrated circuits.

ON EXACT OPTIMAL SOLUTION TO GEOMETRIC PROGRAMMING PROBLEMS

ON EXACT OPTIMAL SOLUTION TO GEOMETRIC PROGRAMMING PROBLEMS

Choosing Optimal Supply Radius of Transformer Substations (TSs) in Iraq’s Cities Using Geometric Programming with Neutrosophic Coefficients

Numerous uncertainties exist in various electricity power system problems due to the size, complexity, geographical distribution, and influence of unforeseen events in these systems, making it difficult for traditional mathematics tools based on crisp set theory to have an impact on and solve many power system problems. As a new branch of mathematical uncertainty techniques, the neutrosophic expert systems approach has therefore emerged with the development of electric power systems and has proven successful when correctly linked. The expert typically uses ambiguous or neutrosophic language to describe their empirical knowledge, such as "very likely," "quite likely," "if x is large, then y is very likely to occur," "x should not be less than a," etc. To design an optimal radius of power supply in the electrical transformer substation, this article presents a new method for creating primal and dual neutrosophic geometric programming problems. It also provides a numerical example to evaluate the approximate optimal economic power supply radius.

Optimizing elastic optical networks quality of transmission with an empirical optical amplifier gain saturation model

In this paper, we revisit the problem of adjusting transceiver power, amplifier gain, and nodal gain in optical-bypass-enabled networks, taking into account an empirical model of erbium-doped fiber amplifier (EDFA) gain saturation. We assume that network demands have been routed via the shortest possible paths. A heuristic algorithm, which estimates the degradation of the signal-to-noise ratio (SNR) for each demand, is used to assign the modulation level and spectrum slots to the routed demands. Our optimization process aims to maximize the minimum SNR margins of demands, by adjusting EDFA small-signal gain, variable optical attenuator (VOA) attenuation, and transceiver power. Constraints and impairments imposed by the physical-layer include EDFA and VOA gain limits, transceiver power limits, receiver sensitivity, insertion loss of optical switch input ports, EDFA gain saturation, EDFA noise, and fiber non-linear interference. These are modeled using the Gaussian noise model. The proposed optimization problem is formulated as a geometric programming problem and then converted into an equivalent convex problem.Numerical results suggest that in the optimal solution of our optimization, all demands are served with typically low bit error rates (BER) (<10−4). We calculate the operational gains of the EDFA to set the spans to full compensation whenever possible. Pre-amplifiers and line amplifiers do not experience gain saturation due to relatively small input powers, while approximately 10% of boosters undergo more than a 1 dB gain reduction due to high input powers. We observe that overlooking the effect of EDFA gain saturation and physical-layer-imposed constraints such as nodal gain limits or transceiver power limits during network optimization leads to a considerable number of blocked demands due to SNR degradation or high BER (>10−3).

Geometric Programming Problems with Triangular and Trapezoidal Twofold Uncertainty Distributions

Geometric Programming Problems with Triangular and Trapezoidal Twofold Uncertainty Distributions

Joint Task Offloading and Resource Allocation for Vehicular Edge Computing With Result Feedback Delay

In this paper, we study the problem of joint Task offloading and resource Allocation for vehicular edge computing with Result Feedback Delay (TARFD). Specifically, we consider a typical roadside unit (RSU) and vehicles within its coverage area, and optimize the task offloading decisions of vehicles as well as the uplink bandwidth allocation and the computation resources allocation on the RSU. The TARFD problem is formulated as a non-convex mixed integer nonlinear programming (MINLP) to minimize the average delay consisting of task offloading delay, task computation delay, and result feedback delay. We derive a lower bound of the optimum to the TARFD problem, based on which we propose an approximate algorithm of the TARFD problem, called A-TARFD. The A-TARFD algorithm can effectively deliver solutions for small-scale scenarios. To tackle large-scale scenarios, a low-complexity algorithm for the TARFD problem, called L-TARFD, is developed by constructing an iteratively updated sequence of locally tight approximate geometric programming (GP) problems. The L-TARFD algorithm can converge to a Karush-Kuhn-Tucker (KKT) point and forces the offloading decisions arbitrarily close to binary values. By comparison with the lower bound, simulation results show that the proposed two algorithms have near-optimal performance over a wide range of parameter settings.

Orthogonal-Time-Frequency-Space Signal Design for Integrated Data and Energy Transfer: Benefits From Doppler Offsets

Integrated data and energy transfer (IDET) is an advanced technology for enabling energy sustainability for massively deployed low-power electronic consumption components. However, the existing work of IDET using the orthogonal-frequency-division-multiplexing (OFDM) waveforms is designed for static scenarios, which would be severely affected by the destructive Doppler offset in high-mobility scenarios. Therefore, we proposed an IDET system based on orthogonal-time-frequency-space (OTFS) waveforms with the imperfect channel assumption, which is capable of counteracting the Doppler offset in high-mobility scenarios. At the transmitter, the OTFS-IDET system superimposes the random data signals and deterministic energy signals in the delay-Doppler (DD) domain with optimally designed amplitudes. The receiver optimally splits the received signal in the power domain for achieving the best IDET performance. After formulating a non-convex optimisation problem, it is transformed into a geometric programming (GP) problem through inequality relaxations to obtain the optimal solution. The simulation demonstrates that a higher amount of energy can be harvested when employing our proposed OTFS-IDET waveforms than the conventional OFDM-IDET ones in high mobility scenarios.

Multi-sinusoidal waveform shaping for integrated data and energy transfer in aging channels

Integrated data and energy transfer (IDET) is capable of simultaneously delivering on-demand data and energy to low-power Internet of Everything (IoE) devices. We propose a multi-carrier IDET transceiver relying on superposition waveforms consisting of multi-sinusoidal signals for wireless energy transfer (WET) and orthogonal-frequency-division-multiplexing (OFDM) signals for wireless data transfer (WDT). The outdated channel state information (CSI) in aging channels is employed by the transmitter to shape IDET waveforms. With the constraints of transmission power and WDT requirement, the amplitudes and phases of the IDET waveform at the transmitter and the power splitter at the receiver are jointly optimised for maximising the average direct-current (DC) among a limited number of transmission frames with the existence of carrier-frequency-offset (CFO). For the amplitude optimisation, the original non-convex problem can be transformed into a reversed geometric programming problem, then it can be effectively solved with existing tools. As for the phase optimisation, the artificial bee colony (ABC) algorithm is invoked in order to deal with the non-convexity. Iteration between the amplitude optimisation and phase optimisation yields our joint design. Numerical results demonstrate the advantage of our joint design for the IDET waveform shaping with the existence of the CFO and the outdated CSI.

Lower energy consumption in cache-aided cell-free massive MIMO systems

Cell-free (CF) massive MIMO systems provide uniform and fast service to users by the cooperation of multiple distributed access points (APs) at the expense of high energy consumption (EC) due to the heavy use of fronthaul and backhaul links. To tackle this issue, we propose a cache-aided cell-free (CCF) system where all APs are deployed with edge cache to reduce the load on links. In this paper, a scalable total EC model of CCF systems is established with two parts: EC of APs (ECAP) and EC for cache transmission (ECCT). An improved partial zero-forcing precoding scheme is proposed, which can provide high spectral efficiency and transmit few control data to reduce ECAP greatly. Then a minimum ECCT strategy is formulated. We transform ECCT minimization problem into a series of geometric programming problems by the successive convex approximation and an ingenious equivalent transformation. The obtained optimal cache placement makes the ECCT of CCF systems lower than that obtained by maximum hit ratio, equal probability and most popular caching strategies. Simulation results show that the proposed CCF systems are much more energy efficient than CF systems.

Parameterized Synthesis of Discrete-Time Positive Linear Systems: A Geometric Programming Perspective

This letter focuses on optimization problems of discrete-time positive linear systems. To this end, the synthesis problem is presented by introducing parameterized system coefficient matrices and optimizing system parameters directly. Based on results concerning positive linear systems and nonnegative matrix theory, we demonstrate that optimization problems of minimizing the parameter tuning cost while satisfying the H2 norm, H∞ norm, and ℓ1/ℓ∞ Hankel norm constraints can be reduced to corresponding geometric programming problems. In turn, by imposing reasonable assumptions on system matrices, these geometric programming problems can be further transformed into convex optimization problems owing to the convexity of the logarithm transformation on posynomials. Finally, simulation experiments on a numerical example and epidemic spreading process example are used to show the validity of the main results.

Joint Pilot and Data Power Control for Cell-Free Massive MIMO IoT Systems

Although cell-free massive multiple-input–multiple-output (mMIMO) Internet-of-Things (IoT) systems can improve spectral efficiency (SE) significantly compared with conventional cellular mMIMO IoT systems due to high coverage and no intercell interference, the pilot contamination and active sensors (ASs) interference can still seriously limit the further improvement of SE. In this article, we first develop a model of the cell-free mMIMO IoT system and then derive a mathematical expression of the SE to formulate a maximized sum SE optimization problem for jointly controlling pilot and data power. This problem is nonconvex; thus, a successive convex approximation (SCA) iteration algorithm is proposed, which, in fact, approximates the objective and constraint function in each iteration, and converts the initial optimization problem into an easy-to-solve geometric programming (GP) problem. Finally, simulation results show that our proposed algorithm outperforms the existing algorithms related to power control optimization in terms of the sum SE performance.

Energy Efficient HARQ for Ultrareliability via Novel Outage Probability Bound and Geometric Programming

Hybrid automatic repeat 1 request (HARQ) is a key enabler for ultrareliable communications. This paper optimizes transmit power for the initial transmission and the subsequent retransmissions of HARQ with either incremental redundancy or Chase combining, aiming to minimize the expected energy consumption given the target outage probability and the target latency. The main challenge is due to the fact that the outage probability is a complicated function of the power variables which are nested in successive convolutions. The existing works mostly use a classic upper bound to approximate the outage probability by assuming unbounded transmit power, then convert the original problem to a geometric programming (GP) problem. In contrast, we propose a novel and much tighter upper bound by taking the practical power limit into consideration. The new bound and the resulting new GP method are further extended to a broader group of channel models with various fading, multiple antennas, and multiple receivers. As shown in simulations, the GP method based on the new bound significantly outperforms the existing strategies that either fix transmit power or optimize power by the classic bounding technique.

Modeling and Optimizing the System Reliability Using Bounded Geometric Programming Approach

The geometric programming problem (GPP) is a beneficial mathematical programming problem for modeling and optimizing nonlinear optimization problems in various engineering fields. The structural configuration of the GPP is quite dynamic and flexible in modeling and fitting the reliability optimization problems efficiently. The work’s motivation is to introduce a bounded solution approach for the GPP while considering the variation among the right-hand-side parameters. The bounded solution method uses the two-level mathematical programming problems and obtains the solution of the objective function in a specified interval. The benefit of the bounded solution approach can be realized in that there is no need for sensitivity analyses of the results output. The demonstration of the proposed approach is shown by applying it to the system reliability optimization problem. The specific interval is determined for the objective values and found to be lying in the optimal range. Based on the findings, the concluding remarks are presented.

Solving geometric programming problems with triangular and trapezoidal uncertainty distributions

The geometric programming problem is an important optimization technique that is often used to solve different nonlinear optimization problems and engineering problems. The geometric programming models that are commonly used are generally based on deterministic and accurate parameters. However, it is observed that in real-world geometric programming problems, the parameters are frequently inaccurate and ambiguous. In this paper, we consider chance-constrained geometric programming problems with uncertain coefficients and with geometric programming techniques in the uncertain-based framework. We show that the associated chance-constrained uncertain geometric programming problem can be converted into a crisp geometric programming problem by using triangular and trapezoidal uncertainty distributions for the uncertain variables. The main aim of this paper is to provide the solution procedures for geometric programming problems under triangular and trapezoidal uncertainty distributions. To show how well the procedures and algorithms work, two numerical examples and an application in the inventory model are given.

Proportional Fair Scheduling for Downlink mmWave Multi-User MISO-NOMA Systems

In this paper, we study non-orthogonal multiple access (NOMA) user scheduling and resource allocation problem for a generic downlink single-cell multiple input and single output (MISO) millimeter wave (mmWave) system. The larger number of packed antennas and the highly directional property of mmWave communications enable directional beamforming to achieve spatial diversity. Toward this end, we consider two different hybrid precoding schemes which are based on orthogonal matching pursuit (OMP). Users are assigned into different clusters and the base station (BS) transmits superposed signals that share the same precoding vector. Moreover, both fixed number of users per cluster and dynamic number of users per cluster are investigated. We aim to jointly optimize the user clustering, service scheduling, and power allocation strategy, in maximizing the proportional fairness (PF) among the users and exploring the multiuser diversity and multiplexing gain. Since the formulated joint user clustering, scheduling and power allocation problem is a mixed integer non-convex optimization problem, we propose a two-fold methodology. First, we apply a hybrid precoding and user clustering scheme, where the hybrid precoder is constructed by singular vector division (SVD) or minimum mean square error (MMSE). Then, with the obtained result, we approximate the proportional fairness power allocation problem by a sequence of Geometric Programming (GP) problems which are solved iteratively. The proposed scheme strikes a balance between the spectral efficiency and service fairness. Results show that the proposed MISO-NOMA scheme which is based on MMSE hybrid precoder and the proposed user scheduling and power allocation strategy under proportional fairness metric can outperform various conventional MISO schemes. Furthermore, our proposed dynamic number of users per cluster scheme outperforms the fixed scheme and can be considered as an upper bound in several aspects, including spectral efficiency and fairness.

Estimating Noise Floor in Sentinel-1 Images With Linear Programming and Least Squares

Sentinel-1 is a synthetic aperture radar platform that provides free and open-source images of the Earth. A product type of Sentinel-1 is ground range detected (GRD), which records intensity while discarding phase information from the radar backscatter. Especially in cross-polarized GRD images, there are noticeable intensity changes throughout the image that are caused by amplifying the noise floor of the signal, which varies due to the nonuniform radiation pattern of the satellite's antenna. While Sentinel-1 has instrument processing facility (IPF) software to estimate the noise floor, even in the newer versions (3.1 or above) of the IPF software there are still instances where the estimates provided do not fit the actual noise floor in the image, which is particularly noticeable in transitions between adjacent subswaths. In this work, we propose a method that reduces the impact of the varying noise-floor throughout the image. The method models the intensity of the noise floor to be a power function of the radiation pattern power. The method divides the swath into several sections depending on the location of the local minimum and maximum of the radiation pattern power with respect to the range. The parameter estimation is portrayed as a geometric programming problem that is transformed into a linear programming problem by logarithmic transformation. Affine offsets are computed for each subswath by a weighted least squares approach. Vast improvement is found on extra-wide (EW) and interferometric wide (IW) Sentinel-1 modes over cross-polarized images. Code implementation is available at https://github.com/PeterQLee/sentinel1_denoise_rs.

Secure Transmission in NOMA-Aided Multiuser Visible Light Communication Broadcasting Network With Cooperative Precoding Design

In this paper, we study the secrecy performance of non-orthogonal multiple access (NOMA) enabled visible light communication (VLC) broadcast channels in the presence of an active eavesdropper (Eve). The considered VLC system consists of multiple separately distributed light-emitting diodes arrays and multiple randomly located users (UEs) in an indoor room. User clustering is conducted to reduce the implementation complexity of successive interference cancellations. Two cooperative precoding strategies based on zero-forcing (ZF) and maximum ratio transmission (MRT) are designed using the effective channel of each cluster. Based on each precoding strategy, a sum secrecy rate maximization problem is developed to obtain the near-optimal power allocation (PA) to strengthen UEs&#x2019; confidential transmission and degrade Eve&#x2019;s reception under minimum secrecy rate requirement, peak amplitude, non-negativity, and power constraints. To tackle the challenging non-convex problem for each precoding strategy, equivalent transformations and arithmetic-geometric mean approximation are conducted to convert the original problem into a series of geometric programming (GP) problems. Based on the reformulated problems, iterative algorithms are proposed to obtain near-optimal solutions by solving the GP problems through successive convex approximations. The convergence and complexity analysis of the proposed algorithms are studied. Simulation results show that the sum security performance of the proposed PA approach outperforms the conventional PA approaches in both ZF-based and MRT-based precoder schemes. The effectiveness of applying NOMA compared with the orthogonal multiple access-based scheme is also validated for the proposed system.

Energy-Efficient Interference-Aware Cognitive Machine-to-Machine Communications Underlaying Cellular Networks

Machine-to-machine (M2M) communications can effectively utilize cognitive radio (CR) to coexist with cellular users in what is known as cognitive M2M (CM2M) communications. In this system, underlay CR is used to manage spectrum sharing among machine type communication devices (MTCDs) and cellular user equipment (CUE) where the CUE are considered to be primary users (PUs) and the MTCDs are secondary users (SUs). Moreover, due to the limited battery capacity of MTCDs and diverse quality-of-service (QoS) requirements of both MTCDs and CUE, energy efficiency (EE) is critical to prolonging network lifetime. This paper investigates the power allocation problem for energy-efficient CM2M communications underlaying cellular networks. Underlay CR is employed to manage the coexistence of MTCDs and CUE and exploit spatial spectrum opportunities to improve spectrum utilization. Two power allocation problems are proposed where the first targets MTCD power consumption minimization while the second considers MTCD EE maximization subject to MTCD transmit power constraints, MTCD minimum data rate requirements, and CUE interference limits. The proposed power consumption minimization problem is transformed into a geometric programming (GP) problem and solved iteratively. The proposed EE maximization problem is a nonconvex fractional programming problem. Hence, a parametric transformation is used to convert it into an equivalent convex form and this is solved using an iterative approach. Simulation results are presented which show that the proposed algorithms provide MTCD power allocation with lower power consumption and higher EE than the equal power allocation (EPA) scheme while satisfying the constraints.

The maximal angle between 5×5 positive semidefinite and 5×5 nonnegative matrices

The paper is devoted to the study of the maximal angle between the $5\times 5$ semidefinite matrix cone and $5\times 5$ nonnegative matrix cone. A signomial geometric programming problem is formulated in the process to find the maximal angle. Instead of using an optimization problem solver to solve the problem numerically, the method of Lagrange Multipliers is used to solve the signomial geometric program, and therefore, to find the maximal angle between these two cones.

Cell-Free Massive MIMO Systems With Oscillator Phase Noise: Performance Analysis and Power Control

Cell-free massive multiple-input multiple-output (MIMO) is considered as a promising technology for satisfying higher rate requirement of users in beyond-5G networks. This paper investigates the impact of phase noise on the performance of cell-free MIMO systems which employ a large number of multi-antenna access points (APs). For both downlink and uplink transmissions, we derive the tractable closed-form achievable rate expressions when signal detection with channel state information (CSI) and channel distribution information (CDI), respectively, by utilizing the extended deterministic equivalence method in random matrix theory. The closed-form achievable rate expressions provide efficient evaluation for the impact of phase noise on system performance. The analytical results indicate that the phase noise influences the power of the desired signal, the beamforming uncertainty gain, and the inter-user interference increment caused by the pilot contamination. In addition, the special cases that only APs have phase noise and only users have phase noise are discussed. It is found that the impact of phase noise at users is more severe than that at APs. Besides, we discuss the system performance at low and high SNR regimes. It demonstrates that the phase noise is the major limitation of the system performance in high SNR regime while it can be neglected in low SNR regime. Furthermore, we formulate the max-min power control problem for both downlink and uplink transmissions to guarantee the users’ fairness and prove that the downlink max-min problem is a quasi-concave problem while the uplink max-min problem is a geometric programming (GP) problem. Then, the phase noise-aware max-min power control schemes for both downlink and uplink transmissions are proposed to ensure the users’ fairness. Finally, numerical results are provided to validate the analytical results and the proposed phase noise-aware power control scheme.