Geometric Programming Research Articles

A non-linear multi-objective technique for hybrid peer-to-peer communication

This work proposes a strategy management technique based on hybrid peer-to-peer communication system. The main techniques used in the P2PC are: (i) Multi-objective optimization, (ii) Game theory technique, (iii) Non-linear geometric programming, and (iv) Intuitionistic fuzzy logic. Multi-objective optimization is used to design multiple non-linear objective functions. Game theory is used to model the conflicting strategies of the nodes using payoff matrices. Non-linear geometric programming is used to estimate uncertainty related parameters. Intuitionistic fuzzy logic is used to reduce the imprecise parameters of the nodes with the support of some objective functions. The set of stated techniques gives effective mathematical modelling to analyze the conflicting situation of the nodes. Therefore, this modelling helps to derive intelligent communication between source and destination nodes. Mathematical analysis and simulation results are used to validate the P2PC method. The simulation results are achieved in the Lingo optimization simulator using the existing methodologies in terms of some performance metrics.

An Optimization Framework for Federated Edge Learning

The optimal design of federated learning (FL) algorithms for solving general machine learning (ML) problems in practical edge computing systems with quantized message passing remains an open problem. This paper considers an edge computing system where the server and workers have possibly different computing and communication capabilities and employ quantization before transmitting messages. To explore the full potential of FL in such an edge computing system, we first present a general FL algorithm, namely GenQSGD, parameterized by the numbers of global and local iterations, mini-batch size, and step size sequence. Then, we analyze its convergence for an arbitrary step size sequence and specify the convergence results under three commonly adopted step size rules, namely the constant, exponential, and diminishing step size rules. Next, we optimize the algorithm parameters to minimize the energy cost under the time constraint and convergence error constraint, with the focus on the overall implementing process of FL. Specifically, for any given step size sequence under each considered step size rule, we optimize the numbers of global and local iterations and mini-batch size to optimally implement FL for applications with preset step size sequences. We also optimize the step size sequence along with these algorithm parameters to explore the full potential of FL. The resulting optimization problems are challenging non-convex problems with non-differentiable constraint functions. We propose iterative algorithms to obtain KKT points using general inner approximation (GIA) and tricks for solving complementary geometric programming (CGP). Finally, we numerically demonstrate the remarkable gains of GenQSGD with optimized algorithm parameters over existing FL algorithms and reveal the significance of optimally designing general FL algorithms.

Applying Multi-objective Geometric Programming to the Fuzzy Inventory of Kerosene in Al-Dora Refinery

Amulti-objective inventory model was formulated bydecreasingthe unit cost on which orderdepends. The model was formulated with three constraints: storage space, number of orders, and production cost. We note in the research that the cost parameters in the objective function and constraints are imprecise in nature. Hence,the cost criteria are superimposed on the objective function and constraints in an ambiguous environment. Geometricprogramming was used to solve the production model without fuzzydeficit. It turned out that this model is better, as different possibilities were taken, and the total cost at these possibilities was shown. The best order volume was (1544117.647) liters, because it achieved the lowest total cost of storage, which amounted to (94,577,205.91) Iraqi dinars

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.

Physical Layer Security in Two-Way SWIPT Relay Networks with Imperfect CSI and a Friendly Jammer.

In this paper, the security of two-way relay communications in the presence of a passive eavesdropper is investigated. Two users communicate via a relay that depends solely on energy harvesting to amplify and forward the received signals. Time switching is employed at the relay to harvest energy and obtain user information. A friendly jammer is utilized to hinder the eavesdropping from wiretapping the information signal. The eavesdropper employs maximal ratio combining and selection combining to improve the signal-to-noise ratio of the wiretapped signals. Geometric programming (GP) is used to maximize the secrecy capacity of the system by jointly optimizing the time switching ratio of the relay and transmit power of the two users and jammer. The impact of imperfect channel state information at the eavesdropper for the links between the eavesdropper and the other nodes is determined. Further, the secrecy capacity when the jamming signal is not perfectly cancelled at the relay is examined. The secrecy capacity is shown to be greater with a jammer compared to the case without a jammer. The effect of the relay, jammer, and eavesdropper locations on the secrecy capacity is also studied. It is shown that the secrecy capacity is greatest when the relay is at the midpoint between the users. The closer the jammer is to the eavesdropper, the higher the secrecy capacity as the shorter distance decreases the signal-to-noise ratio of the jammer.

Strong solvability of restricted interval systems and its applications in quadratic and geometric programming

We consider interval systems of linear equations and inequalities with a restriction to some a priori given set. We focus on a characterization of strong solvability, that is, solvability for each realization of interval values, and we compare this with an existence of a strong solution defined analogously. The motivation comes from the area of interval-valued optimization problems, where strong solvability means guaranteed feasibility of any realization of the problem. Strong solvability with strict inequalities implies the robust Slater condition, which ensures that standard optimality conditions can be used. We apply the issues particularly in two optimization classes, convex quadratic programming with quadratic constraints and posynomial geometric programming. For the former, we also utilize the presented result to improve a characterization of the worst case optimal value. Eventually, we state several open problems that emerged while deriving the results.

A GEOMETRIC PROGRAMMING APPROACH TO CONVEX MULTI-OBJECTIVE PROGRAMMING PROBLEMS

Over the past few years, convex optimization has played a vital role in the study of complex engineering problems in different fields. Geometric programming is one of the available techniques particularly used for solving nonconvex programming problems. But in this article, a suitable attempt has been made to solve a real-life model on convex multi-objective using geometric programming technique with help of the E-constraint method and result is compared with the solutions obtained by fuzzy technique. Finally, a conclusion is presented by analyzing the solutions to a numerical problem.

Geometric Programming Assisted Conversion Time Reduction Technique Applied to a Multiregime-Based Digitizer for Wide Span Resistive Sensors

This manuscript proposes a geometric-programming (GP)-aided approach for reducing the conversion time of digitizers for resistive sensors. The digitizer, comprising a relaxation oscillator circuit, partitions the sensor span into multiple regimes. An optimization problem with the objective of minimizing the maximum conversion time for the multiple regimes, while maintaining lower limits to on and off durations is formulated. This constrained optimization problem is shown to be a GP and solved numerically. Optimal thresholds for partitioning the span as well as the optimal circuit parameters have been derived from the GP-based solution. It has been shown that the usage of GP, in the digitizer, can reduce the conversion time by at least a factor of three. The performance of the digitizer, designed with GP-solutions, has been evaluated in simulation and experimental studies. Results show that the digital interface circuit requires only around 17 ms during the measurement range of 1 kΩ to 1 MΩ and provides a nonlinearity of 0.2 %.

Incorporating Promotional Effects in Sales Planning of Retail Industry Using Geometric Programming

Incorporating Promotional Effects in Sales Planning of Retail Industry Using Geometric Programming

Design Optimization of Battery-Electric Marine Vessels via Geometric Programming

Design Optimization of Battery-Electric Marine Vessels via Geometric Programming

AoI Minimization for WSN Data Collection With Periodic Updating Scheme

In this paper, we consider the design of a wireless sensor network (WSN) that aims at monitoring the environment and collecting data periodically. In view of the limited energy and computational capability of the sensor nodes, a mobile edge computing (MEC) server is deployed in the WSN as a data processing unit. The goal of the design is to maintain the freshness of the data, which is characterized by the criterion of the age of information (AoI). Therefore, we analyze the long-term average AoI of the considered network. Then, the energy and time constraints for the WSN are modeled with consideration of transmission and computation. Next, a non-convex average AoI minimization problem is formulated subject to the energy and time constraints by jointly optimizing the sampling rate, computing scheduling, and transmit power. To tackle the challenging problem, the geometric programming and successive convex approximation (SCA) technique are applied to develop an algorithm with convergence guarantee. Moreover, to exhibit the benefits of the MEC server, a joint design is investigated for the WSN without the MEC server. Finally, the numerical results demonstrate the efficiency of our proposed SCA-based algorithm and show the impact of the sampling rate on the AoI performance.

Sum Rate Maximization of MIMO Monostatic Backscatter Networks by Suppressing Residual Self-Interference

Monostatic backscatter (MBS) networks provide connectivity for ultra-low power and low-cost tags for numerous applications. However, the reader operates in the full-duplex (FD) mode and experiences self-interference (SI) levels much higher (e.g., 160 dB) than the desired signal. However, hardware-based SI cancellation (SIC) techniques can remove SI partially only. Therefore, residual SI (RSI) dramatically degrades system performance. To remedy this problem, we develop a sum-rate maximization algorithm that suppresses the RSI and ensures that the tags harvest sufficient energy. It jointly optimizes the reader precoder and combiners and the tags reflection coefficients. Because of the non-convexity of this problem, we utilize alternating optimization (AO) to split it into three parts. They are then solved using successive convex approximation (SCA) and semidefinite relaxation (SDR) techniques to yield the precoder, a generalized Rayleigh quotient-based closed-form solution for the combiners, and geometric programming (GP) to get the reflection coefficients. Simulation results validate the fast convergence of the algorithm and show significant sum rate improvements (more than 21%) over the baselines.

Resource Allocation for Uplink Cell-Free Massive MIMO Enabled URLLC in a Smart Factory

Smart factories need to support the simultaneous communication of multiple industrial Internet-of-Things (IIoT) devices with ultra-reliability and low-latency communication (URLLC). Meanwhile, short packet transmission for IIoT applications incurs performance loss compared to traditional long packet transmission for human-to-human communications. On the other hand, cell-free massive multiple-input and multiple-output (CF mMIMO) technology can provide uniform services for all devices by deploying distributed access points (APs). In this paper, we adopt CF mMIMO to support URLLC in a smart factory. Specifically, we first derive the lower bound (LB) on achievable uplink data rate under the finite blocklength (FBL) with imperfect channel state information (CSI) for both maximum-ratio combining (MRC) and full-pilot zero-forcing (FZF) decoders. The derived LB rates based on the MRC case have the same trends as the ergodic rate, while LB rates using the FZF decoder tightly match the ergodic rates, which means that resource allocation can be performed based on the LB data rate rather the exact ergodic data rate under FBL. The log-function method and successive convex approximation (SCA) are then used to approximately transform the non-convex weighted sum rate problem into a series of geometric program (GP) problems, and an iterative algorithm is proposed to jointly optimize the pilot and payload power allocation. Simulation results demonstrate that CF mMIMO significantly improves the average weighted sum rate (AWSR) compared to centralized mMIMO. An interesting observation is that increasing the number of devices improves the AWSR for CF mMIMO whilst the AWSR remains relatively constant for centralized mMIMO.

Transmission Scheme, Detection and Power Allocation for Uplink User Cooperation With NOMA and RSMA

In this paper, we propose two novel cooperative-non-orthogonal-multiple-access (C-NOMA) and cooperative-rate-splitting-multiple-access (C-RSMA) schemes for uplink user cooperation. At the first mini-slot of these schemes, each user transmits its signal and receives the transmitted signal of the other user in full-duplex mode, and at the second mini-slot, each user relays the other user’s message with amplify-and-forward (AF) protocol. At both schemes, to achieve better spectral efficiency, users transmit signals in the non-orthogonal mode in both mini-slots. In C-RSMA, we also apply the rate-splitting method in which the message of each user is divided into two streams. In the proposed detection schemes for C-NOMA and C-RSMA, we apply a combination of maximum-ratio-combining (MRC) and successive-interference-cancellation (SIC). Then, we derive the achievable rates for C-NOMA and C-RSMA, and formulate two optimization problems to maximize the minimum rate of two users by considering the proportional fairness coefficient. We propose two power allocation algorithms based on successive-convex-approximation (SCA) and geometric-programming (GP) to solve these non-convex problems. Next, we derive the asymptotic outage probability of the proposed C-NOMA and C-RSMA schemes, and prove that they achieve diversity order of two. Finally, the above-mentioned performance is confirmed by simulations.

User Scheduling and Power Allocation for Precoded Multi-Beam High Throughput Satellite Systems With Individual Quality of Service Constraints

For extensive coverage areas, multi-beam high throughput satellite (HTS) communication is a promising technology that plays a crucial role in delivering broadband services to many users with diverse Quality of Service (QoS) requirements. This article focuses on multi-beam HTS systems where all beams reuse the same spectrum. In particular, we propose a novel user scheduling and power allocation design capable of providing guarantees in terms of the individual QoS requirements while maximizing the system throughput under a limited power budget. Precoding is employed in the forward link to mitigate mutual interference among the users in multiple-access scenarios over different coherence time intervals. The combinatorial optimization structure from user scheduling requires an extremely high cost to obtain the global optimum even when a reduced number of users fit into a time slot. Therefore, we propose a heuristic algorithm yielding a good trade-off between performance and computational complexity, applicable to a static operation framework of geostationary (GEO) satellite networks. Although the power allocation optimization is signomial programming, non-convex on a standard form, the solution can be lower bounded by the global optimum of a geometric program with a hidden convex structure. A local solution to the joint user scheduling and power allocation problem is consequently obtained by a successive optimization approach. Numerical results demonstrate the effectiveness of our algorithms on GEO satellite networks by providing better QoS satisfaction combined with outstanding overall system throughput.

MmWave Networking and Edge Computing for Scalable 360° Video Multi-User Virtual Reality.

We investigate a novel multi-user mobile Virtual Reality (VR) arcade system for streaming scalable 8K 360° video with low interactive latency, while providing high remote scene immersion fidelity and application reliability. This is achieved through the integration of embedded multi-layer 360° tiling, edge computing, and wireless multi-connectivity that comprises sub-6 GHz and mmWave (millimeter wave) links. The sub-6 GHz band is used for broadcast of the base layer of the entire 360° panorama to all users, while the directed mmWave links are used for high-rate transmission of VR-enhancement layers that are specific to the viewports of the individual users. The viewport-specific enhancements can comprise compressed and raw 360° tiles, decoded first at the edge server. We aim to maximize the smallest immersion fidelity for the delivered 360 content across all VR users, given rate, latency and computing constraints. We characterize analytically the rate-distortion trade-offs across the spatiotemporal 360° panorama and the computing power required to decompress 360° tiles. The proposed solution consists of geometric programming algorithms and an intermediate step of graph-theoretic VR user to mmWave access point assignment. The results reveal a significant improvement (8 - 10 dB) in delivered VR user immersion fidelity and spatial resolution (8K vs. 4K) compared to a state-of-the-art method based on sub-6 GHz transmission only. We also show that an increasing number of raw 360° tiles are sent, as the mmWave network link data rate or the edge server/user computing power increase. Finally, we demonstrate that in order to hypothetically deliver the same immersion fidelity, the reference method would incur a much higher (2.5-4.5x) system latency.

Geometric Programming for Optimal Hosting Capacity Allocation in Radial Grids

Geometric Programming for Optimal Hosting Capacity Allocation in Radial Grids

A Single-valued Pentagonal Neutrosophic Geometric Programming Approach to Optimize Decision Maker’s Satisfaction Level

A Single-valued Pentagonal Neutrosophic Geometric Programming Approach to Optimize Decision Maker’s Satisfaction Level

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.

Multi-Objective Energy Efficient Resource Allocation in Massive Multiple Input Multiple Output-Aided Heterogeneous Cloud Radio Access Networks

In this work, a novel energy efficient multi-objective resource allocation algorithm for heterogeneous cloud radio access networks (H-CRANs) is proposed where the trade-off between increasing throughput and decreasing operation cost is considered. H-CRANs serve groups of users through femto-cell access points (FAPs) and remote radio heads (RRHs) equipped with massive multiple input multiple output (MIMO) connected to the base-band unit (BBU) pool via front-haul links with limited capacity. We formulate an energy-efficient multi-objective optimization (MOO) problem with a novel utility function. Our proposed utility function simultaneously improves two conflicting goals as total system throughput and operation cost. With this MOO, we jointly assign the sub-carrier, transmit power, access point (AP)(RRH/FAP), RRH, front-haul link, and BBU. To address the conflicting objectives, we convert the MOO problem into a single-object optimization problem using an elastic-constraint scalarization method. With this approach, we flexibly adjust trade-off parameters to choose between two objective functions. To propose an efficient algorithm, we deploy successive convex approximation (SCA) and complementary geometric programming (CGP) approaches. Finally, via simulation results we discuss how to select the values of trade-off parameters, and we study their effects on conflicting objective functions (i.e., throughput and operation cost in MOO problem). Simulation results also show that our proposed approach can offload traffic from C-RANs to FAPs with low transmit power and thereby reduce operation costs by switching off the under-utilized RRHs and BBUs. It can be observed from the simulation results that the proposed approach outperforms the traditional approach in which each user is associated to the AP (RRHs/FAPs) with the largest average value of signal strength. The proposed approach reduces operation costs by 30% and increases throughput index by 25% which in turn leads to greater energy efficiency (EE).