Geometric Programming Research Articles

Decentralized Joint Pilot and Data Power Control Based on Deep Reinforcement Learning for the Uplink of Cell-Free Systems

While the problem of jointly controlling the pilot-and-data power in cell-based systems has been extensively studied, this problem is difficult to solve in cell-free systems due to two reasons. First, both the large- and small-scale fading are markedly different between a served user and the multiple serving access points. Second, due to the user-centric architecture, there is a need for decentralized algorithms that scale well in the cell-free environment. In this work, we study the impact of joint pilot-and-data power control and receive filter design in the uplink of cell-free systems. The problem is formulated as optimization tasks considering two different objectives: 1) maximization of the minimum spectral efficiency (SE) and 2) maximization of the total SE. Since these problems are non-convex, we resort to successive convex approximation and geometric programming to obtain a local optimal centralized solution for benchmarking purposes. We also propose a decentralized solution based on actor-critic deep reinforcement learning, in which each user acts as an agent to locally obtain the best policy relying on minimum information exchange. Practical signaling aspects are provided for such a decentralized solution. Finally, numerical results indicate that the decentralized solution performs very close to the centralized one and outperforms state-of-the-art algorithms in terms of minimum SE and total system SE.

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.

Optimal Resource Management for NOMA-Based Visible Light Communication Systems With Shot Noise

In this paper, the problem of power allocation is investigated for downlink visible light communication (VLC) systems with non-orthogonal multiple access. In the system, both input-dependent shot noise and input-independent Gaussian noise are considered due to the properties of practical VLC channels. We consider two scenarios for the transmitter with a single light emitting diode (LED) and multiple LEDs, respectively. For single-LED scenarios, the problem is posed as a joint optimization problem of alternating current and direct current power control, whose goal is to maximize the minimum signal-to-interference-plus-noise-ratio (SINR) of all users. Although the original problem is non-convex, a geometric programming based algorithm is proposed to acquire the global optimal solution. For multi-LED scenarios, the transmit and dimming beamforming vectors are jointly optimized to maximize the minimum SINR of all users. A joint optimization algorithm that alternately optimizes transmit and dimming beamforming vectors is proposed. Particularly, the transmit beamforming problem is converted into a second-order cone programming problem through the successive convex approximation method. The dimming beamforming problem is transformed into a feasibility problem utilizing a low-complexity bisection algorithm. Numerical results reveal the superior performance of the proposed algorithm compared with its orthogonal multiple access counterpart and conventional schemes without shot noise.

On the Maximum Energy Efficiency of Random Access-Based OMA and NOMA in Multirate Environment

The paradigm of the upcoming 5G and beyond is the machine type communications (MTC), where a large number of devices automatically operate wireless communications. Due to their automatic and energy-intensive operations, energy efficiency (EE) becomes very important, but there is a lack of investigation for EE of random access networks, which is the underlying platform for MTC. In this paper, we focus on the EE of random access orthogonal and non-orthogonal multiple access (OMA and NOMA) networks. We first construct mathematical models of their EE performances. Information-theoretic theories are employed to transform probabilistic decodability into deterministic form for OMA, and along with the unique features of NOMA decoding process, by pivoting around the decodability of a specific packet, we realize the modeling for NOMA. Then, based on the established models, exploiting the nature of random access networks, a complementary geometric programming approach is used for performance optimization. From our investigation, the following results are discovered: (1) NOMA has significant advantage over OMA in EE performance due to its strong data recovery; (2) to maximize the EE performance, devices with higher data rate should transmit more frequently, but with lower transmission power; (3) lower data rates have beneficial effects on EE.

Pricing‐based power allocation in cloud radio access network with multiple access technology selection

AbstractIn this article, we consider an uplink economy‐efficient resource allocation framework in a multicellular cloud radio access network (C‐RAN) architecture with network virtualization, where a mobile network operator (MNO) interacts with a number of mobile virtual network operators (MVNOs) with a predetermined business model. OFDMA and Massive MIMO multiple access technologies have been assumed to be available for each MVNO at two different prices. In this setup, we propose a multi‐access technology selection approach (MATSA) with the objective of maximizing delivered rate to end users, reducing operation costs and maximizing MVNOs' profit subject to a set of constraints, which leads to a nonconvex resource allocation problem with very high computational complexity. The utility function is defined as the difference between the total throughput and the utilization cost for each technology. To tackle this problem, we apply complementary geometric programming (CGP) and successive convex approximation (SCA), that results in a two‐step iterative solution. Simulation results demonstrate superiority of the proposed approach compared to a similar scenario with predetermined multi‐access technology, especially for large numbers of users.

Power Allocation of Superimposed Pilots for URLLC With Short-Packet Transmission in IIoT

Intelligent manufacturing in future factories requires ultra-reliable and low-latency communications (URLLC) between central controllers and actuators (e.g., robots). This letter studies a superimposed pilot (SP)-aided uplink transmission system in industrial scenarios, where each actuator sends short-packet messages containing a superimposition of pilot and data signals to the central controller. A practical setting is considered where the blocklength is finite and imperfect pilot removal (IPPR) is available. Then, a closed-form lower bound of the achievable rate with a maximum ratio combining receiver is derived, which can be easily extended to the special cases of perfect pilot removal (PPR) and regular pilot (RP). Based on the derived lower bound, the pilot and data power allocation is optimized by exploiting successive convex approximation method to transform the weighted sum rate maximization problem into a series of geometric programs. Simulation results reveal that the SP-aided transmission outperforms the RP-aided transmission in both the PPR and IPPR cases.

A neural network approach to solve geometric programs with joint probabilistic constraints

We study a dynamical neural network approach to solve nonconvex geometric programs with joint probabilistic constraints (GPPC for short) with normally distributed coefficients and independent matrix row vectors. The main feature of our framework is to solve the biconvex deterministic equivalent problem of GPPC without the use of any convex approximation unlike the state-of-the-art solving methods. The second feature of our approach is to solve the dynamical system without the use of any iterative method but only by solving an ordinary differential equation system. We prove the stability together with the convergence of our neural network in the sense of Lyapunov. We also prove the equivalence between the optimal solution of the deterministic equivalent problem of GPPC and the solution of the dynamical system. We provide numerical experiments to show the performances of our approach compared to the state-of-art.

On a Geometric Programming Approach to Profit Maximization: The Case of CES Technology

The profit maximization problem takes a central place in the theory of the firm, especially when conditions for perfect competition hold. In this paper, we solve the profit maximization problem of a perfectly competitive firm when the constant elasticity of substitution (CES) production function with n≥2 inputs describes its technology. Commonly, this problem is solved by using multivariable differential calculus. However, to avoid tedious algebraic manipulations and bypass checking nontrivial necessary and sufficient conditions, we employ geometric programming (GP), and the power mean inequality (PMI) as an elegant complementary tool to multivariable calculus. Since the GP and the PMI are simple optimization techniques without derivatives, they can provide new insights into the given problem to managers, students, and other audiences who may be unfamiliar with multivariable differential calculus. Additionally, by using the properties of limits, we show that the solution to the profit maximization problem with Cobb-Douglas technology is a limiting case of our result.

Geometric Programming in Imprecise Domain with Application

Geometric Programming in Imprecise Domain with Application

EOQ model with price, marketing, service and green dependent neutrosophic demand under uncertain resource constraint: A geometric programming approach

EOQ model with price, marketing, service and green dependent neutrosophic demand under uncertain resource constraint: A geometric programming approach

Maximizing Energy Efficiency With Channel Uncertainty Under Mutual Interference

We study the problem of channel uncertainty on wireless transmissions from different users with mutual interference. Specifically, the channel gains from the transmitters to the receivers are available only through their mean and covariance rather than complete distributions. Our goal is to maximize the energy efficiency among all transmitter-receiver pairs while guaranteeing their capacity requirements. For this problem, we employ chance-constrained programming (CCP), which allows occasional violation of target capacity threshold as long as the probability of such violation is below a small tolerable constant (risk level). We propose a solution based on a novel reformulation technique that converts the original CCP into a deterministic optimization problem without relaxation errors. Then the deterministic optimization problem is approximated into a Geometric Program (GP) based on tight polynomial approximations, which can be solved optimally. We prove that our proposed solution achieves near-optimal performance with polynomial time complexity.

Capacity analysis and sum rate maximization for the SCMA cellular network coexisting with D2D communications

Sparse code multiple access (SCMA) is the most concerning scheme among non-orthogonal multiple access (NOMA) technologies for 5G wireless communication new interface. Another efficient technique in 5G aimed to improve spectral efficiency for local communications is device-to-device (D2D) communications. Therefore, we utilize the SCMA cellular network coexisting with D2D communications for the connection demand of the Internet of things (IOT), and improve the system sum rate performance of the hybrid network. We first derive the information-theoretic expression of the capacity for all users and find the capacity bound of cellular users based on the mutual interference between cellular users and D2D users. Then we consider the power optimization problem for the cellular users and D2D users jointly to maximize the system sum rate. To tackle the non-convex optimization problem, we propose a geometric programming (GP) based iterative power allocation algorithm. Simulation results demonstrate that the proposed algorithm converges fast and well improves the sum rate performance.

Log–log convexity of an optimal control problem for positive linear systems

This paper investigates the finite-time optimal control problems for positive linear systems with a time-varying control input. Cost functional of the system state is constructed. Under some assumptions on the designed parameters and cost functionals, the optimization problem with piecewise-constant matrix functions is first proved to be log–log convex and can be solved via geometric programming. Then, the log–log convex result is further extended to the optimization problem with continuous functions. Finally, an optimal control problem is investigated to verify the effectiveness of the proposed optimization framework.

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.

Minimization of the Worst Case Average Energy Consumption in UAV-Assisted IoT Networks

The Internet of Things (IoT) brings connectivity to a massive number of devices that demand energy-efficient solutions to deal with limited battery capacities, uplink-dominant traffic, and channel impairments. In this work, we explore the use of unmanned aerial vehicles (UAVs) equipped with configurable antennas as a flexible solution for serving low-power IoT networks. We formulate an optimization problem to set the position and antenna beamwidth of the UAV, and the transmit power of the IoT devices subject to average-signal-to-average-interference-plus-noise ratio ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\bar {\text {S}}\overline {\text {IN}}\text {R}$ </tex-math></inline-formula> ) Quality-of-Service (QoS) constraints. We minimize the worst case average energy consumption of the latter, thus targeting the fairest allocation of the energy resources. The problem is nonconvex and highly nonlinear; therefore, we reformulate it as a series of three geometric programs that can be solved iteratively. Results reveal the benefits of planning the network compared to a random deployment in terms of reducing the worst case average energy consumption. Furthermore, we show that the target <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\bar {\text {S}}\overline {\text {IN}}\text {R}$ </tex-math></inline-formula> is limited by the number of IoT devices, and highlight the dominant impact of the UAV hovering height when serving wider areas. Our proposed algorithm outperforms other optimization benchmarks in terms of minimizing the average energy consumption at the most energy-demanding IoT device, and convergence time.

A method of sequential log-convex programming for engineering design

A method of Sequential Log-Convex Programming (SLCP) is constructed that exploits the log-convex structure present in many engineering design problems. The mathematical structure of Geometric Programming (GP) is combined with the ability of Sequential Quadratic Program (SQP) to accommodate a wide range of objective and constraint functions, resulting in a practical algorithm that can be adopted with little to no modification of existing design practices. Three test problems are considered to demonstrate the SLCP algorithm, comparing it with SQP and the modified Logspace Sequential Quadratic Programming (LSQP). In these cases, SLCP shows up to a 77% reduction in number of iterations compared to SQP, and an 11% reduction compared to LSQP. The airfoil analysis code XFOIL is integrated into one of the case studies to show how SLCP can be used to evolve the fidelity of design problems that have initially been modeled as GP compatible. Finally, a methodology for design based on GP and SLCP is briefly discussed.

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.

Computational Simulation: Multidisciplinary Teaching of Dynamic Models from the Linear Algebra Perspective

The technological development has stablish a new learning perspective in the Linear Algebra Concepts. In progress is being made in multidisciplinary teaching in engineering careers, since the proposal is that students learn Math taking into account the benefits provided from the new technologies that analyse different aspects of an engineering system. In this paper are presented a classroom experience in the Computer and Multidisciplinary Laboratory of Basic Sciences where a fluid flow system is modelled when we want to deal with the Eigenvalues and Eigenvectors topic, content that belongs to Algebra and Analytic Geometrical program. The possibilities that the computational systems offers have unchained a reformulation of methodological focus of educative programs, since they propose to intensify learning through the acquisition of skills by students. The developed experience proposes a learning methodology that guarantees an academic knowledge according to the new developments, and assert that Engineering students approach new mathematical subjects trough adequate applications.

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.

Conflict-Directed Diverse Planning for Logic-Geometric Programming

Robots operating in the real world must combine task planning for reasoning about what to do with motion planning for reasoning about how to do it -- this is known as task and motion planning. One promising approach for task and motion planning is Logic Geometric Programming (LGP) which integrates a logical layer and a geometric layer in an optimization formulation. The logical layer describes feasible high-level actions at an abstract symbolic level, while the geometric layer uses continuous optimization methods to reason about motion trajectories with geometric constraints. In this paper we propose a new approach for solving task and motion planning problems in the LGP formulation, that leverages state-of-the-art diverse planning at the logical layer to explore the space of feasible logical plans, and minimizes the number of optimization problems to be solved on the continuous geometric layer. To this end, geometric infeasibility is fed back into planning by identifying prefix conflicts and incorporating this back into the planner through a novel multi-prefix forbidding compilation. We further leverage diverse planning with a new novelty criteria for selecting candidate plans based on the prefix novelty, and a metareasoning approach which attempts to extract only useful conflicts by leveraging the information that is gathered in the course of solving the given problem.