Convex Subproblems Research Articles

An Augmented Lagrangian Decomposition Method for Chance-Constrained Optimization Problems

Joint chance-constrained optimization problems under discrete distributions arise frequently in financial management and business operations. These problems can be reformulated as mixed-integer programs. The size of reformulated integer programs is usually very large even though the original problem is of medium size. This paper studies an augmented Lagrangian decomposition method for finding high-quality feasible solutions of complex optimization problems, including nonconvex chance-constrained problems. Different from the current augmented Lagrangian approaches, the proposed method allows randomness to appear in both the left-hand-side matrix and the right-hand-side vector of the chance constraint. In addition, the proposed method only requires solving a convex subproblem and a 0-1 knapsack subproblem at each iteration. Based on the special structure of the chance constraint, the 0-1 knapsack problem can be computed in quasi-linear time, which keeps the computation for discrete optimization subproblems at a relatively low level. The convergence of the method to a first-order stationary point is established under certain mild conditions. Numerical results are presented in comparison with a set of existing methods in the literature for various real-world models. It is observed that the proposed method compares favorably in terms of the quality of the best feasible solution obtained within a certain time for large-size problems, particularly when the objective function of the problem is nonconvex or the left-hand-side matrix of the constraints is random.

Sparse Array Beamforming Design for Coherently Distributed Sources

Recently, nonstructured sparse array designs have attracted wide interest due to their capability of providing optimum performance for environment-dependent adaptive beamforming. In this article, we develop a sparse array design approach for adaptive beamforming in the presence of spatially coherently distributed (CD) sources. The proposed approach formulates the design problem via maximizing the output signal-to-interference-plus-noise ratio (SINR), but it exploits the generalized array manifold of CD sources. Moreover, sequential convex programming is utilized to convert the nonconvex optimization problem into a series of convex subproblems. We also analyze the impact of source distributed shape on optimum array configuration and theoretically prove that when the CD sources are Gaussian-shaped, the uniform linear array (ULA) is exactly the optimum array configuration. The resulting optimum sparse arrays yield higher output SINR and better-shaped beampattern than the structured sparse arrays. Numerical results verify the superiority of the optimum sparse arrays obtained by the proposed approach over the structured sparse arrays.

Intelligent Ubiquitous Network Accessibility for Wireless-Powered MEC in UAV-Assisted B5G

Providing the ubiquitous network accessibility is the key goal for 5G and Beyond 5G (B5G) networks. As the number of devices and data increase, ensuring the Quality of Service(QoS) by the existing network with a singular focus could be challenging. Meanwhile, it always is of the utmost importance to perform the computation-intensive or delay-sensitive tasks welland provide long-term services for the B5G networks. To tackle these challenges, we present an air-ground integrated network in B5G wireless communications, where an UAV is deployed as an aerial radio access platform to formulate system strategy intelligently, as well as provide task offloading and energy harvesting opportunities for terrestrial devices. To get more insight of it, we propose an intelligent charging-offloading scheme and formulate the joint multi-taskcharging-offloading scheduling as an optimization problem aiming to minimizing the system servicelatency of all devices by jointly optimizing the task offloading decisions, connection scheduling, charging and computation resources allocation of UAV. However, the formulated optimization problem is a Mixed-Integer Nonlinear Programming (MINLP) problem which is challenging to solve in general. Therefore, we decompose it into multiple convex sub-problems based on Block-Coordinate Descent (BCD) method and develop an improved greedy algorithm to obtain a feasible optimal solution. To validate the proposed algorithm, it is comprehensively compared with several existing schemes. Performance evaluation demonstrates that our scheme outperforms the benchmarks in terms of the system service latency of all UDs. Moreover, we represent the system working process and the paradigm of industrial applications.

Performance Analysis of NOMA Enabled Fog Radio Access Networks

In the fifth generation (5G) era, mobile networks will provide diversified services such as enhanced Mobile BroadBand (eMBB) and ultra-reliable low-latency communication (URLLC). Furthermore, the 5G architecture will be fog-like, enabling a functional split of network functionalities between cloud and edge nodes. In this article, we propose a non-orthogonal multiple access (NOMA)-enabled fog-cloud structure in a novel density-aware fog-based radio access networks (F-RAN) to tackle different aspects of high and low-density regions, in order to meet the heterogeneous requirements of eMBB and URLLC traffic. A framework of two different problem formulations is considered to cater the high throughput and low-latency requirements in a high and low-density mode respectively. In the first problem, we study the joint caching placement and association strategy aiming at minimizing the average delay. To deal with the first problem, we apply McCormick envelopes and Lagrange partial relaxation method to transform it into three convex sub-problems, which is then solved by proposing a distributed algorithm. The second problem is to jointly optimize transmission mode selection, subchannel assignment and power allocation to maximize the sum data rate of all fog user equipments (F-UEs) while satisfying fronthaul capacity and fog-computing access point (F-AP) power constraints. Moreover, for given transmission mode selection and subchannel assignment, the optimal power allocation is derived in a closed-form expression. Simulation results are provided to validate the effectiveness of the proposed NOMA-enabled F-RAN framework and reveal that the ultra-low latency and high throughput can be achieved by properly utilizing the available resources.

Dual-Use Signal Design for Radar and Communication via Ambiguity Function Sidelobe Control

This article considers the dual-use unimodular signal design for a novel dual-functional radar-communication (DFRC) architecture. The information of downlink communication is modulated via the ambiguity function (AF) sidelobe nulling in the prescribed range-Doppler cells. At the same time, the sidelobe around the AF mainlobe is suppressed to ensure the radar detection performance. User equipment (UE) computes the AF of the received signal to make a judgment of the position of nulling for achieving demodulation. To this end, an objective function to evaluate the depth of nulling in AF is developed as figure of merit accounting for the worst-case sidelobe level ratio over the range-Doppler cells of interest. The resultant design is very challenging to solve due to the non-convex and non-smooth quartic fractional objective function and the NP-hard constant modulus constraint. Herein, we develop a fractional-alternating direction penalty method (FADPM) algorithm that invokes the fractional program theory and the ADPM framework. Specifically, we formulate a new ADPM form through introducing an auxiliary variable and resorting to the Dinkelbach's procedure. In each iteration, we transform the original quartic fractional program into two quadratic optimization subproblems both of which are approximated successively through a series of convex subproblems. We also provide the analytical convergence guarantee of the proposed FADPM algorithm. The simulation results verify the performance of the proposed algorithm and exhibit the effectiveness of the DFRC framework in modulation and demodulation process guaranteeing that a high data rate is achievable.

Sequential Difference-of-Convex Programming

Optimization methods for difference-of-convex programs iteratively solve convex subproblems to define iterates. Although convex, depending on the problem’s structure, these subproblems are very often challenging and require specialized solvers. This work investigates a new methodology that defines iterates as approximate critical points of significantly easier difference-of-convex subproblems approximating the original one. Since there is considerable freedom to choose such more accessible subproblems, several algorithms can be designed from the given approach. In some cases, the resulting algorithm boils down to a straightforward process with iterates given in an analytic form. In other situations, decomposable subproblems can be chosen, opening the way for parallel computing even when the original program is not decomposable. Depending on the problem’s assumptions, a possible variant of the given approach is the Josephy–Newton method applied to the system of (necessary) optimality conditions of the original difference-of-convex program. In such a setting, local convergence with superlinear and even quadratic rates can be achieved under certain conditions.

Accelerating Benders decomposition for short-term hydropower maintenance scheduling

Maintenance of power generators is essential for reliable and efficient electricity production. Because generators under maintenance are typically inactive, optimal planning of maintenance activities must consider the impact of maintenance outages on the system operation. However, in hydropower systems finding a minimum cost maintenance schedule is a challenging optimization problem due to the uncertainty of the water inflows and the nonlinearity of the hydroelectricity production. Motivated by an industrial application problem, we formulate the hydropower maintenance scheduling problem as a two-stage stochastic program, and we implement a parallelized Benders decomposition algorithm for its solution. We obtain convex subproblems by approximating the hydroelectricity production using linear inequalities and indicator variables, which account for the nonlinear effect of the number of active generators in the solution. For speeding up the execution of our decomposition algorithm, we tailor and test seven techniques, including three new applications of special ordered sets, presolve and warm start for Benders acceleration. Given the large number of possible configurations of these acceleration techniques, we illustrate the application of statistical methods and computational experiments to identify the best performing configuration, which achieved a fourfold speedup of the decomposition algorithm. Results in an industrial setting confirm the high scalability on the number of scenarios of our parallelized Benders implementation.

Control Channel Anti-Jamming in Vehicular Networks via Cooperative Relay Beamforming

In vehicular networks, radio-frequency (RF) jamming attacks are considered a major threat to the availability of control channel (CCH). In particular, vehicles may not be able to receive control messages from roadside units (RSUs) due to persistent interference in the CCH, which may claim human lives and result in significant economic losses. In this article, a cooperative anti-jamming beamforming scheme is proposed to address the CCH jamming problems in vehicular networks. This scheme utilizes spatial diversity provided by the multiantenna RSU and relay vehicles to improve the transmission reliability of downlink control messages. In addition, to address the additive effects of the jamming signals and the intergroup interference, the relay selection problem and the beamformer design problem are jointly considered, which is modeled as a mixed-integer nonlinear programming (MINLP) problem. Then, we address this challenging problem by relaxing it into a series of convex subproblems via the semi-definite relaxation (SDR) and convex-concave process (CCP) methods, and then propose to solve these convex subproblems iteratively. The simulation results show that our proposed method convergences rapidly, and compared to the benchmark schemes, significant performance gains can be observed.

Energy Efficiency Optimization for NOMA UAV Network With Imperfect CSI

Unmanned aerial vehicles (UAVs) are developing rapidly owing to flexible deployment and access services as air base stations. However, the channel errors of low-altitude communication links formed by mobile deployment of UAVs cannot be ignored. And the energy efficiency of the UAVs communication with imperfect channel state information (CSI) hasnt been well studied yet. Therefore, we focus on system performance optimization in non-orthogonal multiple access (NOMA) UAV network considering imperfect CSI between the UAV and users. A suboptimal resource allocation scheme including user scheduling and power allocation is designed for maximizing energy efficiency. Because of the nonconvexity of optimization function with an probability constraint for imperfect CSI, the original problem is converted into a non-probability problem and then decoupled into two convex subproblems. First, a user scheduling method is applied in the two-side matching of users and subchannels by the difference of convex programming. Then based on user scheduling, the energy efficiency in UAV cells is optimized through a suboptimal power allocation algorithm by successive convex approximation method. The simulation results prove that the proposed algorithm is effective compared with existing resource allocation schemes.

Robust resource allocation in two-tier NOMA heterogeneous networks toward 5G

This paper proposes a framework of users association and power allocation in the two-tier heterogeneous networks with nonorthogonal multiple access (NOMA). Taking account of practical wireless communication environment, the channel uncertainties are considered and described in the form of probability constraint. In order to maximize the total energy efficiency of the system and reduce the intra-cell and inter-cell interference, the joint optimization problem is formulated with the uncertain channel gain. The probability constraint is transformed to the deterministic one based on the integral transformation. With the aid of relaxation variable and successive convex approximation method, the original integer non-convex optimization problem is divided into two solvable convex subproblems. The user association algorithm and power control algorithm are presented to determine the optimal resource allocations. The simulation results show that the proposed algorithm is effective and robust to the dynamic communication environment.

L1–2 minimization for P- and S-impedance inversion

Amplitude variation with offset (AVO) inversion has been widely used in reservoir characterization to predict lithology and fluids. However, some existing AVO inversion methods that use [Formula: see text] norm regularization may not obtain the block boundary of subsurface layers because the AVO inversion is a severely ill-posed problem. To obtain sparse and accurate solutions, we have introduced the [Formula: see text] minimization method as an alternative to [Formula: see text] norm regularization. We used [Formula: see text] minimization for simultaneous P- and S-impedance inversion from prestack seismic data. We first derived the forward problem with multiangles and set up the inversion objective function with constraints of a priori low-frequency information obtained from well-log data. Then, we introduced minimization of the difference of [Formula: see text] and [Formula: see text] norms, denoted as [Formula: see text] minimization, to solve this objective function. The nonconvex penalty function of the [Formula: see text] minimization method is decomposed into two convex subproblems via the difference of convex algorithm, and each subproblem is solved by the alternating direction method of multipliers. Compared to [Formula: see text] norm regularization, the results indicate that [Formula: see text] minimization has superior performance over [Formula: see text] norm regularization in promoting blocky/sparse solutions. Tests on synthetic and field data indicate that our method can provide sparser and more accurate P- and S-impedance inversion results. The overall results confirm that our method has great potential in the detection and identification of fluids.

Let's Share the Resource When We’re Co-Located: Colocation Edge Computing

Multi-access Edge Computing (MEC) is recently acknowledged as one of the key pillars for the next revolution of mobile communications area, where the convergence of IT and telecommunications network provides the low latency and computation capability for cellular base stations (BSs). As a result, this is a great opportunity for mobile network operators by deploying new services and applications at BSs. Nevertheless, huge capital and operational cost can challenge mobile operators for the deployment of new BSs and MEC micro-datacenters. Colocation Edge Computing (ColoMEC) is a new concept where multiple operators share not only the same BS tower but also their radio and computation resources colocated at the edge sites. In order to reduce the operational cost of a ColoMEC system, the limited bandwidth at over-utilized colocation BSs can be extended by sharing the bandwidth among BSs, while shared MEC micro-datacenters can be scaled based on the arrival traffic loads. Thus, sharing the BS infrastructure, bandwidth, and MEC micro-datacenters among the co-located mobile operators can be an economical solution to provide high-performance services with low expenses by exploiting the temporal and spatial difference in traffic loads. Turning this vision into reality, we study a joint bandwidth allocation sharing and MEC micro-datacenter scaling in ColoMEC management problem ( $\mathbf {ColoMEC-MP}$ ). To solve $\mathbf {ColoMEC-MP}$ problem, we propose an algorithm based on proximal block coordinate descent technique by iteratively solving the decoupled convex subproblems (i.e., user association, bandwidth allocation, and MEC micro-datacenter scaling) with additional proximal terms. To improve the convergence of the proposed algorithm, we propose a greedy initialization for the user association which is based on the link capacity at each user. Our simulation demonstrates the superiority of the algorithm in terms of the operational cost compared with fixed service rate of shared MEC micro-datacenters strategies.

Efficient Directivity Maximization of Time-Modulated Arrays With Two-Stage Convex Optimization

An efficient pattern synthesis approach for time-modulated arrays (TMA) is proposed. The method is intended to maximize the directivity of TMA with sidelobe level as constraint. By applying appropriate transformations/approximations, the synthesis problem can be decomposed into two stages with each stage composed by a sequence of convex subproblems, which can be efficiently solved by convex optimization in an iterative manner. In the first stage, the pattern shape at center frequency is optimized to obtain maximized pattern directivity. In the second stage, based on the optimized result in the first stage, the sideband radiation is further suppressed to obtain maximized TMA directivity. The proposed method is capable to solve pencil-beam, wide-beam, and shaped-beam TMA synthesis problems with arbitrary array geometry. Numerical examples are presented to show the robustness of the proposed method.

Non-Negative Iterative Convex Refinement Approach for Accurate and Robust Reconstruction in Cerenkov Luminescence Tomography.

Cerenkov luminescence tomography (CLT) is a promising imaging tool for obtaining three-dimensional (3D) non-invasive visualization of the in vivo distribution of radiopharmaceuticals. However, the reconstruction performance remains unsatisfactory for biomedical applications because the inverse problem of CLT is severely ill-conditioned and intractable. In this study, therefore, a novel non-negative iterative convex refinement (NNICR) approach was utilized to improve the CLT reconstruction accuracy, robustness as well as the shape recovery capability. The spike and slab prior information was employed to capture the sparsity of Cerenkov source, which could be formalized as a non-convex optimization problem. The NNICR approach solved this non-convex problem by refining the solutions of the convex sub-problems. To evaluate the performance of the NNICR approach, numerical simulations and in vivo tumor-bearing mice models experiments were conducted. Conjugated gradient based Tikhonov regularization approach (CG-Tikhonov), fast iterative shrinkage-thresholding algorithm based Lasso approach (Fista-Lasso) and Elastic-Net regularization approach were used for the comparison of the reconstruction performance. The results of these experiments demonstrated that the NNICR approach obtained superior reconstruction performance in terms of location accuracy, shape recovery capability, robustness and in vivo practicability. It was believed that this study would facilitate the preclinical and clinical applications of CLT in the future.

Multicell Edge Coverage Enhancement Using Mobile UAV-Relay

Unmanned aerial vehicle (UAV)-assisted communication is a promising technology in future wireless communication networks. UAVs can not only help offload data traffic from ground base stations (GBSs), but also improve the quality of service of cell-edge users (CEUs). In this paper, we consider the enhancement of cell-edge communications through a mobile relay, i.e., UAV, in multi-cell networks. During each transmission period, GBSs first send data to the UAV, and then the UAV forwards its received data to CEUs according to a certain association strategy. In order to maximize the sum rate of all CEUs, we jointly optimize the UAV mobility management, including trajectory, velocity, and acceleration, and association strategy of CEUs to the UAV, subject to minimum rate requirements of CEUs, mobility constraints of the UAV and causal buffer constraints in practice. To address the mixed-integer nonconvex problem, we transform it into two convex subproblems by applying tight bounds and relaxations. An iterative algorithm was proposed to solve the two subproblems in an alternating manner. Numerical results show that the proposed algorithm achieves higher rates of CEUs as compared with existing benchmark schemes.

Approximate versions of proximal iteratively reweighted algorithms including an extended IP-ICMM for signal and image processing problems

Iteratively reweighted algorithms are popular methods for solving nonconvex unconstrained minimization problems. Applications are notably mathematical models in image processing or signal processing. They often have a convex subproblem and do not have closed form solutions in general. In this paper, we propose approximate versions of proximal iteratively reweighted algorithms for nonconvex and nonsmooth unconstrained minimization problems. Specifically, we can achieve an approximate solution for the subproblem by applying a computable inexact stopping rule. The convergence of our method is proved based on an inexact unified framework. Numerical applications for image deblurring or denoising problems validate the effectiveness of the proposed algorithms.

Truss optimization applying finite element limit analysis including global and local stability

For practical applications of optimized truss structures, it is essential to include global and local stability in order to obtain stable and realistic structures. The challenge of including both global and local stability has previously been approached in many ways. However, these proposals often lead to ill-conditioned optimization problems, with convergence issues due to the concavity of the problem. In this paper, a new method for handling both global and local stability in truss optimization is presented. The proposed method is based on the finite element limit analysis method. Initially, the global stability problem is solved by a convex semidefinite constraint, and subsequently, the concave local stability problem is included through an iterative process, where the local stability constraints are linearized and solved by a convex sub-problem. This step-wise approach diminishes convergence issues due to the concavity of the problem. The proposed method is demonstrated through three different applications showing significant effects of including global and local stability in the optimized designs, while at the same time demonstrating the validity and potential of the proposed method.

Mobility-Aware Multi-User Offloading Optimization for Mobile Edge Computing

Mobile Edge Computing (MEC) is a new computing paradigm with great potential to enhance the performance of user equipment (UE) by offloading resource-hungry computation tasks to lightweight and ubiquitously deployed MEC servers. In this paper, we investigate the problem of offloading decision and resource allocation among multiple users served by one base station to achieve the optimal system-wide user utility, which is defined as a trade-off between task latency and energy consumption. Mobility in the process of task offloading is considered in the optimization. We prove that the problem is NP-hard and propose a heuristic mobility-aware offloading algorithm (HMAOA) to obtain the approximate optimal offloading scheme. The original global optimization problem is converted into multiple local optimization problems. Each local optimization problem is then decomposed into two subproblems: a convex computation allocation subproblem and a non-linear integer programming (NLIP) offloading decision subproblem. The convex subproblem is solved with a numerical method to obtain the optimal computation allocation among multiple offloading users, and a partial order based heuristic approach is designed for the NLIP subproblem to determine the approximate optimal offloading decision. The proposed HMAOA is with polynomial complexity. Extensive simulation experiments and comprehensive comparison with six baseline algorithms demonstrate its excellent performance.

Energy-efficient resource allocation in wireless powered CCRNs with simultaneous wireless information and power transfer

In this paper, we study the optimal energy efficiency resource allocation for simultaneous wireless information and power transfer (SWIPT) in cooperative cognitive radio networks (CCRNs). The optimal design attempts to deploy each secondary receiver (SR) to harvest the energy and to use the signal received from the secondary transmitter (ST) to decode information. For the information transmission, a traditional multiple access scheme, namely, time division multiple access (TDMA) is addressed. For the TDMA-based information transmission, the users are scheduled in non-overlapping time slots to decode information. To tackle the design with the above scenarios, an optimization problem is formulated to maximize the energy efficiency of the secondary system constrained with the uncertainty of channel state information (CSI). To solve this non-convex problem, nonlinear fractional programming is utilized to transform the objective function into a subtractive form. The nonlinear fractional programming is also used to decompose the problem into two convex subproblems, and the probability constraints of signal to interference plus noise ratio (SINR) are converted into the deterministic ones using integral transformation. A joint optimal time and power allocation (JOTPA) iterative algorithm is proposed to determine the optimum of original non-convex problem. Simulation results have verified the effectiveness of the proposed resource allocation algorithm.

Globally minimizing the sum of a convex–concave fraction and a convex function based on wave-curve bounds

We consider the problem of minimizing the sum of a convex–concave function and a convex function over a convex set (SFC). It can be reformulated as a univariate minimization problem, where the objective function is evaluated by solving convex optimization. The optimal Lagrangian multipliers of the convex subproblems are used to construct sawtooth curve lower bounds, which play a key role in developing the branch-and-bound algorithm for globally solving (SFC). In this paper, we improve the existing sawtooth-curve bounds to new wave-curve bounds, which are used to develop a more efficient branch-and-bound algorithm. Moreover, we can show that the new algorithm finds an $$\epsilon $$-approximate optimal solution in at most $$O\left( \frac{1}{\epsilon }\right) $$ iterations. Numerical results demonstrate the efficiency of our algorithm.