Difference Of Two Convex Research Articles

A UAV-Assisted Secure Communication System by Jointly Optimizing Transmit Power and Trajectory in the Internet of Things

Secure communication technology has attracted more and more attention to improve the security of the Internet of Things (IoT). In this article, we discuss the maximum secrecy capacity problem for a mobile unmanned aerial vehicle (UAV) assisted secure communication in IoT, where the UAV relays the covert information from the transmitter (Alice) to the legal receiver (Bob) without detection by an eavesdropper (Eve) though sharing the spectrum with the ground device-to-device (D2D) communications. Our design aims to maximize the sum secrecy capacity Bob received by optimizing the transmit power and the UAV’s trajectory jointly while guaranteeing that the normal communication behavior between ground D2D pairs is not affected. Therefore, we propose a jointly optimizing the transmit power and UAV trajectory algorithm (JOTUTA) based on the block coordinate descent (BCD) method. Specifically, we first transform the non-convex problem of transmit power based on a fixed UAV’s trajectory into a convex problem by D.C. (difference of two convex functions) programming. Secondly, we solve the problem of UAV’s trajectory optimization with given transmit power by the successive convex approximation (SCA) method. Experiments results show the effectiveness of JOTUTA and the advantages of improving the secrecy capacity are verified compared with existing work.

Trajectory Optimization and Power Allocation for Cell-Free Satellite-UAV Internet of Things

Massive access outside the coverage of terrestrial cellular networks will be the main feature of the sixth generation (6G) networks. In order to cope with it, the cognitive satellite-UAV network (CSUN) has drawn a lot of attentions. In this paper, we investigate the UAV trajectory optimization and power allocation for the cell-free CSUN consisting of one satellite and a swarm of UAVs. Indeed, due to the on-board energy constraints of UAVs, both the trajectory optimization and the power allocation can significantly save the energy to improve the energy efficiency. The joint trajectory optimization and power allocation problem is formulated as a mixed-integer non-convex optimization problem which is extremely difficult to solve. In order to reduce the computational complexity, we decompose the original optimization problem into two subproblems in terms of the trajectory optimization and power allocation. For the trajectory optimization subproblem, we model it as a Traveling Salesman Problem (TSP), and the PSO is adopted to solve it. When the trajectory variables are fixed, the power allocation subproblem is still difficult to tackle due to its non-convexity and large scale. Firstly, we present a kind of centralized algorithm in which the DC (difference of two convex functions) algorithm is applied to optimize it, and then a distributed algorithm based on auxiliary variables is proposed to reduce the signaling overhead and computational complexity. The simulation results demonstrate the effectiveness of the proposed joint trajectory optimization and power allocation algorithm for the cell-free CSUN.

A smooth approximation approach for optimization with probabilistic constraints based on sigmoid function

Many practical problems, such as computer science, communications network, product design, system control, statistics and finance, etc.,can be formulated as a probabilistic constrained optimization problem (PCOP) which is challenging to solve since it is usually nonconvex and nonsmooth. Effective methods for the probabilistic constrained optimization problem mostly focus on approximation techniques, such as convex approximation, D.C. (difference of two convex functions) approximation, and so on. This paper aims at studying a smooth approximation approach. A smooth approximation to the probabilistic constraint function based on a sigmoid function is analyzed. Equivalence of PCOP and the corresponding approximation problem are shown under some appropriate assumptions. Sequential convex approximation (SCA) algorithm is implemented to solve the smooth approximation problem. Numerical results suggest that the smooth approximation approach proposed is effective for optimization problems with probabilistic constraints.

Solving nonnegative sparsity-constrained optimization via DC quadratic-piecewise-linear approximations

In this paper, we propose a novel algorithm that is based on quadratic-piecewise-linear approximations of DC functions to solve nonnegative sparsity-constrained optimization. A penalized DC (difference of two convex functions) formulation is proved to be equivalent to the original problem under a suitable penalty parameter. We employ quadratic-piecewise-linear approximations to the two parts of the DC objective function, resulting in a nonconvex subproblem. This is the key ingredient of our main algorithm. This nonconvex subproblem can be solved by a globally convergent alternating variable algorithm. Under some mild conditions, we prove that the proposed main algorithm for the penalized problem is globally convergent. Some preliminary numerical results on the sparse nonnegative least squares and logistic regression problems demonstrate the efficiency of our algorithm.

DC programming and DCA for enhancing physical layer security via relay beamforming strategies

Beside of cryptography-the primary traditional methods for ensuring information security and confidentiality, the appearance of the physical layer security approach plays an important role for not only enabling the data transmission confidentially without relying on higher-layer encryption, but also enhancing confidentiality of the secret key distribution in cryptography. Many techniques are employed in physical layers to improve secure transmission including cooperative relaying and beamforming technique. In this paper, we consider the secrecy rate maximization problems using two techniques mentioned above with two different relaying protocols: Amplify-and-Forward and Decode-and-Forward. The optimization problems with the aim of maximizing secrecy rate subject to total and individual relay power constraints are formulated as nonconvex problems, which can be reformulated as DC (difference of two convex functions) programs and thus can be solved by DC Algorithms (DCA). The special structure of feasible set is exploited which results to an efficient DC decomposition in the sense that it leads to convex subproblems that can be explicitly solved. The numerical results show that the proposed DCA schemes are better than the existing methods in terms of both runtime and secrecy rate.

Achievable Secrecy Rate Region for Buffer-Aided Multiuser MISO Systems

In this paper, we consider a buffer-aided multiuser multiple-input single-output (MISO) network consisting of one multi-antenna access point (AP) and multiple single-antenna users, in which the AP is provisioned with data buffers for temporarily storing the data for each user either from the upper layer applications or the message by the AP. The data for one specific user must be kept confidential from all other unintended users. For such a system, we aim at maximizing the long-term average achievable secrecy rate region by carefully designing the flow control, the information signal and artificial noise beamforming, as well as the user selection. To address this issue, we first transform the time average optimization problem into a real-time one by using the Lyapunov optimization framework. Then it is proposed to decompose the optimization problem into several sub-problems by using the optimization decomposition technique. Although the information signal and artificial noise beamforming sub-problem is non-convex, we show that it can be decomposed into a two-stage optimization problem to effectively solve it by using the exact line search and DC (difference of two convex functions) algorithms. Moreover, we extend the average secrecy rate region maximization problem to the worst-case scenario, in which all unintended users are colluding in eavesdropping. Our analysis discloses that, there exists an inherent tradeoff between the average achievable secrecy rate region and the average queueing length. It is shown that a better average secrecy rate region can be realized by fully taking advantage of the buffer-aided transmission potentials in the MISO network, if a certain queueing delay is tolerable.

Robust Multicategory Support Matrix Machines.

We consider the classification problem when the input features are represented as matrices rather than vectors. To preserve the intrinsic structures for classification, a successful method is the Support Matrix Machine (SMM) in [19], which optimizes an objective function with a hinge loss plus a so-called spectral elastic net penalty. However, the issues of extending SMM to multicategory classification still remain. Moreover, in practice, it is common to see the training data contaminated by outlying observations, which can affect the robustness of existing matrix classification methods. In this paper, we address these issues by introducing a robust angle-based classifier, which boils down binary and multicategory problems to a unified framework. Benefitting from the use of truncated hinge loss functions, the proposed classifier achieves certain robustness to outliers. The underlying optimization model becomes nonconvex, but admits a natural DC (difference of two convex functions) representation. We develop a new and efficient algorithm by incorporating the DC algorithm and primal-dual first-order methods together. The proposed DC algorithm adaptively chooses the accuracy of the subproblem at each iteration while guaranteeing the overall convergence of the algorithm. The use of primal-dual methods removes a natural complexity of the linear operator in the subproblems and enables us to use the proximal operator of the objective functions, and matrix-vector operations. This advantage allows us to solve large-scale problems efficiently. Theoretical and numerical results indicate that for problems with potential outliers, our method can be highly competitive among existing methods.

Max–Min Weighted Achievable Rate for Full-Duplex MIMO Systems

In this letter, we aim to maximize the minimum weighted achievable rate of the two links in multiple-input multiple-output full-duplex systems via precoder design. With bidirectional communication and unidirectional communication fashion considered, our system model involves limited dynamic range effect and channel estimation error. We propose D.C. (difference of two convex functions) based algorithms to solve the highly non-convex optimization problem, and the complexity, convergence and optimality of proposed algorithms are discussed.

Spectrum Sharing Planning for Full-Duplex UAV Relaying Systems With Underlaid D2D Communications

In this paper, we consider the spectrum sharing planning problem for a full-duplex unmanned aerial vehicle (UAV) relaying systems with underlaid device-to-device (D2D) communications, where a mobile UAV employed as a full-duplex relay assists the communication link between separated nodes without direct link. Our design aims to maximize the sum throughput under the transmit power budget, while guaranteeing the coexistence with terrestrial D2D pairs, satisfying the information causality and UAV’s trajectory constraints. First, the transmit power planning with a given trajectory is investigated, where a successive convex algorithm is developed by leveraging the D.C. (difference of two convex) programming. Then, we propose a two-step trajectory design method for the given transmit power since the constraints of D2D pairs result in a non-convex feasible set. Furthermore, an efficient spectrum sharing method for an aerial UAV and terrestrial D2D communications is designed by alternately optimizing the transmit power and UAV’s trajectory. Finally, simulation results under various parameter configurations are provided to show the effectiveness of the proposed algorithms.

Power optimization of LTE-800 and coexistence with DVB-T services

The emergence of cognitive radio technologies and the reallocation of the upper Ultra High Frequency (UHF) band (790−862 MHz) for broadband services has led to interference concerns for the DVB-T (Digital Video Broadcasting-Terrestrial) services. The DVB-T receivers need to be protected from aggregate interference coming from in-band and out-of-band transmissions, especially from the LTE-800 (Long-Term Evolution at 800 MHz) networks due to the operation in near spectral and location proximity. Although operating in different, but neighboring bands, the LTE-800 networks can be a serious threat to the DVB-T services due to excessive aggregate out-of-band interference caused to the DVB-T receivers. The regulators have assessed and defined the protection criteria for these cases. The power optimization of the LTE cellular networks, operating in near one spectral reuse factor, also remains an open issue due to the non-convex optimization problems. This paper proposes a novel algorithm for LTE-800 power allocation algorithm for capacity maximization of an LTE-800 network under DVB-T protection criteria. The algorithm uses k-means clustering to find the most critical geographical areas based on the DVB-T receivers protection ratios. It also adapts a local D.C. (Difference of two Convex functions/sets) programming approach to make the power optimization problem convex and perform the LTE-800 coverage and capacity optimization. The simulation analysis of the algorithm include realistic data for DVB-T modeling and Poisson Point Process for LTE-800 base stations (BS) spatial modeling. The proposed algorithm is compared to two extreme approaches, i.e. the max LTE BS power approach totally ignoring the DVB-T protection and the uniform power scaling approach that certainly protects the DVB-T service. The results prove the performance increase introduced by the proposed algorithm in terms of spatial capacity and coverage and DVB-T service protection.

Energy-Efficient Joint Power and Bandwidth Allocation for NOMA Systems

In this letter, we consider a downlink non-orthogonal multiple access (NOMA) system where a base station (BS) simultaneously communicates with multiple users. These users are grouped on multiple unequal-bandwidth subchannels. The power and bandwidth allocation are jointly optimized in order to maximize the system energy efficiency (EE) under transmit power constraint at the BS and minimum rate constraints for individual users. The optimization problem is nonconvex because of the fractional objective function and the nonconvex constraints. To deal with the original problem, it is first converted into an equivalent DC (difference of two convex functions) program by introducing slack variables, and then an iterative algorithm based on the constrained concave-convex procedure is proposed to solve it. Our numerical results demonstrate that the proposed algorithm has better EE performance compared to the NOMA scheme with equal-bandwidth allocation and conventional orthogonal multiple access scheme.

Resource Allocation for Energy Harvesting-Powered D2D Communication Underlaying UAV-Assisted Networks

In this paper, we investigate the resource allocation problem for unmanned aerial vehicle (UAV)-assisted networks, where a UAV acting as an energy source provides radio frequency energy for multiple energy harvesting-powered device-to-device (D2D) pairs with much information to be transmitted. The goal is to maximize the average throughput within a time horizon while satisfying the energy causality constraint under a generalized harvest-transmit-store model, which results in a non-convex problem. By introducing the Lagrangian relaxation method, we analytically show that the behavior of all D2D pairs at each time slot is exclusive: harvesting energy or transmitting information signals. The formulated non-convex optimization problem is thus transformed into a mixed integer nonlinear programming (MINIP). We then design an efficient resource allocation algorithm to solve this MINIP, where D.C. (difference of two convex functions) programming and golden section method are combined to achieve a suboptimal solution. Furthermore, we provide an idea to reduce the computational complexity for facilitating the application in practice. Simulations are conducted to validate the effectiveness of the proposed algorithm and evaluate the system throughput performance.

DC formulations and algorithms for sparse optimization problems

We propose a DC (Difference of two Convex functions) formulation approach for sparse optimization problems having a cardinality or rank constraint. With the largest-k norm, an exact DC representation of the cardinality constraint is provided. We then transform the cardinality-constrained problem into a penalty function form and derive exact penalty parameter values for some optimization problems, especially for quadratic minimization problems which often appear in practice. A DC Algorithm (DCA) is presented, where the dual step at each iteration can be efficiently carried out due to the accessible subgradient of the largest-k norm. Furthermore, we can solve each DCA subproblem in linear time via a soft thresholding operation if there are no additional constraints. The framework is extended to the rank-constrained problem as well as the cardinality- and the rank-minimization problems. Numerical experiments demonstrate the efficiency of the proposed DCA in comparison with existing methods which have other penalty terms.

Cooperative MIMO relay beamforming in overloaded DS/CDMA multicast networks

A new beamforming scheme is proposed for multi-input multi-output (MIMO) non-regenerative relays in an overloaded synchronous direct sequence code division multiple access (DS-CDMA) network in which the number of users is larger than the processing gain. In this scheme, several transmitters multicast their data to their groups of receivers through MIMO relays and DS-CDMA technique is utilised to decrease the level of multiple access interference (MAI) at the relays and the receivers. At the relays, the data of transmitters are despread, beamformed and re-spread. The aim of the beamformer is to minimise the total relays transmission power subject to the signal to interference and noise ratio (SINR) constraints at the receivers. This optimisation problem is an indefinite quadratic (non-convex) programming. Semi-definite programming (SDP) and second-order cone programming (SOCP) are two existing methods to solve the problem. However, they, respectively, suffer from high computational complexity and performance degradation when the number of antennas or users increase. To effectively mitigate these deficiencies, we propose a novel DC (difference of two convex functions) programming method. Our simulation results demonstrate the computational advantage of the proposed DC programming over SDP, while a better performance over SOCP is achieved. Moreover, the superiority of the proposed MIMO-CDMA scheme over other multiplexing schemes is shown by several scenarios.

Convergence analysis on a smoothing approach to joint chance constrained programs

This paper aims to solve the joint chance constrained programs (JCCP) by a DC (difference of two convex functions) function approach, which was established by Hong et al. [Oper. Res. 2011;59:617–630]. They used a DC function to approximate the chance constraint function and constructed a sequential convex approximation method to solve the approximation problem. A disadvantage of this method is perhaps that the DC function they used is nonsmooth. In this article, we first propose a class of smoothing functions to approximate the maximum function and the indicator function . Then, we construct the conservative smooth DC approximation function to and obtain the smooth DC approximation problems to JCCPs. We show that the solutions of a sequence of smooth approximation problems converge to some Karush–Kuhn–Tucker point of JCCPs under a certain asymptotic regime.

Joint optimization of spectrum and energy efficiency in cognitive radio networks

In this paper, we discuss the joint improvement of the energy efficiency (EE) and the spectrum efficiency (SE) in OFDM-based cognitive radio (CR) networks. A multi-objective resource allocation task is formulated to optimize the EE and the SE of the CR system simultaneously with the consideration of the mutual interference and the spectrum sensing errors. We first exploit the EE–SE relations and demonstrate that the EE is a quasiconcave function of the SE, based on which the Pareto optimal set of the multi-objective optimization problem is characterized. To find a unique globally optimal solution, we propose a unified EE–SE tradeoff metric to transform the multi-objective optimization problem into a single-objective one which has a D.C. (difference of two convex functions/sets) structure and yields a standard convex optimization problem. We derive a fast method to speed up the time-consuming computation by exploiting the structure of the convex problem. Simulation results validate the effectiveness and efficiency of the proposed algorithms, which can produce the unique globally optimal solution of the original multi-objective optimization problem.

LOCAL AND GLOBAL OPTIMALITY CONDITIONS FOR DC INFINITE OPTIMIZATION PROBLEMS

We consider the optimality conditions for the DC (difference of two convex functions) optimization problem with the objective and constraint functions given as DC functions. Adopting convexification technique, the local and global KKT type conditions for this optimization problem are defined. By using properties of the subdifferentials of the involved functions, some sufficient and/or necessary conditions for these two types of optimality conditions are provided.

A Smoothing Function Approach to Joint Chance-Constrained Programs

In this article, we consider a DC (difference of two convex functions) function approach for solving joint chance-constrained programs (JCCP), which was first established by Hong et al. (Oper Res 5...

A DC programming approach for sensor network localization with uncertainties in anchor positions

The sensor network localization with uncertainties in anchor positions has been studied in this paper. We formulate this problem as a DC (difference of two convex functions) programming. Then, a DC programming based algorithm has been proposed to solve such a problem. Simulation results obtained by our proposed method are better performance than those obtained by existing ones.

DC models for spherical separation

We propose two different approaches for spherical separation of two sets. Both methods are based on minimizing appropriate nonconvex nondifferentiable error functions, which can be both expressed in a DC (Difference of two Convex) form. We tackle the problem by adopting the DC-Algorithm. Some numerical results on classical binary datasets are reported.