Concave Programming Research Articles

Secrecy Rate Maximization for Multicarrier-Based Cognitive Radio Networks With an Energy Harvesting Jammer

Cognitive radio (CR) is becoming an increasingly significant component of Internet-of-Things technologies because it overcomes the challenge of spectrum resource scarcity for wireless communications. For the cooperative CR network (CRN) based on the multicarrier system, this article investigates the physical layer security (PLS) to guarantee information privacy for the secondary network while guaranteeing the quality of service (QoS) of the primary network. First, we propose one communication scheme for this type of CRN. The secondary transmitter (ST) and secondary jammer (SJ) first harvest RF energy from the primary signal transmitted by the primary transmitter (PT). Then, ST transmits confidential information with the spectrum vacated by the primary network to the secondary receiver (SR), and SJ simultaneously generates the jamming signal to interfere with the eavesdropper (ED). Second, for the scenarios with the instantaneous channel state information (CSI) and the statistical CSI of the ED, respectively, we aim to maximize the secrecy rate of the secondary network while guaranteeing the QoS requirement of the primary network by jointly optimizing the time allocation between the two phases and the transmit power allocation for ST and SJ over subcarriers. For each scenario, the secrecy rate maximization is formulated as the nonconvex optimization problem. Finally, we design efficient algorithms for both two nonconvex problems. The optimal time allocation is obtained by the 1-D search method at the outer layer, and the optimal transmit power allocation is obtained by the difference of concave (DC) programming at the inner layer, where the Lagrange duality method is used to solve the convex approximation problem. Simulation results verify that the proposed scheme essentially improves the secrecy rate of the secondary network compared to benchmark schemes.

Optimization Method for Wide Beam Sonar Transmit Beamforming.

Imaging and mapping sonars such as forward-looking sonars (FLS) and side-scan sonars (SSS) are sensors frequently used onboard autonomous underwater vehicles. To acquire information from around the vehicle, it is desirable for these sonar systems to insonify a large area; thus, the sonar transmit beampattern should have a wide field of view. In this work, we study the problem of the optimization of wide transmission beampatterns. We consider the conventional phased-array beampattern design problem where all array elements transmit an identical waveform. The complex weight vector is adjusted to create the desired beampattern shape. In our experiments, we consider wide transmission beampatterns (≥20) with uniform output power. In this paper, we introduce a new iterative-convex optimization method for narrowband linear phased arrays and compare it to existing approaches for convex and concave–convex optimization. In the iterative-convex method, the phase of the weight parameters is allowed to be complex as in disciplined convex–concave programming (DCCP). Comparing the iterative-convex optimization method and DCCP to the standard convex optimization, we see that the former methods archive optimized beampatterns closer to the desired beampatterns. Furthermore, for the same number of iterations, the proposed iterative-convex method achieves optimized beampatterns, which are closer to the desired beampattern than the beampatterns achieved by optimization with DCCP.

Unmanned Aerial Vehicle (UAV)-Assisted Path Planning for Unmanned Ground Vehicles (UGVs) via Disciplined Convex-Concave Programming

Thelast decade has witnessed growing research interests in the path-planning problem for unmanned ground vehicles (UGVs) equipped with RGB-D cameras or LiDARs. However, apart from their extremely high cost, these sensors may be obscured by obstacles, which makes them impractical for many complex scenarios. Besides, most of the existing works for path planning have not considered energy efficiency for UGVs that are usually constrained by limited on-board batteries. Distinct from existing methods, this work presents a vision-based unmanned aerial vehicle (UAV)-assisted cooperative system for multiple UGVs. The proposed system harnesses the broad vision of UAV and operates in both general outdoor and global positioning system (GPS)-denied indoor environments. In sharp contrast to the conventional heuristic algorithms such as the rapidly exploring random tree (RRT) algorithm and the Dijkstra methods, the proposed energy-efficient path planning for UGVs is formulated as a non-convex optimization problem by considering the collision-aware obstacle-avoidance. More specifically, the proposed path-planning scheme contains two stages, namely the semantic segmentation that localizes UGVs and obstacles in the environment by exploiting the wide-angle camera mounted on the UAV, followed by the trajectory generation in which a disciplined convex and concave programming (DCCP) algorithm is devised for the non-convex energy-minimization problem. Extensive experiments validate the effectiveness of the UAV-UGV cooperative system and the proposed DCCP-based path-planning scheme.

Open-Loop Equilibrium Strategies for Dynamic Influence Maximization Game Over Social Networks

We consider the problem of budget allocation for competitive influence maximization over social networks. In this problem, multiple competing parties (players) want to distribute their limited advertising resources over a set of social individuals to maximize their long-run cumulative payoffs. It is assumed that the individuals are connected via a social network and update their opinions based on the classical DeGroot model. The players must decide on the budget distribution among the individuals at a finite number of campaign times to maximize their overall payoff as a function of individuals’ opinions. Under some assumptions, we show that i) the optimal investment strategy for a single player can be found in polynomial time by solving a concave program, and ii) the open-loop equilibrium strategies for the multiplayer dynamic game can be computed efficiently by following natural regret-minimization dynamics. Our results extend earlier work on the static version of the problem to a dynamic multistage game.

Energy efficient resource allocation in cache‐enabled fog networks

AbstractTo gain high energy efficiency and low latency, fog computing can be considered as a promising enabling technology for supporting future IoT networks. Exponentially increasing data rate demand of smart IoT devices in the fifth‐Generation (5G) networks require new resource management and power allocation schemes. Fog computing involves the deployment of data storage devices (fog nodes) near the proximity of IoT nodes. The fog nodes collaborate to process requests generated from users. These nodes can store data in their storage capacity and supports in achieving quality‐of‐service (QoS) requirements. A joint node association and energy efficiency maximization problem is formulated in this article. The proposed problem is a non‐linear concave programming problem under cache size, association, and optimal power allocation constraints. A mesh adaptive direct search algorithm (MADS) is used to solve the formulated problem to find the sub‐optimal results. The efficient performance of the algorithm can be seen in comparison with more complex algorithms namely, outer approximation algorithm and exhaustive search algorithm. Extensive simulations are done to observe the detailed performance analysis of the IoT‐Fog network. As a result, energy‐efficient resource allocation is done under cache and QoS constraints with very little complexity and low computational power and delays.

Sustainable multi-commodity capacitated facility location problem with complementarity demand functions

We investigate a multi-commodity capacitated facility location problem involving sustainability concerns (e.g., restrained carbon emission). In addition, this problem incorporates a complementarity demand function. We show that this incorporation can lead to superior decisions both analytically and computationally. The resultant formulation is a 0–1 mixed-integer non-concave quadratic program with equilibrium constraints. We adopt the piecewise-linear envelope method to transform the formulation into a 0–1 mixed-integer concave program. We then propose an efficient branch-and-refine algorithm with global convergence. Numerical examples demonstrate the effect of carbon emission limit and carbon trading on company decisions.

A successive linear approximation algorithm for the global minimization of a concave quadratic program

In this work, we propose an algorithm for finding an approximate global minimum of a concave quadratic function with a negative semi-definite matrix, subject to linear equality and inequality constraints, where the variables are bounded with finite or infinite bounds. The proposed algorithm starts with an initial extreme point, then it moves from the current extreme point to a new one with a better objective function value. The passage from one basic feasible solution to a new one is done by the construction of certain approximation sets and solving a sequence of linear programming problems. In order to compare our algorithm with the existing approaches, we have developed an implementation with MATLAB and conducted numerical experiments on numerous collections of test problems. The obtained numerical results show the accuracy and the efficiency of our approach.

Partial Knowledge Data-Driven Event Detection for Power Distribution Networks

The power system has been incorporating increasing amount of unconventional generations and loads, such as distributed renewable resources, electric vehicles, and controllable loads. The induced dynamic and stochastic power flow require high-resolution monitoring technology and agile decision support techniques for system diagnosis and control. This paper discusses the application of micro-phasor measurement unit ( $\mu$ PMU) data for power distribution network event detection. A novel data-driven event detection method, namely hidden structure semi-supervised machine (HS3M), is established. HS3M only requires partial expert knowledge: it combines unlabeled data and partly labeled data in a large margin learning objective to bridge the gap between supervised learning, semi-supervised learning, and learning with hidden structures. To optimize the non-convex learning objective, a novel global optimization algorithm, namely parametric dual optimization procedure, is established through its equivalence to a concave programming. Finally, the proposed method is validated on an actual distribution feeder with installed $\mu$ PMUs, and the result justifies the effectiveness of the learning-based event detection framework, as well as its potential to serve as one of the core algorithms for power system security and reliability.

Energy-efficient power allocation for massive MIMO-enabled multi-way AF relay networks with channel aging

In this paper, we consider a massive MIMO-enabled multi-way amplify-and-forward relay network with channel aging, where multiple users mutually exchange information via an intermediate relay equipped with massive antennas. For this system, we propose an energy-efficient power allocation scheme for the optimization of energy efficiency (EE). Specifically, we firstly derive accurate closed-form expressions of the system sum rate with aged channel state information (CSI) and predicted CSI. Secondly, based on the derived analytical results, a unified power allocation optimization problem with aged/predicted CSI is formulated for maximizing the system EE. To solve this challenging problem, the successive convex approximation technique is invoked to transform the original optimization problem into a tractable concave fractional programming problem. Then, Dinkelbach’s algorithm and Lagrangian dual method are adopted to find the optimal solution. In addition, to strike a balance between the computational complexity and the optimality, the EE maximization problem using the equal power allocation scheme is solved by extreme value theorem, leading to a closed-form optimal solution. Numerical results demonstrate the accuracy of our analytical results and the effectiveness of the proposed algorithms. Moreover, the impact of several important system parameters on the system performance achieved by the proposed algorithms is also illustrated.

Secrecy energy efficiency optimization for MISO SWIPT systems

In this paper, we investigate the secrecy energy efficiency (SEE) optimization of the multiple-input single-output (MISO) system. First, transmission beamforming vector is designed to achieve the SEE maximization subject to the constraints of transmission power, minimum secrecy rate threshold and harvested energy threshold. The optimization problem belongs to the category of fractional optimization, which is non-convex and is very difficult to tackle. In order to solve the optimization problem, we propose an algorithm that can obtain a near-optimal solution, which consists of outer-tier and inner-tier iterations. For the outer-tier iteration, we first employ the Dinkelbach method to convert the fractional objective function into a polynomial form, and then transform the optimization problem into a difference of concave (DC) programming. For the inner-tier iteration, we employ the first-order Taylor expansion and successive convex approximation (SCA) method to solve the DC optimization problem. Then, we analyze the computational complexity of the proposed algorithm. In addition,we prove that the rank relaxation is tight. The simulation results show that the SEE performance of our proposed algorithm is obviously superior to that of secrecy rate maximization scheme and zero-forcing scheme.

Tractable Consideration Set Structures for Assortment Optimization and Network Revenue Management

Discrete‐choice models are widely used to model consumer purchase behavior in assortment optimization and revenue management. In many applications, each customer segment is associated with a consideration set that represents the set of products that customers in this segment consider for purchase. The firm has to make a decision on what assortment to offer at each point in time without the ability to identify the customer's segment. A linear program called the Choice‐based Deterministic Linear Program ( CDLP) has been proposed to determine these offer sets. Unfortunately, its size grows exponentially in the number of products and it is NP‐hard to solve when the consideration sets of the segments overlap. The Segment‐based Deterministic Concave Program with some additional consistency equalities ( SDCP+) is an approximation of CDLP that provides an upper bound on CDLP's optimal objective value. SDCP+ can be solved in a fraction of the time required to solve CDLP and often achieves the same optimal objective value. This raises the question under what conditions can one guarantee equivalence of CDLP and SDCP+. In this study, we obtain a structural result to this end, namely that if the segment consideration sets overlap with a certain tree structure or if they are fully nested, CDLP can be equivalently replaced with SDCP+. We give a number of examples from the literature where this tree structure arises naturally in modeling customer behavior.

Achieving Maximum Effective Capacity in OFDMA Networks Operating Under Statistical Delay Guarantee

We propose a spectrally efficient design that guarantees the statistical delay quality-of-service (QoS) for delay-sensitive traffic in the downlink of orthogonal frequency-division multiple-access (OFDMA) networks. This design is based on the so-called effective capacity (EC) concept, which describes the maximum throughput, a system can achieve under a specific statistical delay-QoS violation probability constraint. We investigate the EC maximization problem, in which, the statistical delay profile of the traffic is characterized by the QoS-exponent $\theta $ determining the exponential decay rate of the delay-QoS violation probability. By exploiting the properties of concave programming and Slater’s condition, the Lagrangian dual decomposition method is applied and an iterative algorithm that does not depend on the instantaneous channel state information (CSI) is proposed for solving the concave problem formulated. Extensive simulations demonstrate the efficacy and robustness of the proposed iterative algorithm. Furthermore, we show that the system’s achievable EC does not depend on the specific choice of the subcarrier allocation, but rather on the number of subcarriers allocated to each user. This is, because, the EC is calculated using the channel’s statistics, instead of the instantaneous CSI, implying that the EC is more of a long term channel capacity metric.

One algorithm for branch and bound method for solving concave optimization problem

The article describes the algorithm for branch and bound method for solving the concave programming problem, which is based on the idea of similarity the necessary and sufficient conditions of optimum for the original problem and for a convex programming problem with another feasible set and reverse the sign of the objective function. To find the feasible set of the equivalent convex programming problem we construct an algorithm using the idea of the branch and bound method. We formulate various branching techniques and discusses the construction of the lower objective function evaluations for the node of the decision tree. The article discusses the results of experiments of this algorithm for some famous test problems of a particular form.

Decomposition Procedure for Solving NLP and QP Problems based on Lagrange and Sander's Method

This paper develops a decompose procedure for finding the optimal solution of convex and concave Quadratic Programming (QP) problems together with general Non-linear Programming (NLP) problems. The paper also develops a sophisticated computer technique corresponding to the author's algorithm using programming language MATHEMATICA. As for auxiliary by making comparison, the author introduces a computer-oriented technique of the traditional Karush-Kuhn-Tucker (KKT) method and Lagrange method for solving NLP problems. He then modify the Sander's algorithm and develop a new computational technique to evaluate the performance of the Sander's algorithm for solving NLP problems. The author observe that the technique avoids some certain numerical difficulties in NLP and QP. He illustrates a number of numerical examples to demonstrate his method and the modified algorithm.

Analysis of a Joint Pricing and Seat Allocation Model in a Hub-to-Hub Airline Network

In airline revenue management, after differentiating products on each itinerary according to various restrictions, management needs to set the prices for the products. A deterministic joint pricing and seat allocation model is proposed. It is reduced to a separable concave programming problem and thus readily solvable. Focusing on a special hub-to-hub airline network, monotonicity of the pricing decisions is explored. Using a nonlinear primal–dual technique, it is shown that some itineraries’ optimal prices are decreasing, while some other itineraries’ optimal prices are increasing with a capacity change in either a side leg or the middle leg. Such structural properties add important managerial insights into the pricing model for revenue managers in an airline company.

CDF formulation for solving an optimal reinsurance problem

An innovative cumulative distribution function (CDF)-based method is proposed for deriving optimal reinsurance contracts to maximize an insurer’s survival probability. The optimal reinsurance model is a non-concave constrained stochastic maximization problem, and the CDF-based method transforms it into a functional concave programming problem of determining an optimal CDF over a corresponding feasible set. Compared to the existing literature, our proposed CDF formulation provides a more transparent derivation of the optimal solutions, and more interestingly, it enables us to study a further complex model with an extra background risk and more sophisticated premium principle.

Joint optimisation of buffers and network population for closed finite queueing systems

The buffer allocation problem for closed finite queueing networks is a nonlinear integer programming problem which is -Hard. A queueing network decomposition methodology based upon a mean value analysis algorithm is coupled with a nonlinear sequential quadratic programming algorithm to compute the simultaneous optimal buffer allocations and network population via a branch-and-bound scheme for various network topologies. The joint optimisation problem is essentially multi-objective where the decision variable on the population will be treated as a constraint linking the buffer allocation and the throughput which is to be maximised. The fact that the optimisation problem is a concave programming problem assists in the search for an optimal solution. The material handling or transportation system for transferring the finite customer population between the nodes in the network is also examined. Extensive numerical results demonstrate the efficacy of the optimisation methodology.

Concave minimization for sparse solutions of absolute value equations

Based on concave function, the problem of finding the sparse solution of absolute value equations is relaxed to a concave programming, and its corresponding algorithm is proposed, whose main part is solving a series of linear programming. It is proved that a sparse solution can be found under the assumption that the connected matrixes have range space property(RSP). Numerical experiments are also conducted to verify the efficiency of the proposed algorithm.

Energy Efficient Sparse Beamforming for Multi-user MIMO Systems with Non-ideal Power Amplifiers

In this paper, we consider energy-efficiency (EE) optimization problems with nonideal power amplifier (PA) models, per-antenna power constraints, and minimal spectral efficiency (SE) requirements in block-diagonalization-based multiuser multiple-input–multiple-output (MU-MIMO) systems. In the problem formulation, a joint optimization of the transmit covariance and the active transmit antenna set at the base station is considered. In general, the problem is a mixed-integer fractional programming problem, and it is difficult to solve it globally optimally. By reformulating the original problem as a sparse beamforming design problem and using a successive convex approximation method, we propose an iterative algorithm to solve it locally. In each iteration step, a concave fractional programming is solved, and the solution can be expressed in closed form with the help of the Lagrange dual method. Simulation results are used to verify the performance of our proposed algorithms. We also study the EE–SE tradeoff for the considered system. In fact, the relationship between EE and SE under an ideal PA model is not accurate for realistic systems. We characterize the EE–SE tradeoff under nonideal PA models by solving the EE maximization problems under different SE values. In this way, we can also shed light on the design of the system parameters to optimize the EE–SE curve. We find that with nonideal PA considered, the SE corresponding to the optimal EE will increase, compared with the ideal case. If we design the system parameters according to the optimal SE under an ideal PA model to maximize the EE, both the EE and the SE of the realistic system will decrease. We also find that when the proposed antenna selection algorithm is applied, the achieve region of the EE and the SE is expanded, particularly when the SE is low. In other words, the EE–SE win–win region is enlarged, and the EE–SE tradeoff is improved.

Inscribed ball and enclosing box methods for the convex maximization problem

Many important classes of decision models give rise to the problem of finding a global maximum of a convex function over a convex set. This problem is known also as concave minimization, concave programming or convex maximization. Such problems can have many local maxima, therefore finding the global maximum is a computationally difficult problem, since standard nonlinear programming procedures fail. In this article, we provide a very simple and practical approach to find the global solution of quadratic convex maximization problems over a polytope. A convex function achieves its global maximum at extreme points of the feasible domain. Since an inscribed ball does not contain any extreme points of the domain, we use the largest inscribed ball for an inner approximation while a minimal enclosing box is exploited for an outer approximation of the domain. The approach is based on the use of these approximations along with the standard local search algorithm and cutting plane techniques.