Convex Second-order Cone Research Articles

Energy-Efficient Beamforming Design for User-Centric Networks With Full-Duplex Wireless Fronthaul

In this paper, we investigate the joint beamforming design for access and fronthaul links in a user-centric network with full-duplex fronthaul. We formulate an optimization problem to maximize the network energy efficiency while guaranteeing fronthaul rate requirements. Particularly, the power consumed by self-interference cancellation at full-duplex access points (APs) is taken into consideration to comprehensively reflect the effect of full-duplex technology. Due to the non-convexity and the nonlinear fractional form of the formulated problem, we resort to the successive convex approximation method. The original problem is approximated as a convex second-order cone program. We then extend the study to the scenario of imperfect and partial channel state information (CSI). To avoid the intractable expression of the achievable rate in this case, we derive its lower bound for further beamforming design. Simulation results demonstrate the superiority of the proposed algorithm in terms of energy efficiency and power consumption under both CSI scenarios. For our considered full-duplex fronthaul networks, moderate maximum transmit power of macro base station (MBS) and APs is sufficient to obtain desired energy efficiency performance. Moreover, the lower bound is verified to be tight at the low-maximum power region.

A penalized inequality-constrained approach for robust beamforming with DoF limitation

A well-known challenge in beamforming is how to optimally utilize the degrees of freedom (DoF) of the array to design a robust beamformer, especially when the array DoF is limited. In this paper, we leverage the tool of constrained convex optimization and propose a penalized inequality-constrained minimum variance (P-ICMV) beamformer to address this challenge. Specifically, a well-targeted objective function and inequality constraints are proposed to achieve the design goals. By penalizing the maximum gain of the beamformer at any interfering directions, the total interference power can be efficiently mitigated with limited DoF. Multiple robust constraints on the target protection and interference suppression can be introduced to increase the robustness of the beamformer against steering vector mismatch. By integrating the noise reduction, interference suppression, and target protection, the proposed formulation can efficiently obtain a robust beamformer design while optimally trading off various design goals. To numerically solve this problem, we formulate the P-ICMV beamformer design as a convex second-order cone program (SOCP) and propose a low complexity iterative algorithm based on the alternating direction method of multipliers (ADMM). Three applications are simulated to demonstrate the effectiveness of the proposed beamformer.

Air-breathing hypersonic vehicle trajectory optimization with uncertain no-fly zones

The air-breathing hypersonic planes are considered to be the future of commercial airlines, which can fly from NYC to London in under an hour. For air-breathing vehicles, 3D trajectory planning will become extremely important due to its significant impact on flight performance. Past research on this issue was not comprehensive enough. Thus, we proposed a hybrid method for generating the optimal trajectories efficiently. No-fly zones are specified for geopolitical restrictions in air-breathing hypersonic vehicle missions. However, previous studies focused on no-fly zone constraints with fixed locations and boundaries. For robust execution, we must take into account no-fly zones’ uncertainties, which arise due to uncertain localization, measurement errors, and no-fly zones’ movement. A chance-constrained approach is presented here to deal with such uncertainties. The key idea of this method is controlling risk when the flight path is close to the uncertain no-fly zones. First, the trajectory planning of air-breathing hypersonic vehicles is modeled as a chance-constrained optimization (CCOPT) problem. With the help of the convexification and linearization techniques, we can approximate the non 3.2.2 CCOPT problem as a convex second-order cone programing under the Gaussian distribution assumption. It can be solved to global optimality by the interior point method. Finally, we give numerical simulation results to prove the effectiveness of this method.

The augmented Lagrangian method can approximately solve convex optimization with least constraint violation

There are many important practical optimization problems whose feasible regions are not known to be nonempty or not, and optimizers of the objective function with the least constraint violation prefer to be found. A natural way for dealing with these problems is to extend the nonlinear optimization problem as the one optimizing the objective function over the set of points with the least constraint violation. This leads to the study of the shifted problem. This paper focuses on the constrained convex optimization problem. The sufficient condition for the closedness of the set of feasible shifts is presented and the continuity properties of the optimal value function and the solution mapping for the shifted problem are studied. Properties of the conjugate dual of the shifted problem are discussed through the relations between the dual function and the optimal value function. The solvability of the dual of the optimization problem with the least constraint violation is investigated. It is shown that, if the least violated shift is in the domain of the subdifferential of the optimal value function, then this dual problem has an unbounded solution set. Under this condition, the optimality conditions for the problem with the least constraint violation are established in term of the augmented Lagrangian. It is shown that the augmented Lagrangian method has the properties that the sequence of shifts converges to the least violated shift and the sequence of multipliers is unbounded. Moreover, it is proved that the augmented Lagrangian method is able to find an approximate solution to the problem with the least constraint violation and it has linear rate of convergence under an error bound condition. The augmented Lagrangian method is applied to an illustrative convex second-order cone constrained optimization problem with least constraint violation and numerical results verify our theoretical results.

Dynamic planning of a two-dose vaccination campaign with uncertain supplies

The ongoing COVID-19 pandemic has led public health authorities to face the unprecedented challenge of planning a global vaccination campaign, which for most protocols entails the administration of two doses, separated by a bounded but flexible time interval. The partial immunity already offered by the first dose and the high levels of uncertainty in the vaccine supplies have been characteristic of most of the vaccination campaigns implemented worldwide and made the planning of such interventions extremely complex. Motivated by this compelling challenge, we propose a stochastic optimization framework for optimally scheduling a two-dose vaccination campaign in the presence of uncertain supplies, taking into account constraints on the interval between the two doses and on the capacity of the healthcare system. The proposed framework seeks to maximize the vaccination coverage, considering the different levels of immunization obtained with partial (one dose only) and complete vaccination (two doses). We cast the optimization problem as a convex second-order cone program, which can be efficiently solved through numerical techniques. We demonstrate the potential of our framework on a case study calibrated on the COVID-19 vaccination campaign in Italy. The proposed method shows good performance when unrolled in a sliding-horizon fashion, thereby offering a powerful tool to help public health authorities calibrate the vaccination campaign, pursuing a trade-off between efficacy and the risk associated with shortages in supply.

Power allocation in D2D enabled cellular network with probability constraints: A robust Stackelberg game approach

In this paper, we develop a Stackelberg game with a non-uniform pricing in a two-tier network environment, where multiple device-to-device (D2D) users reuse the uplink spectrum resources of the cellular user (CU). Assuming that the base station (BS) is protected by pricing the interference from D2D users, we propose a Stackelberg game framework where BS and D2D users are treated as the leader and followers respectively. Besides, the channel state information (CSI) uncertainty in the actual communication environment is considered to be suppressed the interference fluctuation effects. Specifically, the probability constraint is included to optimize the user utility. The complex probability constraint is transformed to the deterministic form of the convex second-order cone constraint by a novel method. Furthermore, a norm approximation method is proposed to reduce the computational complexity. Finally, the analytical solution of Nash equilibrium is acquired to generate the best response to the leader and all followers. Simulation results show that the proposed scheme is more effective comparing to the algorithms without the CSI probability constraint.

Energy Management and Control of Photovoltaic and Storage Systems in Active Distribution Grids

The evolution of power distribution grids from passive to active systems creates reliability and efficiency challenges to the distribution system operators. In this paper, an energy management and control scheme for managing the operation of an active distribution grid with prosumers is proposed. A multi-objective optimization model to minimize (i) the prosumers electricity cost and (ii) the cost of the grid energy losses, while guaranteeing safe and reliable grid operation is formulated. This is done by determining the active and reactive power set-points of the photovoltaic and storage systems integrated in the grid buildings. The resulting optimization model is non-convex, thus a convex second-order cone program is developed by appropriately relaxing the non-convex constraints which yields optimal results in most operating conditions. The convexified model is further utilized to develop an algorithm that yields feasible solutions to the non-convex problem under any operating conditions. Moreover, a second novel algorithm to find the operating point that provides fairness between the prosumers and the grid costs is proposed. Simulation results demonstrate the effectiveness and superiority of the proposed scheme in managing an industrial distribution grid compared to a self-consumption approach.

Acoustic-Electrical Joint Localization Method of Partial Discharge in Power Transformer Considering Multi-Path Propagation Impact

Efficient and accurate localization of partial discharge (PD) is of paramount importance to ensure the safe operation of power transformers. However, the multi-path propagation effect introduced by the reflection, refraction and diffraction of the ultrasonic signal may add significant computational complexity to the localization process and degrade the localization accuracy. This paper proposes an acoustic- electrical joint method for partial discharge location in the power transformer with the full consideration of the multi-path propagation impact. Unlike the conventional error analysis methods, a partial discharge localization model is proposed for characterizing the multipath propagation impact without the prior knowledge of the transcendental error probability. Based on the matrix inequality transformation and relaxation, the high-dimensional nonlinear localization equations are transformed into a set of second-order convex optimization equations that can be solved using the convex second-order cone program (SOCP). The proposed solution can significantly reduce the computational complexity and improve the localization accuracy as well as avoid the local optimum and slow convergence. The solution is assessed through extensive experiments based on simulations, testbed and trial deployment in comparison with the existing solutions with the localization error of about 0.1 m.

A new projection neural network for linear and convex quadratic second-order cone programming

A new projection neural network approach is presented for the linear and convex quadratic second-order cone programming. In the method, the optimal conditions of the linear and convex second-order cone programming are equivalent to the cone projection equations. A Lyapunov function is given based on the G-norm distance function. Based on the cone projection function, the descent direction of Lyapunov function is used to design the new projection neural network. For the proposed neural network, we give the Lyapunov stability analysis and prove the global convergence. Finally, some numerical examples and two kinds of grasping force optimization problems are used to test the efficiency of the proposed neural network. The simulation results show that the proposed neural network is efficient for solving some linear and convex quadratic second-order cone programming problems. Especially, the proposed neural network can overcome the oscillating trajectory of the exist projection neural network for some linear second-order cone programming examples and the min-max grasping force optimization problem.

Adaptive channel estimation for cognitive fully adaptive radar

The paper addresses channel state information estimation for a cognitive radar system. The underlying radar signal model considered in this paper is based on a stochastic Green's function approach. In the radar model, the clutter returns can be expressed as a convolution of the stochastic Green's function impulse response with the transmit radar waveform. The paper proposes and solves constrained channel estimation algorithms that exploit the cosine similarity constraint and the inner product constraint to improve conventional least squares solutions. Using the fact that adjacent channel transfer functions have a close relationship in terms of a cosine similarity metric, the cosine similarity constraint can be exploited in the least squares problem. It turns out that the derived optimisation problem under the cosine similarity constraint is non-convex. However, since strong duality holds for the non-convex optimisation problem, it can be solved using the semi-definite relaxation (SDR) approach. The paper also proposes a new optimisation problem with an additional inner product constraint, which results in a convex second-order cone programming (SOCP) and is robust to the signal-to-noise ratio (SNR) in terms of the cosine similarity measurement. Numerical simulations are provided to verify the performance of the proposed methods on a challenge dataset generated using the high-fidelity modelling and simulation tool, RFView®.

Hybrid robust/stochastic transmission expansion planning considering uncertainties in generators’ offer prices: A second-order cone program approach

The aim of this paper is to propose a hybrid robust/stochastic model for transmission expansion planning (TEP) problem. The uncertainties related to demand and wind are modelled via stochastic programming (SP), while the uncertainties associated with the generators’ offer prices are modelled via robust optimization (RO). In this sense, an ellipsoidal uncertainty set is considered which the offer price values are known to lie in these given ellipsoids. The model presented is then divided into a master problem and several independent subproblems using a tailored implementation of Bender's decomposition (BD) method. Robust linear constraints of the lower level problems are then transformed into a convex second-order cone programming (SOCP) problem. We will apply the proposed model to IEEE-118 bus power system showing that generators’ offer price and their correlation can considerably affect the optimal plan of the transmission planning. It is observed that considering correlation among generators’ offer prices would not necessarily yield more expensive expansion plan in comparison with the case where correlation is ignored. Indeed, increase or decrease in investment cost depends on the entries of the variance-covariance matrix, network topology and the location of generators. Moreover, the results show the computational efficiency of the robust convex model.

A quadratic voltage model for optimal power flow of a class of meshed networks

A quadratic voltage model is proposed to formulate the optimal power flow problem of the meshed networks with maximum phase angle differences between buses less than π/2. The quadratic voltage model uses the real and imaginary parts of node voltage directly so as to avoid the problem of inexact angle relaxation, which is the Achilles’ Heel in previous researches. The nonlinear equalities between quadratic terms of voltages’ variables are relaxed into convex second-order cone constraints. A penalty term is added into the objective function to guarantee the exactness of the relaxation, which is proved with the strict inequality on the upper and lower bounds of power generation nodes. For the proper value of the penalty factors, analytical representation and mathematical proof are provided. The applicability of the proposed model is tested over several systems, which shows better efficiency and less computational time.

Energy-Efficient Joint Association and Precoding in Ultra-Dense C-RAN

In this work, we consider the joint optimization of virtual cell association and the precoder design for a multi-user downlink cloud radio access network (C-RAN). The optimal design of cell associations and precoders are particularly relevant due to the anticipated ultra-dense networks suffering from heavy interference. As the resulting optimization is a non-convex and combinatorial problem, we propose an iterative low complexity algorithm, employing the recently developed penalty dual decomposition (PDD) framework. In contrast to predominantly used sparsity inducing methods, the proposed method incorporates the penalty/re-weighting strategy into the algorithm, which is then solved as a sequence of convex second-order cone programs. The aforementioned feature leads to a high convergence rate and subsequently a low computational cost for the proposed algorithm. Furthermore, the proposed algorithm meets the requirements to converge to a KKT solution. Numerical simulations verify the superior performance of our proposed method in ultra-dense networks, in comparison to commonly used methods, such as the reweighted $\ell _1$ -norm, in terms of system energy efficiency.

Robust Ellipse Fitting With Laplacian Kernel Based Maximum Correntropy Criterion.

The performance of ellipse fitting may significantly degrade in the presence of outliers, which can be caused by occlusion of the object, mirror reflection or other objects in the process of edge detection. In this paper, we propose an ellipse fitting method that is robust against the outliers, and thus maintaining stable performance when outliers can be present. We formulate an optimization problem for ellipse fitting based on the maximum entropy criterion (MCC), having the Laplacian as the kernel function from the well-known fact that the l1 -norm error measure is robust to outliers. The optimization problem is highly nonlinear and non-convex, and thus is very difficult to solve. To handle this difficulty, we divide it into two subproblems and solve the two subproblems in an alternate manner through iterations. The first subproblem has a closed-form solution and the second one is cast as a convex second-order cone program (SOCP) that can reach the global solution. By so doing, the alternate iterations always converge to an optimal solution, although it can be local instead of global. Furthermore, we propose a procedure to identify failed fitting of the algorithm caused by local convergence to a wrong solution, and thus, it reduces the probability of fitting failure by restarting the algorithm at a different initialization. The proposed robust ellipse fitting method is next extended to the coupled ellipses fitting problem. Both simulated and real data verify the superior performance of the proposed ellipse fitting method over the existing methods.

A Thinned Irregular Array Synthesis Approach Based on Benders Decomposition

A thinned irregular tiled array synthesis approach is proposed for 5G base station array design. To fit the needs of multiusers, different sizes of vertical tiles with thinned elements ae considered. The tiling configuration and excitations are optimized to achieve high gain, low sidelobes, and wide-angle scanning simultaneously. A mixed integer second-order cone programming (MISOCP) model is obtained from convexity analysis and transformation. Besides, a generalized Benders decomposition algorithm is built to decouple the complex original problem into mixed integer linear programming (MILP) master problems and convex second-order cone programming (SOCP) subproblems, which is used to solve the large-scale problems efficiently. Irregular partition optimization is based on full-wave simulations of regular finite arrays. The joint effects of active VSWR and radiation patterns are considered to reduce the mismatch between element simulations and final performances. Numerical results show that the 5G array synthesized by the proposed approach is superior to those reported by the state-of-the-art methods in terms of both E- and H-plane scans and the number of T/R modules.

Cooperative Interference Management for Over-the-Air Computation Networks

Recently, over-the-air computation (AirComp) has emerged as an efficient solution for access points (APs) to aggregate distributed data from many edge devices (e.g., sensors) by exploiting the waveform superposition property of multiple access (uplink) channels. While prior work focuses on the single-cell setting where inter-cell interference is absent, this article considers a multi-cell AirComp network limited by such interference and investigates the optimal policies for controlling devices' transmit power to minimize the mean squared errors (MSEs) in aggregated signals received at different APs. First, we consider the scenario of centralized multi-cell power control. To quantify the fundamental AirComp performance tradeoff among different cells, we characterize the Pareto boundary of the multi-cell MSE region by minimizing the sum MSE subject to a set of constraints on individual MSEs. Though the sum-MSE minimization problem is non-convex and its direct solution intractable, we show that this problem can be optimally solved via equivalently solving a sequence of convex second-order cone program (SOCP) feasibility problems together with a bisection search. This results in an efficient algorithm for computing the optimal centralized multi-cell power control, which optimally balances the interference-and-noise-induced errors and the signal misalignment errors unique for AirComp. Next, we consider the other scenario of distributed power control, e.g., when there lacks a centralized controller. In this scenario, we introduce a set of interference temperature (IT) constraints, each of which constrains the maximum total inter-cell interference power between a specific pair of cells. Accordingly, each AP only needs to individually control the power of its associated devices for single-cell MSE minimization, but subject to a set of IT constraints on their interference to neighboring cells. By optimizing the IT levels, the distributed power control is shown to provide an alternative method for characterizing the same multi-cell MSE Pareto boundary as the centralized counterpart. Building on this result, we further propose an efficient algorithm for different APs to cooperate in iteratively updating the IT levels to achieve a Pareto-optimal MSE tuple, by pairwise information exchange. Last, simulation results demonstrate that cooperative power control using the proposed algorithms can substantially reduce the sum MSE of AirComp networks compared with the conventional single-cell approaches.

Dual Quaternion-Based Powered Descent Guidance with State-Triggered Constraints

This paper presents a numerical algorithm for computing six-degree-of-freedom free-final-time powered descent guidance trajectories. The trajectory generation problem is formulated using a unit dual quaternion representation of the rigid body dynamics and several standard path constraints. Our formulation also includes a special line-of-sight constraint that is enforced only within a specified band of slant ranges relative to the landing site: a novel feature that is especially relevant to terrain and hazard relative navigation. We use the newly introduced state-triggered constraints to formulate these range constraints in a manner that is amenable to real-time implementations. The resulting nonconvex optimal control problem is solved iteratively as a sequence of convex second-order cone programs that locally approximate the nonconvex problem. Each second-order cone program is solved using a customizable interior point method solver. Also introduced are a scaling method and a new heuristic technique that guide the convergence process toward dynamic feasibility. To demonstrate the capabilities of our algorithm, two numerical case studies are presented. The first studies the effect of including a slant-range-triggered line-of-sight constraint on the resulting trajectories. The second study performs a Monte Carlo analysis to assess the algorithm’s robustness to initial conditions and real-time performance.

Successive Convexification for Real-Time Six-Degree-of-Freedom Powered Descent Guidance with State-Triggered Constraints

This paper presents a real-time implementable successive convexification algorithm for a generalized free-final-time six-degree-of-freedom powered descent guidance problem. Building on our previous research, the following contributions are introduced: 1) a free-ignition-time modification that allows the algorithm to determine the optimal engine ignition time, 2) a tractable formulation of nonlinear aerodynamic lift and drag, and 3) a continuous state-triggered constraint formulation that emulates conditionally enforced constraints. In particular, contribution 3 effectively allows constraints to be enabled or disabled by if statements conditioned on the solution variables of the parent continuous optimization problem. To the best of our knowledge, this represents a novel formulation in the optimal control literature and enables a number of interesting applications in powered descent guidance, such as velocity-triggered angle-of-attack constraints. Our algorithm converts the resulting generalized powered descent guidance problem from a nonconvex free-final-time optimal control problem into a sequence of convex second-order cone programming subproblems. With the aid of virtual control and trust region modifications, these subproblems are solved iteratively until convergence is attained. Simulations demonstrate the real-time capabilities of the proposed algorithm, with a maximum solver runtime of less than 0.7 s over a multitude of problem feature combinations on a 2.2 GHz Intel processor.

On the Sensitivity of Local Flexibility Markets to Forecast Error: A Bi-Level Optimization Approach

The large-scale integration of intermittent distributed energy resources has led to increased uncertainty in the planning and operation of distribution networks. The optimal flexibility dispatch is a recently introduced, power flow-based method that a distribution system operator can use to effectively determine the amount of flexibility it needs to procure from the controllable resources available on the demand side. However, the drawback of this method is that the optimal flexibility dispatch is inexact due to the relaxation error inherent in the second-order cone formulation. In this paper we propose a novel bi-level optimization problem, where the upper level problem seeks to minimize the relaxation error and the lower level solves the earlier introduced convex second-order cone optimal flexibility dispatch (SOC-OFD) problem. To make the problem tractable, we introduce an innovative reformulation to recast the bi-level problem as a non-linear, single level optimization problem which results in no loss of accuracy. We subsequently investigate the sensitivity of the optimal flexibility schedules and the locational flexibility prices with respect to uncertainty in load forecast and flexibility ranges of the demand response providers which are input parameters to the problem. The sensitivity analysis is performed based on the perturbed Karush–Kuhn–Tucker (KKT) conditions. We investigate the feasibility and scalability of the proposed method in three case studies of standardized 9-bus, 30-bus, and 300-bus test systems. Simulation results in terms of local flexibility prices are interpreted in economic terms and show the effectiveness of the proposed approach.

Optimal Multi-Period Dispatch of Distributed Energy Resources in Unbalanced Distribution Feeders

This paper presents an efficient algorithm for the multi-period optimal dispatch of deterministic inverter-interfaced energy storage in an unbalanced distribution feeder with significant solar PV penetration. The three-phase, non-convex loss-minimization problem is formulated as a convex second-order cone program (SOCP) for the dispatch of batteries in a receding-horizon fashion in order to counter against the variable, renewable generation. The solution of the SOCP is used to initialize a nonlinear program (NLP) in order to ensure a physically realizable solution. The phenomenon of simultaneous charging and discharging of batteries is rigorously analyzed and conditions are derived that guarantee it is avoided. Simulation scenarios are implemented with GridLab-D for the IEEE-13 and IEEE-123 node test feeders and illustrate not only AC feasibility of the solution, but also near-optimal performance and solve-times within a minute.