Closed-form Optimal Solution Research Articles

Codesign of transmit waveform and reflective beamforming for active reconfigurable intelligent surface-aided MIMO ISAC system

This article discusses the active reconfigurable intelligent surface (ARIS)-aided integrated sensing and communication (ISAC) system for non-line-of-sight (NLoS) target sensing in cluttered environments while performing multi-user communication. To optimize sensing and communication performance simultaneously, we jointly design the shared transmit waveform, ARIS reflection coefficients and radar receive filter by using the multi-user interference and the reciprocal of radar output signal-to-interference-plus-noise ratio as metrics. Limited by practical requirements, the transmit waveform suffers from constant modulus or total energy constraints and the ARIS is subject to both maximum power and amplification gain constraints. Based on these considerations, the proposed codesign is formulated into a nonconvex constrained fractional function minimization problem. To tackle it effectively, we first translate the fractional objective into an integral form by employing Dinkelbach transform and then propose an alternating optimization-based algorithm, where the transmit waveform and ARIS reflection coefficients are respectively optimized by the customized algorithms based on the consensus alternating direction method of multipliers, and the receive filter has a closed-form optimal solution. Numerical results demonstrate that the ARIS-aided ISAC concurrently achieve superior NLoS sensing and communication performance to passive reconfigurable intelligent surface-aided and traditional ISACs in cluttered environments, regardless of waveform constraints and sensing-communication trade-off factor.

Optimal co-development contracts for companion diagnostics

The market for companion diagnostics is expected to be a US$10.07 billion by 2026. Companion diagnostics have the potential to make expensive drugs cost-effective by identifying patients who would benefit from them. We consider the contract design problem between a pharmaceutical company which owns a drug that is effective for a particular subset of the patient population and a biotech company which owns some technology that could facilitate the development of a companion diagnostic. We obtain theoretical and practical results. We determine when both parties enter such a contract and fully characterize the optimal solutions in closed-form. We find sufficient conditions under which the optimal contract exhibits a particular structure. We show that the first-best can be achieved in some cases and identify sufficient conditions under which the biotech company would not work alone but participates in the project with the pharmaceutical company’s subsidy. We find that heuristics based on practical preferences could be costly to the pharmaceutical company and hence the principal should use the second-best solution; and contract type depends heavily on the biotech company’s workforce level, unit cost of workforce and information level.

Adaptive Bayesian speckle reduction in the 2D DOST domain using 2D MC-GARCH-M model: Preserving image edges and textural information

Image denoising and image quality enhancement are major issues in image processing. Additive and multiplicative noise removal is one of the main image enhancement approaches. In this paper, we propose a novel 2D Merged Complex Generalized Autoregressive Conditional Heteroscedasticity Mixture (MC-GARCH-M) model for 2D complex stochastic processes. This model effectively captures non-Gaussian statistics and all three types of dependencies within the processes (RRD, IID, and RID) using the proposed linear mapping transformation T. Our statistical analysis shows that the 2D MC-GARCH-M model provides the best fit to the non-Gaussian statistics of the mapped matrix of 2D DOST coefficients compared to Gaussian, Generalized Gaussian, and Stable distributions. We confirm the presence of heteroscedasticity in the mapped matrix using the Engle hypothesis test, verifying ARCH/GARCH effects. The 2D MC-GARCH-M model, with its location-dependent conditional variances, provides a flexible approach for noise removal and modeling complex stochastic processes. Experimental results on artificially speckled aerial and actual SAR images demonstrate that our method outperforms state-of-the-art approaches in terms of speckle removal and preserving the edges and textural information of the image. Key innovations include: a new statistical analysis using linear mapping T showing non-Gaussian heavy-tailed distributions; an adaptive Bayesian estimator, MCMAP, based on the heteroscedastic model for speckle noise removal; and the method's applicability to images with varying mutual information (MI) between real and imaginary parts of 2D DOST coefficients. Our method is adaptive and robust to initial parameter settings, effectively utilizing magnitude and phase information, providing an optimal closed-form solution, reducing memory and computational requirements, and demonstrating robustness to speckle power levels. While requiring more computational resources compared to thresholding methods, our approach is less demanding than other Bayesian algorithms, making it suitable for offline processing.

Low complexity beamforming design in reconfigurable intelligent surface-assisted communication systems

Due to the possibility of blockage during communication, the communication quality may be poor. Therefore, Reconfigurable Intelligent Surface (RIS) technology can effectively solve this problem. In order to increase the capacity of RIS-assisted wireless communication systems, joint optimization of beamforming design is crucial. However, the complexity of the optimization algorithm increases with the increase in the number of base station antennas and RIS elements deployed. Therefore, we propose a joint low-complexity optimization beamforming design based on fractional programming (FP) to address this issue. Specifically, we first use perfect channel state information to maximize the system's sum rate, as this problem is non-convex, we decompose the original problem into three sub-problems, and then introduce appropriate auxiliary variables. We derive optimal closed-form solutions for active and passive beamforming types, respectively. As the optimal solution obtained leads to higher computational complexity with an increase in the number of base station antennas and RIS elements, we reduce the computational complexity of obtaining the optimal solution based on the Woodbury transformation and scalar transformation. The proposed algorithm is also extended to the case where there is channel state information error. Finally, simulation results show that the proposed algorithm has certain advantages over other algorithms.

Joint Transmit and Receive Beamforming Design for DPC-Based MIMO DFRC Systems

This paper proposes an optimal beamforming strategy for a downlink multi-user multi-input–multi-output (MIMO) dual-function radar communication (DFRC) system with dirty paper coding (DPC) adopted at the transmitter. We aim to achieve the maximum weighted sum rate of communicating users while adhering to a predetermined transmit covariance constraint for radar performance assurance. To make the intended problem trackable, we leverage the equivalence of the weighted sum rate and the weighted minimum mean squared error (MMSE) to reframe the issue and devise a block coordinate descent (BCD) approach to iteratively calculate transmit and receive beamforming solutions. Through this methodology, we demonstrate that the optimal receive beamforming aligns with the traditional MMSE approach, whereas the optimal transmit beamforming design can be cast into a quadratic optimization problem defined on a complex Stiefel manifold. Based on the majorization–minimization (MM) method, an iterative algorithm is then developed to compute the optimal transmit beamforming design by solving a series of orthogonal Procrustes problems (OPPs) that admit closed-form optimal solutions. Numerical findings serve to validate the efficacy of our scheme. It is demonstrated that our approach can achieve at least 73% higher spectral efficiency than the existing methods in a high signal-to-noise ratio (SNR) regime.

Secure User Pairing and Power Allocation for Downlink Non-Orthogonal Multiple Access against External Eavesdropping.

We propose a secure user pairing (UP) and power allocation (PA) strategy for a downlink Non-Orthogonal Multiple Access (NOMA) system when there exists an external eavesdropper. The secure transmission of data through the downlink is constructed to optimize both UP and PA. This optimization aims to maximize the achievable sum secrecy rate (ASSR) while adhering to a limit on the rate for each user. However, this poses a challenge as it involves a mixed integer nonlinear programming (MINLP) problem, which cannot be efficiently solved through direct search methods due to its complexity. To handle this gracefully, we first divide the original problem into two smaller issues, i.e., an optimal PA problem for two paired users and an optimal UP problem. Next, we obtain the closed-form optimal solution for PA between two users and UP in a simplified NOMA system involving four users. Finally, the result is extended to a general 2K-user NOMA system. The proposed UP and PA method satisfies the minimum rate constraints with an optimal ASSR as shown theoretically and as validated by numerical simulations. According to the results, the proposed method outperforms random UP and that in a standard OMA system in terms of the ASSR and the average ASSR. It is also interesting to find that increasing the number of user pairs will bring more performance gain in terms of the average ASSR.

Coverage Increase at THz Frequencies: A Cooperative Rate-Splitting Approach

Numerous studies claim that terahertz (THz) communication will be an essential piece of sixth-generation wireless communication systems. Its promising potential also comes with major challenges, in particular the reduced coverage due to harsh propagation loss, hardware constraints, and blockage vulnerability. To increase the coverage of THz communication, we revisit cooperative communication. We propose a new type of cooperative rate-splitting (CRS) called extraction-based CRS (eCRS). Furthermore, we explore two extreme cases of eCRS, namely, identical eCRS and distinct eCRS. To enable the proposed eCRS framework, we design a novel THz cooperative channel model by considering unique characteristics of THz communication. Through mathematical derivations and convex optimization techniques considering the THz cooperative channel model, we derive local optimal solutions for the two cases of eCRS and a global optimal closed form solution for a specific scenario. Finally, we propose a novel channel estimation technique that not only specifies the channel value, but also the time delay of the channel from each cooperating user equipment to fully utilize the THz cooperative channel. In simulation results, we verify the validity of the two cases of our proposed framework and channel estimation technique.

Consumption and portfolio optimization with generalized stochastic differential utility in incomplete markets

This paper examines continuous time intertemporal consumption and portfolio choice problems of an investor in a generalized stochastic differential utility preference of Epstein–Zin type with subjective beliefs and with ambiguity, respectively. In these two situations, the aggregators of the utility both contain the “variability” process. Based on the generalized verification theorem which formulates Hamilton–Jacobi-Bellman equations under a weak non-Lipschitz condition, we show the explicit closed-form optimal consumption and portfolio solutions with subjective beliefs and the numerical solutions with ambiguity for Heston model in incomplete market, respectively.

Energy Minimization for UAV-Enabled Wireless Power Transfer and Relay Networks

In this paper, we consider an unmanned aerial vehicle (UAV)-enabled wireless power transfer (WPT) and relay communication network consisting of a base station (BS), a UAV, and multiple ground users. The UAV acts as both a wireless power transmission source and an uplink communication relay. Specifically, an entire transmission period of the considered system is divided into two stages. In the first stage, the UAV transfers the power to the ground users along a well-optimized flight trajectory and meanwhile the users transmit data to the UAV using the harvested energy. Subsequently, in the second stage, the UAV flies to the vicinity of the BS and forwards the data to the BS. For the purpose of minimizing the energy consumed by the UAV, we jointly optimize the time durations of the two stages, the UAV’s transmit powers for WPT and data forwarding, as well as its flight trajectory, subject to the constraints of the quality of service (QoS), the information forwarding, the energy causality, and the mobility of the UAV. The involved optimization problem is non-convex and highly intractable. To this end, we propose an efficient alternating algorithm to iteratively solve two subproblems with respect to the time durations of the two stages and the UAV’s transmit powers and trajectory, respectively. The first subproblem has a closed-form optimal solution and the second subproblem is handled by addressing a surrogate convex problem based on the technique of successive convex approximation. Finally, simulation results confirm the superiority of our proposed algorithm.

Hybrid analytical H-norm optimization approach for dynamic vibration absorbers

Conventionally, single-targeted H∞ or H2 optimization methods are adopted to determine the optimal parameters of different Dynamic Vibration Absorbers (DVAs) individually. A generic analytical framework for various DVAs considering a dual-target of H∞ and H2 norms is lacking. Addressing these issues, a hybrid analytical H∞-H2 optimization approach is proposed, leading to closed-form optimal solutions characterized by equal-height fixed points and conditional minimum H2-norms, combining the advantages of H∞ and H2 solutions. The proposed method is completely based on arbitrary filter coefficients of the targeted transfer function. Therefore, it can be generically applied for various DVAs. With the proposed method, the analytical optimal parameters of Negative-Stiffness Inerter-based Tuned Mass Systems (NS-ITMSs) are obtained based on an extended generic representation model considering multiple parameters. In extension, the method is illustratively applied on three complicated DVAs, including a Lever-type Inerter-based Vibration Absorber with Negative Stiffness (NS-LIVA) considering a leverage effect, a Piezoelectric Shunt Damping System (PSDS) considering an electro-mechanical coupling effect, and a Negative-Stiffness Tuned Viscous Mass Damper (NS-TVMD) for vibration isolator considering a transmissibility target. The closed-form solutions for these advanced DVAs are obtained for engineering reference. Furthermore, this method can be extended for more complicated energy dissipation or harvesting techniques.

RIS-Aided MIMO Systems With Hardware Impairments: Robust Beamforming Design and Analysis

Reconfigurable intelligent surface (RIS) has been anticipated to be a novel cost-effective technology to improve the performance of future wireless systems. In this paper, we investigate a practical RIS-aided multiple-input-multiple-output (MIMO) system in the presence of transceiver hardware impairments, RIS phase noise and imperfect channel state information (CSI). Joint design of the MIMO transceiver and RIS reflection matrix to minimize the total average mean-square-error (MSE) of all data streams is particularly considered. This joint design problem is non-convex and challenging to solve due to the newly considered practical imperfections. To tackle the issue, we first analyze the total average MSE by incorporating the impacts of the above system imperfections. Then, in order to handle the tightly coupled optimization variables and non-convex NP-hard constraints, an efficient iterative algorithm based on alternating optimization (AO) framework is proposed with guaranteed convergence, where each subproblem admits a closed-form optimal solution by leveraging the majorization-minimization (MM) technique. Moreover, via exploiting the special structure of the unit-modulus constraints, we propose a modified Riemannian gradient ascent (RGA) algorithm for the discrete RIS phase shift optimization. Furthermore, the optimality of the proposed algorithm is validated under line-of-sight (LoS) channel conditions, and the irreducible MSE floor effect induced by imperfections of both hardware and CSI is also revealed in the high signal-to-noise ratio (SNR) regime. Numerical results show the superior MSE performance of our proposed algorithm over the adopted benchmark schemes, and demonstrate that increasing the number of RIS elements is not always beneficial under the above system imperfections.

Stop clicking around and book direct: Impact of best rate guarantee on hotel pricing

Hotel chains sell their rooms through both their own distribution channels and third-party online travel agencies (OTAs). However, with the growing popularity of OTAs over the last decade, major hotel chains have started to fight against OTAs for control and market share by offering Best Rate Guarantee (BRG) in an attempt to incentivize customers to bypass OTAs and book directly with the hotels. BRG ensures that guests who book direct will get the best price. We study a stylized model where a hotelier that works closely with an OTA offers BRG to its customers to maximize its revenue over a selling horizon. Motivated by common practices in the hotel industry, we model the hotelier-OTA decision making process as a sequential game. We focus on the case where there is a sufficient inventory, derive closed-form optimal solutions for the problem, prove some structural properties, and identify necessary and sufficient conditions under which offering BRG always benefits the hotel (resp. the OTA). While in general BRG makes the hotelier better off and the OTA worse off, we show that when the commission rate is high, BRG can make both channels better off. On the other hand, when the commission rate and the fraction of swing customers who claim BRG are both low, BRG can make both channels worse off. We also consider the case with a limited inventory in the appendix and derive various managerial insights regarding the impact of BRG through computational tests. Our findings will shed some lights for hotel managers when deciding whether to practice BRG and what relevant actions to take when BRG is practiced.

A periodic review inventory model facing different disruption profiles

We study a periodic review inventory system with alternations of periods with supply availability and unavailability. The system is controlled by a periodic review base stock policy (S,T). While periodic review inventory models have been studied assuming supply disruption, they mainly invoke the limiting disruption probabilities overlooking that these probabilities may correspond to different disruption profiles (ranging from rare-long disruptions to frequent-short ones). In this context, we use a continuous-time Markov chain (CTMC), in order to formulate a periodic review inventory model that can distinguish the different disruption profiles. Through theoretical results, we determine the policy variables (order-up-to level and review interval) that minimize the long-run average cost. We show that when the disruption profile is considered, the optimal order-up-to level, although Newsvendor-styled form, is not always linear in the review interval. We present analytical properties demonstrating the impact of the different disruption profiles (under the same limiting disruption probability) and the impact of the limiting probability on the optimal policy. Furthermore, we propose an approximation to obtain a closed-form (near-)optimal solution for the review interval. Finally, numerical comparisons highlight the impact on system's cost performance of deploying approaches that either use the above mentioned closed-form solution for the review interval or the limiting disruption probabilities instead of the time dependent ones in the cost formulation.

The Economic Order Quantity Model under Compound Interest with Planned Backorders

In the classical EOQ model, the annual inventory holding cost per unit is defined as a fixed percentage of the unit price of the item. A portion of the inventory holding cost is the opportunity cost of capital tied up in the inventory, which is based on the best interest rate or the rate of return for the best alternative investment, and assumed as simple interest. In finance and banking, compound interest is the standard and simple interest is very rare; so it is not realistic to use an opportunity cost based on simple interest. To overcome this problem, a number of net present value (NPV)-based approaches have been proposed in the literature but they all recommend the standard EOQ formula as an approximate optimal solution. In this research, we propose an extension of the basic model that uses compound interest for the opportunity cost and allows planned backorders. A closed-form optimal solution is not possible for this model due to the exponential terms in the total cost function. We develop a reasonable approximate model and derive the optimal solution that is intuitive and different from the standard EOQ solution. We show that our solution is very close to the solution of the exact model.

Analysis and Optimization of a Double-IRS Cooperatively Assisted System With a Quasi-Static Phase Shift Design

The analysis and optimization of single intelligent reflecting surface (IRS)-assisted systems have been extensively studied, whereas little is known regarding multiple-IRS-assisted systems. This paper investigates the analysis and optimization of a double-IRS cooperatively assisted downlink system (D-IRS-C), where a multi-antenna base station (BS) serves a single-antenna user with the help of two multi-element IRSs, connected by an inter-IRS channel. The channel between any two nodes is modeled with Rician fading. The BS adopts the instantaneous CSI-adaptive maximum-ratio transmission (MRT) beamformer, and the two IRSs adopt a cooperative quasi-static phase shift design. The goal is to maximize the average achievable rate, which can be reflected by the average channel power of the equivalent channel between the BS and user at low channel estimation and phase adjustment costs and computational complexity. First, we obtain tractable expressions of the average channel power of the equivalent channel in the general (Rician factor), pure line of sight (LoS), and pure non-line of sight (NLoS) regimes, respectively. Then, we jointly optimize the phase shifts of the two IRSs to maximize the average channel power of the equivalent channel in these regimes. The optimization problems are challenging non-convex problems. We obtain globally optimal closed-form solutions for some cases and propose computationally efficient iterative algorithms to obtain stationary points for the other cases. Next, we compare the computational complexity for optimizing the phase shifts and the optimal average channel power of D-IRS-C with those of a counterpart double-IRS non-cooperatively assisted system (D-IRS-NC) and a counterpart single-IRS-assisted system (S-IRS) at a large number of reflecting elements in the three regimes. Finally, we numerically demonstrate notable gains of the proposed solutions over the existing solutions at different system parameters. To our knowledge, this is the first work that optimizes the quasi-static phase shift design of D-IRS-C and characterizes its advantages over the optimal quasi-static phase shift design of the counterpart D-IRS-NC and S-IRS.

On the Egalitarian–Utilitarian spectrum in stochastic capacitated resource allocation problems

In this paper, we study a generalized resource allocation problem where a limited resource is to be allocated to a set of agencies with stochastic receiving capacities while taking into account the decision-maker’s preference towards equity in allocation. Taking the empiric distributional nature of the capacities into account, we formulate the problem in a chance-constrained based Mixed-Integer Programming framework with user-specified reliability and tolerance levels as well as equity preference level. We show that the problem can be conveniently reduced to a linear equivalent with adjusted capacity constraints. Deriving the tightest lower and upper bounds corresponding to the utilitarian and egalitarian perspectives, respectively, we give the closed-form optimal solutions for such cases. Using real data, we test the behavior of our model with a view towards the inherent equity-efficiency as well as reliability-efficiency trade-offs. We further characterize the cost and value of information in a detailed analysis.

The optimum inerter-based additional viscoelastic mass dampers for dynamic response mitigation of structures

The additional inerter-based viscoelastic mass dampers (AIVMD) and additional viscoelastic mass damper inerters (AVMDI) are introduced in this article. H 2 and H ∞ optimization schemes are utilized to derive the optimal closed-form solutions for these novel dampers analytically. A parametric study performs to investigate the sensitivity of the optimal design parameters with other system parameters such as damper mass ratio, inerter mass ratio, and stiffness ratio. Thus, a higher damper mass ratio, a higher inerter mass ratio, and a higher stiffness ratio are recommended to design optimum novel dampers for achieving robust vibration reduction capacities. Therefore, H 2 optimized AIVMD and AVMDI have 53.23 % and 57.73 % dynamic response reduction capacities while H ∞ optimized AIVMD and AVMDI can provide 72.97 % and 75.57 % dynamic response reduction capacities subjected to harmonic excitation, respectively. In addition, random-white noise excitations are also applied instead of harmonic excitation to cross-check the accuracy of the optimal design parameters. The overall result shows that 74.24 % and 82.17 % dynamic response reduction capacities for H 2 optimized AIVMD and AVMDI, furthermore, 92.14 % and 94.36% dynamic response reduction capacities for H ∞ optimized AIVMD and AVMDI. These optimal closed-form solutions are mathematically accurate and relevant for practical applications.

The optimum enhanced viscoelastic tuned mass dampers: Exact closed-form expressions

This paper introduces the inertial amplifier viscoelastic tuned mass dampers (IAVTMD). The viscoelastic materials are implanted inside the core material of the inertial amplifier tuned mass dampers. The standard linear solid models are applied mathematically to formulate the viscoelastic material. H2 and H ∞ optimization mechanisms apply to derive the exact mathematical closed-form formulations for optimal design parameters for novel inertial amplifier viscoelastic tuned mass dampers. The optimum IAVTMD installs at the top of the structures to mitigate the dynamic responses, determining the dynamic responses analytically through transfer function formation. At first, IAVTMD’s dynamic response reduction capacity compares with the conventional tuned mass damper’s (CTMD) dynamic response reduction capacity. As a result, H2 optimized IAVTMD’s dynamic response reduction capacity is significantly 20.87% and 26.47% superior to two well-established H2 optimized conventional tuned mass damper’s dynamic response reduction capacity. In addition, H ∞ optimized IAVTMD has 15.48% more dynamic response reduction capacity than H ∞ conventional tuned mass damper. H2 and H ∞ optimized tuned mass damper inerters with optimal closed-form solutions are introduced in this paper. A higher damper mass ratio and a lower inerter mass ratio are recommended to produce H2 and H ∞ optimized tuned mass damper inerter (TMDI) with a lower frequency and damping ratio in an affordable range. Accordingly, H2 and H ∞ optimized IAVTMD’s dynamic response reduction capacities are significantly 6.94% and 23.29% superior to H2 and H ∞ optimized TMDI’s dynamic response reduction capacity. The closed-form expressions for optimal design parameters of inertial amplifier viscoelastic tuned mass dampers and tuned mass damper inerters are mathematically correct and effective for practical applications.

Contention or Symbiosis: The Model-Based and Model-Less Transmission Mode Selections for Internet of Things in Unlicensed Band

In this work, we investigate a spectrum-sharing network, where an Internet-of-Things (IoT) system and a WiFi system coexist in the unlicensed band. The IoT devices can actively transmit signal by contending for the channel with the WiFi users. Alternatively, they can also convey information by passively backscattering the ambient WiFi signal, and thus form symbiosis with the WiFi during the transmission. Since the active and the passive transmission mode have their own merits under specific circumstances, how to select one of them adaptively and efficiently with guaranteeing the normal service of the WiFi system is of high importance. To solve this problem, we firstly investigate the effective throughput of the IoT system from the cross-layer perspective. Then, by jointly considering the effective throughput and the power consumption of the IoT system, we formulate the transmission mode selection problem. Two solutions are given depending on whether the signaling between the two systems is available. By entailing the inter-system parameter sharing, the model-based transmission mode selection scheme is proposed, which gives the closed-form optimal solution. When the inter-system parameter sharing is absent, the model-less transmission mode selection scheme is proposed by using deep reinforcement learning (DRL) technique. Extensive simulation results validate the theoretical analysis, and demonstrate the effectiveness of the proposed transmission mode selection schemes.

Fully Linear Graph Convolutional Networks for Semi-Supervised and Unsupervised Classification

This article presents FLGC, a simple yet effective fully linear graph convolutional network for semi-supervised and unsupervised learning. Instead of using gradient descent, we train FLGC based on computing a global optimal closed-form solution with a decoupled procedure, resulting in a generalized linear framework and making it easier to implement, train, and apply. We show that (1) FLGC is powerful to deal with both graph-structured data and regular data, (2) training graph convolutional models with closed-form solutions improve computational efficiency without degrading performance, and (3) FLGC acts as a natural generalization of classic linear models in the non-Euclidean domain (e.g., ridge regression and subspace clustering). Furthermore, we implement a semi-supervised FLGC and an unsupervised FLGC by introducing an initial residual strategy, enabling FLGC to aggregate long-range neighborhoods and alleviate over-smoothing. We compare our semi-supervised and unsupervised FLGCs against many state-of-the-art methods on a variety of classification and clustering benchmarks, demonstrating that the proposed FLGC models consistently outperform previous methods in terms of accuracy, robustness, and learning efficiency. The core code of our FLGC is released at https://github.com/AngryCai/FLGC .