Convex Approach Research Articles

Nonconvex regularization for sparse neural networks

Convex ℓ 1 regularization using an infinite dictionary of neurons has been suggested for constructing neural networks with desired approximation guarantees, but can be affected by an arbitrary amount of over-parametrization. This can lead to a loss of sparsity and result in networks with too many active neurons for the given data, in particular if the number of data samples is large. As a remedy, in this paper, a nonconvex regularization method is investigated in the context of shallow ReLU networks: We prove that in contrast to the convex approach, any resulting (locally optimal) network is finite even in the presence of infinite data (i.e., if the data distribution is known and the limiting case of infinite samples is considered). Moreover, we show that approximation guarantees and existing bounds on the network size for finite data are maintained.

Computationally efficient infinite-horizon indefinite model predictive control with disturbance preview information

This paper proposes a model predictive control (MPC) scheme for maximising the benefit of a useful disturbance by exploiting preview information of the disturbance, in the context of goal-oriented operation. For a constrained system, subject to a persistent, bounded, and predictable disturbance, rather than attenuating the influence of disturbance, the proposed MPC aims to utilise the disturbance to optimise high-level economic criteria, e.g., profitability and productivity, which are normally represented by an indefinite cost function. For linear time-invariant systems, after examining the influence of the future disturbance profile, a computationally efficient finite-horizon convex approach is proposed to approximate the solution of the original possibly non-convex infinite-horizon optimisation problem. Then, a receding-horizon implementation is developed, taking into account the recursively updated disturbance prediction, and the recursive feasibility and input-to-state stability of the implementation are established. Numerical examples are provided to verify the efficacy of the proposed method.

Convex Approach to Covariance Control with Application to Stochastic Low-Thrust Trajectory Optimization

This paper outlines a novel approach to the design of optimal space trajectories under significant uncertainty. Finite-horizon covariance control, i.e., the steering of a system from an initial probability distribution to a desired one at a prescribed time, is employed to plan an optimal nominal path along with a robust feedback controller that compensates for exogenous in-flight disturbances. A mindful convexification strategy is devised to recast the nonlinear covariance control problem as a deterministic convex optimization problem. The convexification is based on a convenient change of variables that allows for relaxing the covariance matrix discrete-time propagation into a set of semidefinite cone constraints. While featuring a larger feasible space, the relaxed problem shares the same optimal solution as the original one. The lossless property of the relaxation is analytically proven by leveraging optimal control theory and discussed on the basis of numerical experiments. Monte Carlo campaigns are carried out to validate the in-flight performance of the attained control policies.

A Convex Approach for Trajectory Optimization of Super-Synchronous-Transfer-Orbit Large Launch Systems with Time-Position Constraints

This paper presents a trajectory optimization algorithm for super-synchronous-transfer-orbit (SSTO) large launch systems by convex optimization. The payload of SSTO launch systems is typically a geostationary equatorial orbit (GEO) satellite, and the time and position of orbital injection are constrained, which is quite different from the case of general satellites. In this paper, the optimal control problem of SSTO large launch systems is formulated considering the terminal constraints including orbital elements and the time-position equation. To improve the computational performance of the algorithm, the terminal orbital element constraints are expressed in the perifocal coordinate system with second-order equations. And then, several convexification techniques and their modified strategies are applied to transform the original trajectory optimization problem into a series of convex optimization problems, which can be solved iteratively with high accuracy and computational efficiency. Considering the time-position constraint of the payload, the flight time updater design method is proposed to correct the error of time during the flight, which lays solid foundation for the subsequent flight phase, guaranteeing that the GEO satellite settles into the required position. Finally, simulation results indicate the high efficiency and accuracy and strong robustness of the proposed algorithm in different special situations including engine failure and time delay. The algorithm proposed in this paper has great development potential and application prospect in onboard trajectory optimization of SSTO launch missions and similar situations.

State Estimation for Complex-Valued Inertial Neural Networks with Multiple Time Delays

In this paper, the problem of state estimation for complex-valued inertial neural networks with leakage, additive and distributed delays is considered. By means of the Lyapunov–Krasovskii functional method, the Jensen inequality, and the reciprocally convex approach, a delay-dependent criterion based on linear matrix inequalities (LMIs) is derived. At the same time, the network state is estimated by observing the output measurements to ensure the global asymptotic stability of the error system. Finally, two examples are given to verify the effectiveness of the proposed method.

Optimal Signal-Adaptive Trading with Temporary and Transient Price Impact

We study optimal liquidation in the presence of linear temporary and transient price impact along with taking into account a general price predicting finite-variation signal. We formulate this problem as minimization of a cost-risk functional over a class of absolutely continuous and signal-adaptive strategies. The stochastic control problem is solved by following a probabilistic and convex analytic approach. We show that the optimal trading strategy is given by a system of four coupled forward-backward SDEs, which can be solved explicitly. Our results reveal how the induced transient price distortion provides together with the predictive signal an additional predictor about future price changes. As a consequence, the optimal signal-adaptive trading rate trades off exploiting the predictive signal against incurring the transient displacement of the execution price from its unaffected level. This answers an open question from [C. A. Lehalle and E. Neuman, Finance Stoch., 23 (2019), pp. 275--311] as we show how to derive the unique optimal signal-adaptive liquidation strategy when price impact is not only temporary but also transient.

Bi-level convex optimization of eco-driving for connected Fuel Cell Hybrid Electric Vehicles through signalized intersections

Eco-driving for connected Fuel Cell Hybrid Electric Vehicles (FCHEVs) is a coupled problem of speed planning and energy management. To reduce the computational burden, bi-level optimization decouples and hierarchically solves the upper-level and lower-level subproblems. This paper proposes a bi-level convex approach for eco-driving of a connected FCHEV proceeding through multiple signalized intersections. On the upper level, the non-linear traffic light constraints are transformed into time-varying linear state constraints and the cost function becomes quadratic after using the average speed. On the lower level, model convexification is carried out for the fuel cell system and battery. Then the upper-level speed planning and lower-level energy management are sequentially solved by the MOSEK solver and the Alternating Direction Method of Multipliers (ADMM) algorithm. The results show that the proposed bi-level convex approach greatly reduces the computational cost while maintaining high energy efficiency, with only 6.59% computational time and almost the same fuel economy compared to the bi-level Dynamic Programming (DP) method.

Event-Triggered Mechanism for Multiple Frequency Services of Electric Vehicles in Smart Grids

This paper considers an event-triggered mechanism (ETM) for multiple frequency services of electric vehicles (EVs) in smart grids (SGs). A new model of a SG with an ETM and multiple agents of EVs is introduced. ETM is deployed to reduce the load on network bandwidth. EVs are incorporated in both secondary and primary frequency services to suppress frequency fluctuations in the system. The behaviours of EV owners are also considered by using batteries SOC controls. Then, a mathematical model of the system is derived which contains parametric uncertainties due to the incorporation of SOC control, an ETM variable arising from ETM control, and a time-varying delay in both the state and control input vectors. We derive new Lyapunov stability and stabilizability conditions to ensure robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance of the closed-loop system. The conditions are derived based on various mathematical tools including relaxed Jensen inequality, free weighting matrix technique, reciprocally convex approach, and matrix inequality for uncertainties. An effective procedure is proposed based on an approach using the state transformation and tractable linear matrix inequalities to synthesis robust static output feedback controller gains. Extensive simulations of a SG with three agents of EVs are undertaken to validate of our proposed control scheme.

3-D Array Image Data Completion by Tensor Decomposition and Nonconvex Regularization Approach

Various image datasets appear naturally in the form of multi-dimensional arrays (hypermatrices), called tensors. Image with incomplete entries, which often can be formulated as the low-rank tensor completion problem, is practically important in order to process large size 3D array images both efficiently and effectively. To make the low-rank approximation being tractable, most current approaches make use of the lower convex envelope of tensor multi-rank function for the approximation. Due to the gap between the rank function and its lower convex envelope, the adoption of tensor nuclear norm may lead to the approximation of the corresponding tensor tubal-rank being insufficient. In this paper, we introduce a new nonconvex regularization approach, which can better capture the low-rank characteristics than the convex approach. By transforming the original problem to the Fourier domain, we formulate an equivalent optimization problem with more transparent tensor rank characteristics, whose explicit solution can be obtained under our framework. A minimization algorithm, associated with the augmented Lagrangian multipliers and the nonconvex regularizer, is established and is shown to be feasible. The constructed sequence converges to the desirable Karush-Kuhn-Tucker point, which is mathematically validated in detail. Extensive experimental results demonstrate that our proposed approach outperforms the existing state-of-the-art convex approaches consistently.

A convex approach to optimum design of experiments with correlated observations

Optimal design of experiments for correlated processes is an increasingly relevant and active research topic. Present methods have restricted possibilities to judge their quality. To fill this gap, we complement the virtual noise approach by a convex formulation leading to an equivalence theorem comparable to the uncorrelated case and to an algorithm giving an upper performance bound against which alternative design methods can be judged. Moreover, a method for generating exact designs follows naturally. We exclusively consider estimation problems on a finite design space with a fixed number of elements. A comparison on some classical examples from the literature as well as a real application is provided.

More about operator order preserving

It is well known that increasing functions do not preserve operator order in general; nor do decreasing functions reverse operator order. However, operator monotone increasing or operator monotone decreasing functions do. In this article, we employ a convex approach to discuss operator order preserving or conversing functions. As an easy consequence of more general results, we find nonnegative constants γ and ψ such that A≤B implies f(B)≤f(A)+γ1ℋ andf(A)≤f(B)+ψ1ℋ, for the self-adjoint operators A, B on a Hilbert space ℋ with identity operator 1ℋ and for the convex function f whose domain contains the spectra of both A and B. The connection of these results to the existing literature will be discussed and the significance will be emphasized by some examples.

State Estimation for Genetic Regulatory Networks with Two Delay Components by Using Second-Order Reciprocally Convex Approach

State Estimation for Genetic Regulatory Networks with Two Delay Components by Using Second-Order Reciprocally Convex Approach

Formula omitted] performance analysis for delayed Markovian jump neural networks via the Lyapunov-Krasovskii functional with delay-product-type terms

This paper studies the problem of H∞ performance analysis for delayed Markovian jump neural networks (MJNNs). Firstly, the third-order Bessel–Legendre (B-L) integral inequality based on reciprocally convex approach is derived, which contains some existing ones as its special cases. Secondly, in order to facilitate the use of more information about time delays, the Lyapunov–Krasovskii functional (LKF) with delay-product-type terms is proposed. As a result, by utilizing the LKF with delay-product-type terms and the third-order B-L integral inequality based on reciprocally convex approach, less conservative sufficient conditions that guarantee the H∞ performance for delayed MJNNs are obtained. Finally, simulation results are given to illustrate the effectiveness of the proposed method.

Equivalent-input-disturbance estimator-based event-triggered control design for master–slave neural networks

This paper investigates the robust synchronization problem for a class of master–slave neural networks (MSNNs) subject to network-induced delays, unknown time-varying uncertainty, and exogenous disturbances. An equivalent-input-disturbance (EID) estimation technique is applied to compensate for the effects of unknown uncertainty and disturbances in the system output. In addition, to reduce the burden of the communication channel in the addressed MSNNs and improve the utilization of bandwidth an event-triggered control protocol is developed to obtain the synchronization of MSNNs. In particular, event-triggering conditions are verified periodically at every sampling instant in both sensors and actuators to avoid the Zeno behavior in the networks. By designing an appropriate low-pass filter in the EID estimator block, the accuracy of disturbance estimation performance is improved. Moreover, by concatenating the synchronization error, observer, and filter states as a single state vector, an augmented system is formulated. Then the tangible delay-dependent stability condition for that augmented system is established by employing the Lyapunov stability theory and reciprocally convex approach. Based on the feasible solutions of the derived stability conditions, the event-triggering parameters, controller, and observer gains are co-designed. Finally, two toy examples are given to illustrate the established theoretical findings.

Output feedback fuzzy control of nonlinear dynamic systems: Event‐triggered case

AbstractThe event‐triggered output feedback controller design problem is investigated in this article for a class of discrete‐time Takagi–Sugeno fuzzy systems. We focus on constructing a closed‐loop system by considering the event generator communication mechanism, interval time‐varying delays, and packet dropouts. Under this newly established system, a less conservative stability condition is obtained using the Wirtinger‐based inequality and reciprocally convex approach, which guarantees that the corresponding augmented system is asymptotically stable in the mean‐squared sense with a prespecified – performance. The solution of the designed output feedback controller parameters can then be obtained by variable substitution method and by conversion of the controller parameters. Finally, two examples are presented to demonstrate the effectiveness of the proposed controller design technique.

Efficient Tuning-Free l1-Regression of Nonnegative Compressible Signals

In compressed sensing the goal is to recover a signal from as few as possible noisy, linear measurements with the general assumption that the signal has only a few non-zero entries. The recovery can be performed by multiple different decoders, however most of them rely on some tuning. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising. If the measurement matrix has the robust null space property with respect to the ℓ2-norm, basis pursuit denoising obeys stable and robust recovery guarantees. In the case of unknown noise levels, nonnegative least squares recovers non-negative signals if the measurement matrix fulfills an additional property (sometimes called the M+-criterion). However, if the measurement matrix is the biadjacency matrix of a random left regular bipartite graph it obeys with a high probability the null space property with respect to the ℓ1-norm with optimal parameters. Therefore, we discuss non-negative least absolute deviation (NNLAD), which is free of tuning parameters. For these measurement matrices, we prove a uniform, stable and robust recovery guarantee. Such guarantees are important, since binary expander matrices are sparse and thus allow for fast sketching and recovery. We will further present a method to solve the NNLAD numerically and show that this is comparable to state of the art methods. Lastly, we explain how the NNLAD can be used for viral detection in the recent COVID-19 crisis.

Quantized output‐feedback guaranteed cost control for discrete‐time large‐scale interconnected systems with actuator faults

AbstractThis paper investigates the problem of designing a decentralized static output feedback control for a class of large‐scale interconnected discrete‐time systems with quantized signals and time‐varying actuator faults wherein the control input is quantized by means of logarithmic quantizer to avoid chattering. Based on Lyapunov stability theory together with the improved reciprocally convex approach, sufficient conditions are derived in terms of linear matrix inequalities. Specifically, with the aid of cone complementarity linearization technique, the control law is derived which ensures the asymptotic stabilization of considered system with prescribedH∞performance index and also guarantees optimum cost bound in spite of quantization and fault occurs. Finally, numerical examples are provided to demonstrate the significance of the designed control method.

Trans-base and trans-vault low-velocity penetrating brain injury: A retrospective comparative study of characteristics, treatment, and outcomes

PurposeLow-velocity penetrating brain injury (LVPBI) caused by foreign bodies can pose life-threatening emergencies. Their complexity and lack of validated classification data have prevented standardization of clinical management. We aimed to compare the trans-base and trans-vault phenotypes of LVPBI to help provide guidance for clinical decision-making of such injury type. MethodsA retrospective study on LVPBI patients managed at our institution from November 2013 to March 2020 was conducted. We included LVPBI patients admitted for the first time for surgery, and excluded those with multiple injuries, gunshot wounds, pregnancy, severe blunt head trauma, etc. Patients were categorized into trans-base and trans-vault LVPBI groups based on the penetration pathway. Discharged patients were followed up by outpatient visit or telephone. The data were entered into the Electronic Medical Record system by clinicians, and subsequently derived by researchers. The demography and injury characteristics, treatment protocols, complications, and outcomes were analyzed and compared between the two groups. A t-test was used for analysis of normally distributed data, and a Mann-Whitney U test for non-parametric data. A generalized linear model was further established to determine whether the factors length of stay and performance scale score were influenced by each factor. ResultsA total of 27 LVPBI patients were included in this analysis, comprised of 13 (48.1%) trans-base cases and 14 (51.9%) trans-vault cases. Statistical analyses suggested that trans-base LVPBI was correlated with deeper wounds; while the trans-vault phenotype was correlated with injury by metal foreign bodies. There was no difference in Glasgow Coma Scale score and the risk of intracranial hemorrhage between the two groups. Surgical approaches in the trans-base LVPBI group included subfrontal (n = 5, 38.5%), subtemporal (n = 5, 38.5%), lateral fissure (n = 2, 15.4%), and distal lateral (n = 1, 7.7%). All patients in the trans-vault group underwent a brain convex approach using the foreign body as reference (n = 14, 100%). Moreover, the two groups differed in application prerequisites for intracranial pressure monitoring and vessel-related treatment. Trans-base LVPBI was associated with higher rates of cranial nerve and major vessel injuries; in contrast, trans-vault LVPBI was associated with lower functional outcome scores. ConclusionOur findings suggest that trans-base and trans-vault LVPBIs differ in terms of characteristics, treatment, and outcomes. Further understanding of these differences may help guide clinical decisions and contribute to a better management of LVPBIs.

Convex Approach to Three-Dimensional Launch Vehicle Ascent Trajectory Optimization

This paper deals with the optimization of the ascent trajectory of a multistage launch vehicle, from liftoff to the payload injection into the target orbit, considering inverse-square gravity acceleration and aerodynamic forces. A combination of lossless and successive convexification techniques is adopted to generate a sequence of convex problems that rapidly converges to the original problem solution. An automatic initialization strategy is proposed to make the solution process completely autonomous. In particular, a novel three-step continuation procedure is developed and proved to be more efficient than simpler strategies. This approach relies on the solution of intermediate problems, which either neglect atmospheric drag or fix the time-lengths of the launch vehicle ascent phases, that are solved in succession, gradually passing from easier instances of the optimization problem to the originally intended problem. State-of-the-art techniques to deal with such a complex problem are adopted to enhance the convergence rate, including safeguarding modifications, such as virtual controls and an adaptive trust region. To assess the validity of the proposed approach in a practical scenario, numerical results are presented for two representative practical applications, using as reference a Falcon 9 launch vehicle.

Fuzzy Intermittent Extended Dissipative Control for Delayed Distributed Parameter Systems With Stochastic Disturbance: A Spatial Point Sampling Approach

In this article, the extended dissipative performance of distributed parameter systems (DPSs) with stochastic disturbances and multiple time-varying delays is studied by using a new fuzzy aperiodic intermittent sampled-data control strategy. Different from the previous fuzzy sampled-data control results, the state sampling of the proposed sampled-data controller occurs only in space and is intermittent rather than continuous in the time domain. By introducing a novel multitime-delay-dependent switched Lyapunov functional to explore the dynamic characteristics of the controlled system, and by means of the famous Jensen’s inequality with reciprocally convex approach, Wirtinger’s inequality, the criterion of the system’s mean square stabilization is established based on the LMI technique, which quantitatively reveals the relationship between the control period, the control length, and the upper bound of the control sampling interval. Especially, the optimal control gain is given by designing an optimized algorithm in the article, which greatly reduces the cost. Finally, two numerical examples are presented to demonstrate the effectiveness and superiority of the proposed approach.