Gradient Projection Method Research Articles

A Non-monotone Adaptive Scaled Gradient Projection Method for Orthogonality Constrained Problems

A Non-monotone Adaptive Scaled Gradient Projection Method for Orthogonality Constrained Problems

Private outsourced model predictive control via secure multi-party computation

Protecting data privacy is essential in developing practical cloud-based control systems to benefit from remote and distributed computation while avoiding the risks of disclosing sensitive information to potentially untrusted parties. This paper proposes a privacy-preserving framework to outsource the computation of model predictive control law to several untrusted cloud-based servers. A secret sharing scheme is employed to maintain the privacy of the system’s data in all the stages of the control loop in a secure multi-party computation environment. In this regard, a privacy-preserving algorithm is derived to securely implement a projected gradient method to solve the underlying optimization problem, without revealing private data to external eavesdroppers and curious cloud-based servers. The proposed method is solely based on secret sharing, and all computations can be performed securely by cloud servers without the need to designate target nodes or engage the system’s actuator in computing the intermediate steps. Therefore, the proposed method is less computationally demanding than the existing results based on homomorphic encryption and is applicable to systems with limited computing resources. Information-theoretic privacy assessments based on mutual information measures are provided. The efficacy of the proposed method is investigated on the cruise control problem of a freight train system.

Alternating projection combined with fast gradient projection (FGP-AP) method for intensity-only measurement optical diffraction tomography in LED array microscopy.

Optical diffraction tomography (ODT) is a powerful label-free measurement tool that can quantitatively image the three-dimensional (3D) refractive index (RI) distribution of samples. However, the inherent "missing cone problem," limited illumination angles, and dependence on intensity-only measurements in a simplified imaging setup can all lead to insufficient information mapping in the Fourier domain, affecting 3D reconstruction results. In this paper, we propose the alternating projection combined with the fast gradient projection (FGP-AP) method to compensate for the above problem, which effectively reconstructs the 3D RI distribution of samples using intensity-only images captured from LED array microscopy. The FGP-AP method employs the alternating projection (AP) algorithm for gradient descent and the fast gradient projection (FGP) algorithm for regularization constraints. This approach is equivalent to incorporating prior knowledge of sample non-negativity and smoothness into the 3D reconstruction process. Simulations demonstrate that the FGP-AP method improves reconstruction quality compared to the original AP method, particularly in the presence of noise. Experimental results, obtained from mouse kidney cells and label-free blood cells, further affirm the superior 3D imaging efficacy of the FGP-AP method.

A 1-bit DACs precoding for MU-MIMO based on binary equilibrium constraint: An alternating direction method

High power consumption is one of the main problems in the practical application of massive multi-user multi-input-multi-output (MU-MIMO) systems, and using 1-bit digital-to-analog converters (DACs) instead of high-resolution DACs is an effective way to reduce the power consumption of massive MU-MIMO systems. How to design efficient precoding schemes to improve the performance of 1-bit MU-MIMO systems has become a hot research field. This paper mainly studies the precoding problem of 1-bit DACs based on minimum mean squared error (MMSE). We first transform the non-convex constraints of the traditional optimization problem using binary equilibrium constraints. Then, to solve the transformed objective problem, we establish a framework based on the alternating direction method (ADM), use the fast projection gradient (FPG) method, and propose an alternating maximum minimum (AMM) algorithm to solve the optimization variables alternately. We have verified the convergence of the proposed algorithm and analyzed the accuracy of the proposed ADM framework and the computational complexity of the proposed algorithm in detail. In the simulation section, we analyze the uncoded bit error rate (BER) performance of the proposed algorithm in phase-shift keying (PSK) modulation and quadrature amplitude modulation (QAM) modes. The simulation results show that compared with existing advanced algorithms, the proposed algorithm can achieve performance advantages of about 1 dB and 2 dB in QPSK modulation mode and 16QAM modulation mode, respectively.

Barzilai–Borwein-like rules in proximal gradient schemes for ℓ1-regularized problems

We propose a novel steplength selection rule in proximal gradient methods for minimizing the sum of a differentiable function plus an ℓ 1 -norm penalty term. The proposed rule modifies one of the classical Barzilai–Borwein steplength, extending analogous results obtained in the context of gradient projection methods for constrained optimization. We analyse the spectral properties of the Barzilai–Borwein-like steplength when the differentiable part is quadratic, showing that its reciprocal lies in the spectrum of the submatrix of the Hessian that depends on both the nonzero and the nonoptimal zero components of the current iterate, allowing for acceleration effects when the optimal zero components start to be identified. Furthermore, we insert the modified rule into a proximal gradient method with a nonmonotone line search, for which we prove global convergence towards a stationary point. Numerical experiments show the ability of the proposed rule to sweep the spectrum of the reduced Hessian on a series of quadratic ℓ 1 -regularized problems, as well as its effectiveness in recovering the ground truth in a least squares regularized problem arising in image restoration.

An inertial spectral conjugate gradient projection method for constrained nonlinear pseudo-monotone equations

An inertial spectral conjugate gradient projection method for constrained nonlinear pseudo-monotone equations

Normalized Newton method to solve generalized tensor eigenvalue problems

AbstractThe problem of generalized tensor eigenvalue is the focus of this paper. To solve the problem, we suggest using the normalized Newton generalized eigenproblem approach (NNGEM). Since the rate of convergence of the spectral gradient projection method (SGP), the generalized eigenproblem adaptive power (GEAP), and other approaches is only linear, they are significantly improved by our proposed method, which is demonstrated to be locally and cubically convergent. Additionally, the modified normalized Newton method (MNNM), which converges to symmetric tensors Z‐eigenpairs under the same ‐Newton stability requirement, is extended by the NNGEM technique. Using a Gröbner basis, a polynomial system solver (NSolve) generates all of the real eigenvalues for us. To illustrate the efficacy of our methodology, we present a few numerical findings.

A modified technique of spectral gradient projection method for solving nonlinear equations systems

This study suggests a new line search algorithm modified of Spectral gradient projection method (SGPM) called (D.A), combining the modified spectral gradient method and projection method for solving nonlinear equations systems. This approach can be more efficient because it requires less CPU time, less function evaluation, and fewer iterations. Furthermore, if this equation is Lipschitz continuous (LC) and monotone, we proved that this technique converges globally. The numerical results show how effective and beneficial this method is for solving nonlinear equations systems.

Lipschitz continuity of the metric projection operator and convergence of gradient methods

Various support conditions for a closed subset of a real Hilbert space $\mathcal H$ at a boundary point of this set are considered. These conditions ensure certain local Lipschitz continuity of the metric projection operator as a function of a point. The local Lipschitz continuity of the metric projection as a function of the set in the Hausdorff metric is also proved. This Lipschitz property is used to verify the linear convergence of some gradient methods (the gradient projection method and the conditional gradient method) without assuming that the function must be strongly convex (or even convex) and for not necessarily convex sets. The function is assumed to be differentiable with Lipschitz continuous gradient. Bibliography: 29 titles.

Structured Output Feedback Control for Linear Quadratic Regulator Using Policy Gradient Method

We consider the static output feedback control for Linear Quadratic Regulator problems with structured constraints under the assumption that system parameters are unknown. To solve the problem in the model free setting, we propose the policy gradient algorithm based on the gradient projection method and show its global convergence to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> -stationary points. In addition, we introduce a variance reduction technique and show both theoretically and numerically that it significantly reduces the variance in the gradient estimation. We also show in the numerical experiments that the model free approach efficiently solves the problem.

Comprehensive research and analysis on obstacle–singularity–joint limit avoidance of redundant robot

Redundant robots can complete other tasks while performing main task and have excellent motion performance. Although redundant robots have redundancy characteristics, they may still fall into singular configuration or cannot overcome joint limits due to the limitation of their own structure and motion law. Besides, currently, most of the research mainly focus on obstacle avoidance at the end of the robot. So comprehensive study is conducted on avoidance of obstacle, singularity, and joint limit of redundant robot. Firstly, the kinematics model of robot is obtained by D-H matrix method. Space vector distance method is proposed to calculate the position relationship between robot and obstacle, and potential function is constructed. Based on gradient projection method, using the damped generalized inverse and the weighted norm method, the mathematical models of obstacle avoidance, singularity avoidance, and joint limit avoidance are constructed. The simulation results show that the robot can avoid the obstacle in three-dimensional redundant space. At the same time, manipulability and minimum singular value of the robot are always positive. After adding the function of avoiding joint limit, during the whole control process, the motion range of joint angle is reduced and the angular velocity is relatively small, which ensure the motion stability of robot. Namely, the singular configuration and joint limit of robot are avoided. The research is of great significance for practical application of redundant robot.

Distributed Online Learning Algorithm for Noncooperative Games Over Unbalanced Digraphs.

This article investigates constrained online noncooperative games (NGs) of multiagent systems over unbalanced digraphs, where the cost functions of players are time-varying and are gradually revealed to corresponding players only after decisions are made. Moreover, in the problem, the players are subject to local convex set constraints and time-varying coupling nonlinear inequality constraints. To the best of our knowledge, no result about online games with unbalanced digraphs has been reported, let alone constrained online games. To seek the variational generalized Nash equilibrium (GNE) of the game online, a distributed learning algorithm is proposed based on gradient descent, projection, and primal-dual methods. Under the algorithm, sublinear dynamic regrets and constraint violations are established. Finally, online electricity market games illustrate the algorithm.

On the convergence of gradient projection methods for non-convex optimal control problems with affine system

On the convergence of gradient projection methods for non-convex optimal control problems with affine system

Минимизация гладкой функции на границе внешнего обобщенного сегмента сферы

We consider the problem of minimizing a smooth function on the boundary of the so-called external generalized segment of a sphere, which is constructed in a certain way from a sphere and a convex solid cone with a vertex lying outside the corresponding closed ball. A modification of the gradient projection method is proposed and its convergence to the stationary point of the problem is substantiated.

On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs

We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation (PDE) which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc(ℝ). This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. The convergence of the resulting algorithm is proven in the function space setting. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation functions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero. The paper concludes with numerical experiments which confirm the theoretical findings.

Novel projected gradient methods for solving pseudomontone variational inequality problems with applications to optimal control problems

Novel projected gradient methods for solving pseudomontone variational inequality problems with applications to optimal control problems

Control of the von Neumann Entropy for an Open Two-Qubit System Using Coherent and Incoherent Drives.

This article is devoted to developing an approach for manipulating the von Neumann entropy S(ρ(t)) of an open two-qubit system with coherent control and incoherent control inducing time-dependent decoherence rates. The following goals are considered: (a) minimizing or maximizing the final entropy S(ρ(T)); (b) steering S(ρ(T)) to a given target value; (c) steering S(ρ(T)) to a target value and satisfying the pointwise state constraint S(ρ(t))≤S¯ for a given S¯; (d) keeping S(ρ(t)) constant at a given time interval. Under the Markovian dynamics determined by a Gorini-Kossakowski-Sudarshan-Lindblad type master equation, which contains coherent and incoherent controls, one- and two-step gradient projection methods and genetic algorithm have been adapted, taking into account the specifics of the objective functionals. The corresponding numerical results are provided and discussed.

О задаче формирования портфеля облигаций по вероятностному критерию в модели Кокса-Ингерсолла-Росса

&lt;p&gt;The problem of bond portfolio selection is considered in Cox-Ingersoll-Ross framework. The probability function is chosen as an optimality criterion, which leads to a stochastic optimization problem, The problem is solved using a smooth approximation of the probability function and its derivatives via gradient projection method. An example is provided.&lt;/p&gt;

Traffic offloading for postdisaster rescue in UAV‐assisted networks: A coalition formation game approach

SummaryIn postdisaster rescue scenario, unmanned aerial vehicles (UAVs) are effective tools to help ground users in disaster areas to transmit rescue‐critical data to the relief center in time due to their flexible mobilities and fast deployment. However, how UAVs choose disaster areas to take part in traffic offloading with limited bandwidth resources is challenging. This article investigates UAVs' access selection and disaster areas' bandwidth allocation scheme by jointly optimizing network throughput, bandwidth and UAVs' energy cost. The UAVs form coalitions to participate in traffic offloading cooperatively, where each UAV selects a disaster area independently. Specifically, the bandwidth resources are dynamically allocated to all disaster areas when a new coalition partition forms. The UAVs adjust their access choices for disaster areas when the bandwidth allocation changes. When no UAVs migrate to other disaster areas, the network achieves a stable state. The UAVs fly to the final selected disaster area and provide traffic offloading services. To resolve traffic offloading competitions yet enable cooperations among UAVs, we address the UAVs' access selection issue by coalition formation game. A gradient projection method is then proposed to allocate bandwidth resources which maximizes the benefit of the network. We demonstrate that the UAVs' access selection and disaster areas' bandwidth allocation algorithms are convergent. The simulation results demonstrate that our proposed cooperation order is better than the pareto and selfish orders, thereby increasing the benefit of the network.

Improvement of density inversion efficiency with projected gradient method: Application to FTG data

With the development of instruments for measuring Earth's gravity gradient tensor, full tensor gravity (FTG) data can be obtained validly and accurately. The highly precise performance of FTG data ensures that they can play an important role in many domains. Generally, FTG data can be acquired from airborne platform, and large-scale data can be obtained by long term observations. For this type of data, research methods for large data processing are very important. Three-dimensional data inversion is an effective and significant method for the quantitative interpretation of FTG data, which has received much attention. Large-scale data inversion is inevitable, and may consume large amounts of time with increase in the amount of data and model parameters. The main aim of this study is to improve the calculation efficiency of FTG data inversion compared with general conjugate gradient (CG) method. Accordingly, an improved gradient method with projections onto a convex set (the projected gradient method) was applied to accelerate the recovery rate of the sub-surface rock density, and a corresponding preconditioning factor was introduced to further improve the calculation efficiency of inversion. This new projected gradient method can improve the calculation efficiency of large amounts of FTG data, which is very necessary for practical application of FTG data. The application of synthetic FTG data with Gaussian noise and the real data demonstrated the feasibility and application value of the proposed inversion method.