On the Iteration Complexity of Smoothed Proximal ALM for Nonconvex Optimization Problem with Convex Constraints

It is well-known that the lower bound of iteration complexity for solving nonconvex unconstrained optimization problems is $\Omega(1/\epsilon^2)$, which can be achieved by standard gradient descent algorithm when the objective function is smooth. This lower bound still holds for nonconvex constrained problems, while it is still unknown whether a first-order method can achieve this lower bound. In this paper, we show that a simple single-loop first-order algorithm called smoothed proximal augmented Lagrangian method (ALM) can achieve such iteration complexity lower bound. The key technical contribution is a strong local error bound for a general convex constrained problem, which is of independent interest

An Efficient Alternating Riemannian/Projected Gradient Descent Ascent Algorithm for Fair Principal Component Analysis

Fair principal component analysis (FPCA), a ubiquitous dimensionality reduction technique in signal processing and machine learning, aims to find a low-dimensional representation for a high-dimensional dataset in view of fairness. The FPCA problem involves optimizing a non-convex and non-smooth function over the Stiefel manifold. The state-of-the-art methods for solving the problem are subgradient methods and semidefinite relaxation-based methods. However, these two types of methods have their obvious limitations and thus are only suitable for efficiently solving the FPCA problem in special scenarios. This paper aims at developing efficient algorithms for solving the FPCA problem in general, especially large-scale, settings. In this paper, we first transform FPCA into a smooth non-convex linear minimax optimization problem over the Stiefel manifold. To solve the above general problem, we propose an efficient alternating Riemannian/projected gradient descent ascent (ARPGDA) algorithm, which performs a Riemannian gradient descent step and an ordinary projected gradient ascent step at each iteration. We prove that ARPGDA can find an $\varepsilon$-stationary point of the above problem within $\mathcal{O}(\varepsilon^{-3})$ iterations. Simulation results show that, compared with the state-of-the-art methods, our proposed ARPGDA algorithm can achieve a better performance in terms of solution quality and speed for solving the FPCA problems.Comment: 5 pages, 8 figures, submitted for possible publicatio

Overcoming DoF Limitation in Robust Beamforming: A Penalized Inequality-Constrained Approach

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 smaller than the number of sources in the environment. 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, we propose a beamformer with a well-targeted objective function and inequality constraints to achieve the design goals. The constraints on interferences penalize the maximum gain of the beamformer at any interfering directions. This can efficiently mitigate the total interference power regardless of whether the number of interfering sources is less than the array DoF or not. 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 trade off various design goals. When the array DoF is fewer than the number of interferences, the proposed formulation can effectively align the limited DoF to all of the sources to obtain the best overall interference suppression. $\$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.Comment: submitted to IEEE Transactions on Signal Processin

Robust Antijamming Strategy Design for Frequency-Agile Radar against Main Lobe Jamming

To combat main lobe jamming, preventive measures can be applied to radar in advance based on the concept of active antagonism, and efficient antijamming strategies can be designed through reinforcement learning. However, uncertainties in the radar and the jammer, which will result in a mismatch between the test and training environments, are not considered. Therefore, a robust antijamming strategy design method is proposed in this paper, in which frequency-agile radar and a main lobe jammer are considered. This problem is first formulated under the framework of Wasserstein robust reinforcement learning. Then, the method of imitation learning-based jamming strategy parameterization is presented to express the given jamming strategy mathematically. To reduce the number of parameters that require optimization, a perturbation method inspired by NoisyNet is also proposed. Finally, robust antijamming strategies are designed by incorporating jamming strategy parameterization and jamming strategy perturbation into Wasserstein robust reinforcement learning. The simulation results show that the robust antijamming strategy leads to improved radar performance compared with the nonrobust antijamming strategy when uncertainties exist in the radar and the jammer

Optimal Resource Allocation for Asynchronous Multiple Target Tracking in Heterogeneous Radar Network

In this paper, two optimal resource allocation schemes are developed for asynchronous multiple targets tracking (MTT) in heterogeneous radar networks. The key idea of heterogeneous resource allocation (HRA) schemes is to coordinate the heterogeneous transmit resource (transmit power, dwell time, etc.) of different types of radars to achieve a better resource utilization efficiency. We use the Bayesian Cramér-Rao lower bound (BCRLB) as a metric function to quantify the target tracking performance and build the following two HRA schemes: For a given system resource budget: (1) Minimize the total resource consumption for the given BCRLB requirements on multiple targets and (2) maximize the overall MTT accuracy. Instead of updating the state of each target recursively at different measurement arrival times, we combine multiple asynchronous measurements into a single composite measurement and use it as an input of the tracking filter for state estimation. In such a case, target tracking BCRLB no longer needs to be recursively calculated, and thus, we can formulate the HRA schemes as two convex optimization problems. We subsequently design two efficient methods to solve these problems by exploring their unique structures. Simulation results demonstrate that the HRA processes can either provide a smaller overall MTT BCRLB for given resource budgets or require fewer resources to establish the same tracking performance for multiple target

Influence of Metro Track Irregularities on Pantograph Vibration and Its Interaction with Catenary

The problem of excessive wear of pantograph strips frequently occurs on China’s Z City Metro Line 1. After on-site investigation and analysis by the metro operating company, it was speculated that the problem was related to abnormal track irregularities. Therefore, taking Z City Metro Line 1 as the main research object, the measured track irregularity of the whole line was analyzed and compared with other typical track spectra, and a track-vehicle-pantograph-catenary coupling dynamics model was established to analyze the relationship between the pantograph-catenary dynamic characteristics and the track irregularity. The two frequency ranges of the track irregularity that have a significant impact on the pantograph-catenary contact were found. Finally, after numerical calculation and analysis, it is recommended to focus on the irregularities with the two wavelength ranges of 9~16 m and 3~4 m in the maintenance

Simulation Study of Ultrasonic Elliptical Vibration Cutting of TiC Particle-Reinforced Titanium Matrix Composites

In order to investigate the characteristics of elliptical ultrasonic vibration cutting of TiC particle-reinforced titanium matrix composites, a two-dimensional thermodynamic coupled finite element cutting model was established based on the Johnson-Cook intrinsic structure model using ABAQUS simulation software, and the changes in cutting force, cutting temperature, machined surface shape, and particle fragmentation were obtained under the traditional cutting method and ultrasonic elliptical vibration cutting method. The results show that under the same process parameters, ultrasonic elliptical vibration cutting is better than conventional cutting in terms of surface profile; the stress direction tends to be horizontal during cutting and the TiC particles are mainly removed by cutting off. The average cutting force is significantly lower than conventional cutting, with a maximum reduction of 60%. The cutting temperature is also reduced, with a reduction of approximately 17.6%

Influence of Metro Track Irregularities on Pantograph Vibration and Its Interaction with Catenary

The problem of excessive wear of pantograph strips frequently occurs on China’s Z City Metro Line 1. After on-site investigation and analysis by the metro operating company, it was speculated that the problem was related to abnormal track irregularities. Therefore, taking Z City Metro Line 1 as the main research object, the measured track irregularity of the whole line was analyzed and compared with other typical track spectra, and a track-vehicle-pantograph-catenary coupling dynamics model was established to analyze the relationship between the pantograph-catenary dynamic characteristics and the track irregularity. The two frequency ranges of the track irregularity that have a significant impact on the pantograph-catenary contact were found. Finally, after numerical calculation and analysis, it is recommended to focus on the irregularities with the two wavelength ranges of 9~16 m and 3~4 m in the maintenance