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The augmented Lagrangian method (ALM) is one of the most useful methods for constrained optimization. Its convergence has been well established under convexity assumptions or smoothness assumptions, or under both assumptions. ALM may experience oscillations and divergence when the underlying problem is simultaneously nonconvex and nonsmooth. In this paper, we consider the linearly constrained problem with a nonconvex (in particular, weakly convex) and nonsmooth objective. We modify ALM to use a Moreau envelope of the augmented Lagrangian and establish its convergence under conditions that are weaker than those in the literature. We call it the Moreau envelope augmented Lagrangian (MEAL) method. We also show that the iteration complexity of MEAL is o (ε - 2) to yield an ε -accurate first-order stationary point. We establish its whole sequence convergence (regardless of the initial guess) and a rate when a Kurdyka–Łojasiewicz property is assumed. Moreover, when the subproblem of MEAL has no closed-form solution and is difficult to solve, we propose two practical variants of MEAL, an inexact version called iMEAL with an approximate proximal update, and a linearized version called LiMEAL for the constrained problem with a composite objective. Their convergence is also established. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, ϕ(x0,...,xp,y), subject to coupled linear equality constraints. Our ADMM updates each of the primal variables x0,...,xp,y, followed by updating the dual variable. We separate the variable y from xi's as it has a special role in our analysis. The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, ℓq quasi-norm, Schatten-q quasi-norm (0

We present a type of spherical neural network (SNN) with bounded sigmoidal activation function and study its interpolation capability. We find that the provided SNN can exactly interpolate the training samples. Furthermore, based on the special structure of the presented SNN, we can bound the interpolation error by the modulus of smoothness of the target function, which is different from the previous results on the spherical scattered data interpolation problem. [ABSTRACT FROM AUTHOR]

In this paper, a novel method for synthetic aperture radar (SAR) imaging is proposed. The approach is based on L regularization to reconstruct the scattering field, which optimizes a quadratic error term of the SAR observation process subject to the interested scene sparsity. Compared to the conventional SAR imaging technique, the new method implements SAR imaging effectively at much lower sampling rate than the Nyquist rate, and produces high-quality images with reduced sidelobes and increased resolution. Also, over the prevalent greedy pursuit and L regularization based SAR imaging methods, there are remarkable performance improvements of the new method. On one hand, the new method significantly reduces the number of measurements needed for reconstruction, as supported by a phase transition diagram study. On the other hand, the new method is more robust to the observation noise. These fundamental properties of the new method are supported and demonstrated both by simulations and real SAR data experiments. [ABSTRACT FROM AUTHOR]

The displaced phase center antenna (DPCA) technique is an effective strategy to achieve wide-swath synthetic aperture radar (SAR) imaging with high azimuth resolution. However, traditionally, it requires strict limitation of the pulse repetition frequency (PRF) to avoid non-uniform sampling. Otherwise, any deviation could bring serious ambiguity if the data are directly processed using a matched filter. To break this limitation, a recently proposed spectrum reconstruction method is capable of recovering the true spectrum from the nonuniform samples. However, the performance is sensitive to the selection of the PRF. Sparse regularization based imaging may provide a way to overcome this sensitivity. The existing time-domain method, however, requires a large-scale observation matrix to be built, which brings a high computational cost. In this paper, we propose a frequency domain method, called the iterative spectrum reconstruction method, through integration of the sparse regularization technique with spectrum analysis of the DPCA signal. By approximately expressing the observation in the frequency domain, which is realized via a series of decoupled linear operations, the method performs SAR imaging which is then not directly based on the observation matrix, which reduces the computational cost from O( N) to O( N log N) (where N is the number of range cells), and is therefore more efficient than the time domain method. The sparse regularization scheme, realized via a fast thresholding iteration, has been adopted in this method, which brings the robustness of the imaging process to the PRF selection. We provide a series of simulations and ground based experiments to demonstrate the high efficiency and robustness of the method. The simulations show that the new method is almost as fast as the traditional mono-channel algorithm, and works well almost independently of the PRF selection. Consequently, the suggested method can be accepted as a practical and efficient wide-swath SAR imaging technique. [ABSTRACT FROM AUTHOR]