In this paper, motivated by Moreau's proximalalgorithm, we give several algorithms and related weak and strong convergence theorems for minimization problems under suitable conditions. These algorithms and conver...
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In this paper, motivated by Moreau's proximalalgorithm, we give several algorithms and related weak and strong convergence theorems for minimization problems under suitable conditions. These algorithms and convergence theorems are different from the results in the literatures. Besides, we also study algorithms and convergence theorems for the split feasibility problem in real Hilbert spaces. Finally, we give numerical results for our main results.
作者：
Wang, JiulinXia, YongBeihang Univ
State Key Lab Software Dev Environm LMIB Minist EducSch Math & Syst Sci Beijing 100191 Peoples R China
We present a linear-time approximation scheme for solving the trust region subproblem (TRS). It employs Nesterov's accelerated gradient descent algorithm to solve a convex programming reformulation of (TRS). The t...
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We present a linear-time approximation scheme for solving the trust region subproblem (TRS). It employs Nesterov's accelerated gradient descent algorithm to solve a convex programming reformulation of (TRS). The total time complexity is less than that of the recent linear-time algorithm. The algorithm is further extended to the two-sided trust region subproblem.
L-band digital aeronautical communication system 1(L-DACS1) is a promising candidate data-link for future air-ground communication, but it is severely interfered by the pulse pairs(PPs) generated by distance measure e...
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L-band digital aeronautical communication system 1(L-DACS1) is a promising candidate data-link for future air-ground communication, but it is severely interfered by the pulse pairs(PPs) generated by distance measure equipment. A novel PP mitigation approach is proposed in this paper. Firstly, a deformed PP detection(DPPD) method that combines a filter bank, correlation detection, and rescanning is proposed to detect the deformed PPs(DPPs) which are caused by multiple filters in the receiver. Secondly, a finite impulse response(FIR) model is used to approximate the overall characteristic of filters, and then the waveform of DPP can be acquired by the original waveform of PP and the FIR model. Finally, sparse representation is used to estimate the position and amplitude of each DPP, and then reconstruct each DPP. The reconstructed DPPs will be subtracted from the contaminated signal to mitigate interference. Numerical experiments show that the bit error rate performance of our approach is about 5 dB better than that of recent works and is closer to interference-free environment.
Large-scale generalized linear array models (GLAMs) can be challenging to fit. Computation and storage of its tensor product design matrix can be impossible due to time and memory constraints, and previously considere...
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Large-scale generalized linear array models (GLAMs) can be challenging to fit. Computation and storage of its tensor product design matrix can be impossible due to time and memory constraints, and previously considered design matrix free algorithms do not scale well with the dimension of the parameter vector. A new design matrix free algorithm is proposed for computing the penalized maximum likelihood estimate for GLAMs, which, in particular, handles nondifferentiable penalty functions. The proposed algorithm is implemented and available via the R package glamlasso. It combines several ideas-previously considered separately-to obtain sparse estimates while at the same time efficiently exploiting the GLAM structure. In this article, the convergence of the algorithm is treated and the performance of its implementation is investigated and compared to that of glmnet on simulated as well as real data. It is shown that the computation time for glamlasso scales favorably with the size of the problem when compared to glmnet. Supplementary materials, in the form of R code, data and visualizations of results, are available online.
We consider the problem of minimizing the sum of a smooth function h with a bounded Hessian and a nonsmooth function. We assume that the latter function is a composition of a proper closed function P and a surjective ...
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We consider the problem of minimizing the sum of a smooth function h with a bounded Hessian and a nonsmooth function. We assume that the latter function is a composition of a proper closed function P and a surjective linear map M, with the proximal mappings of tau P, tau > 0, simple to compute. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: the alternating direction method of multipliers and the proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions h and P are semialgebraic. Furthermore, we give simple sufficient conditions to guarantee boundedness of the sequence generated. These conditions can be satisfied for a wide range of applications including the least squares problem with the l(1/2) regularization. Finally, when M is the identity so that the proximal gradient algorithm can be efficiently applied, we show that any cluster point is stationary under a slightly more flexible constant step-size rule than what is known in the literature for a nonconvex h.
Classic multinomial logit model, commonly used in multiclass regression problem, is restricted to few predictors and does not take into account the relationship among variables. It has limited use for genomic data, wh...
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Classic multinomial logit model, commonly used in multiclass regression problem, is restricted to few predictors and does not take into account the relationship among variables. It has limited use for genomic data, where the number of genomic features far exceeds the sample size. Genomic features such as gene expressions are usually related by an underlying biological network. Efficient use of the network information is important to improve classification performance as well as the biological interpretability. We proposed a multinomial logit model that is capable of addressing both the high dimensionality of predictors and the underlying network information. Group lasso was used to induce model sparsity, and a network-constraint was imposed to induce the smoothness of the coefficients with respect to the underlying network structure. To deal with the non-smoothness of the objective function in optimization, we developed a proximal gradient algorithm for efficient computation. The proposed model was compared to models with no prior structure information in both simulations and a problem of cancer subtype prediction with real TCGA (the cancer genome atlas) gene expression data. The network-constrained mode outperformed the traditional ones in both cases.
A novel algorithm is proposed to extract moving targets using the phase information of the radar echoes in multichannel wide-area surveillance radar system. The algorithm is based on the recently proposed relaxed vers...
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A novel algorithm is proposed to extract moving targets using the phase information of the radar echoes in multichannel wide-area surveillance radar system. The algorithm is based on the recently proposed relaxed version of principal component pursuit (PCP) method, which is for the first time introduced into radar signal processing community. In the algorithm, after pre-processing and channel combination, the combined phase matrix of radar echoes can be expressed as the superposition of three matrices: low-rank phase matrix of ground clutter, sparse phase matrix of moving targets and entry-wise phase matrix of the noise component. This expression fits well with recently proposed the relaxed version of PCP method. Therefore accelerated proximal gradient algorithm can be applied to stably extract sparse phase matrix of moving targets. Both simulation and real data processing results verify the effectiveness of this proposed new algorithm.
This paper proposes a generalized accelerated proximalgradient (GAPG) approach for solving total variation (TV)-based image restoration problems. The GAPG algorithm generalizes the original APG algorithm by replacing...
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This paper proposes a generalized accelerated proximalgradient (GAPG) approach for solving total variation (TV)-based image restoration problems. The GAPG algorithm generalizes the original APG algorithm by replacing the Lipschitz constant with an appropriate positive-definite matrix, resulting in faster convergence. For TV-based image restoration problems, we further introduce two auxiliary variables that approximate the partial derivatives. Constraints on the variables can easily be imposed without modifying the algorithm much, and the TV regularization can be either isotropic or anisotropic. As compared with the recently developed APG-based methods for TV-based image restoration, i.e., monotone version of the two-step iterative shrinkage/thresholding algorithm (MTwIST) and monotone version of the fast IST algorithm (MFISTA), our GAPG is much simpler as it does not require to solve an image denoising subproblem. Moreover, the convergence rate of O(k(-2)) is maintained by our GAPG, where k s the number of iterations;the cost of each iteration in GAPG is also lower. As a result, in our experiments, our GAPG approach can be much faster than MTwIST and MFISTA. The experiments also verify that our GAPG converges faster than the original APG and MTwIST when they solve identical problems.
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