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Hybrid Moreau's proximal algorithms and Convergence Theorems for Minimization Problems in Hilbert Spaces with Applications

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2018年第16期39卷 1806-1832页

作者： Chuang, Chih-Sheng Natl Chiayi Univ Dept Appl Math 300 Syuefu Rd Chiayi 60004 Taiwan

In this paper, motivated by Moreau's proximal algorithm, 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.

关键词： Projection proximal gradient algorithm split feasible problem strong convergence

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GLOBAL CONVERGENCE OF SPLITTING METHODS FOR NONCONVEX COMPOSITE OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2015年第4期25卷 2434-2460页

作者： Li, Guoyin Pong, Ting Kei Univ New S Wales Dept Appl Math Sydney NSW 2052 Australia Hong Kong Polytech Univ Dept Appl Math Hong Kong Hong Kong Peoples R China

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.

关键词： global convergence nonconvex composite optimization alternating direction method of multipliers proximal gradient algorithm Kurdyka-Lojasiewicz property

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Applying FISTA to optimization problems (with or) without minimizers

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MATHEMATICAL PROGRAMMING 2020年第1-2期184卷 349-381页

作者： Bauschke, Heinz H. Bui, Minh N. Wang, Xianfu Univ British Columbia Math Kelowna BC V1V 1V7 Canada North Carolina State Univ Dept Math Raleigh NC 27695 USA

Beck and Teboulle's FISTA method for finding a minimizer of the sum of two convex functions, one of which has a Lipschitz continuous gradient whereas the other may be nonsmooth, is arguably the most important optimization algorithm of the past decade. While research activity on FISTA has exploded ever since, the mathematically challenging case when the original optimization problem has no minimizer has found only limited attention. In this work, we systematically study FISTA and its variants. We present general results that are applicable, regardless of the existence of minimizers.

关键词： Convex function FISTA Forward-backward method Nesterov acceleration proximal gradient algorithm

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A two-phase rank-based algorithm for low-rank matrix completion

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OPTIMIZATION LETTERS 2023年第7期17卷 1679-1695页

作者： Araujo, Tacildo de S. Goncalves, Douglas S. Torezzan, Cristiano Univ Estadual Campinas Dept Appl Math IMECC Campinas Brazil Univ Fed Santa Catarina Dept Math Florianopolis SC Brazil Univ Estadual Campinas Sch Appl Sci Limeira Brazil

Matrix completion aims to recover an unknown low-rank matrix from a small subset of its entries. In many applications, the rank of the unknown target matrix is known in advance and this information can be useful in the completion process. In this paper, first, we revisit a recently proposed rank-based heuristic for "known-rank " matrix completion and establish a condition under which the generated sequence is quasi-Fejer convergent to the solution set. Then, by including an acceleration mechanism similar to Nesterov's acceleration, we obtain a new heuristic. Even though the convergence of this new heuristic cannot be granted in general, it turns out that it can be very useful as a warm-start phase (phase one), providing a suitable estimate for the regularization parameter and a good starting point to an accelerated proximal gradient algorithm (phase two) aimed to solve a nuclear-norm regularized problem. Numerical experiments with both synthetic and real data show that the resulting two-phase rank-based algorithm can recover low-rank matrices, with relatively high precision, faster than other well-established matrix completion algorithms.

关键词： Matrix completion proximal gradient algorithm Soft-thresholding

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Joint sparse optimization: lower-order regularization method and application in cell fate conversion

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INVERSE PROBLEMS 2024年第9期40卷 095003-095003页

作者： Hu, Yaohua Hu, Xinlin Kwok Wai Yu, Carisa Qin, Jing Shenzhen Univ Sch Math Sci Shenzhen 518060 Peoples R China Guangdong Univ Technol Sch Math & Stat Guangzhou 510520 Peoples R China Hang Seng Univ Hong Kong Dept Math Stat & Insurance Shatin Hong Kong Peoples R China Sun Yat sen Univ Sch Pharmaceut Sci Shenzhen Shenzhen Campus Shenzhen 518107 Guangdong Peoples R China

Multiple measurement signals are commonly collected in practical applications, and joint sparse optimization adopts the synchronous effect within multiple measurement signals to improve model analysis and sparse recovery capability. In this paper, we investigate the joint sparse optimization problem via l(p,q) regularization (0 <= q <= 1 <= p) in three aspects: theory, algorithm and application. In the theoretical aspect, we introduce a weak notion of joint restricted Frobenius norm condition associated with the l(p,q) regularization, and apply it to establish an oracle property and a recovery bound for the l(p,q) regularization of joint sparse optimization problem. In the algorithmic aspect, we apply the well-known proximal gradient algorithm to solve the l(p,q) regularization problems, provide analytical formulas for proximal subproblems of certain specific l(p,q) regularizations, and establish the global convergence and linear convergence rate of the proximal gradient algorithm under some mild conditions. More importantly, we propose two types of proximal gradient algorithms with the truncation technique and the continuation technique, respectively, and establish their convergence to the ground true joint sparse solution within a tolerance relevant to the noise level and the recovery bound under the assumption of restricted isometry property. In the aspect of application, we develop a novel method, based on joint sparse optimization with lower-order regularization and proximal gradient algorithm, to infer the master transcription factors for cell fate conversion, which is a powerful tool in developmental biology and regenerative medicine. Numerical results indicate that the novel method facilitates fast identification of master transcription factors, give raise to the possibility of higher successful conversion rate and in the hope of reducing biological experimental cost.

关键词： joint sparse optimization lower-order regularization consistency theory proximal gradient algorithm global convergence cell fate conversion

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A linear-time algorithm for the trust region subproblem based on hidden convexity

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OPTIMIZATION LETTERS 2017年第8期11卷 1639-1646页

作者： Wang, Jiulin Xia, Yong Beihang 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 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.

关键词： Trust region subproblem Hidden convexity proximal gradient algorithm Linear time complexity

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Penalized Estimation in Large-Scale Generalized Linear Array Models

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2017年第3期26卷 709-724页

作者： Lund, Adam Vincent, Martin Hansen, Niels Richard Univ Copenhagen Dept Math Sci DK-1165 Copenhagen Denmark

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.

关键词： Generalized linear array models Multidimensional smoothing Penalized estimation proximal gradient algorithm

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Robust Multivariate Lasso Regression with Covariance Estimation

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2023年第3期32卷 961-973页

作者： Chang, Le Welsh, A. H. Australian Natl Univ Res Sch Finance Actuarial Studies & Stat Canberra ACT Australia

Multivariate regression with covariance estimation (MRCE) is a method that performs sparse estimation of multivariate regression coefficients, while taking account the covariance structure of the response variables. MRCE uses a penalized likelihood approach to simultaneously estimate the regression coefficients and the inverse covariance matrix so that prediction accuracy can be significantly improved. However, traditional likelihood-based methods such as MRCE can produce very misleading results in the presence of outliers. In this work, we propose an extension of MRCE, namely, a robust multivariate lasso regression with covariance estimation (RMLC) to handle potential outliers within the data. By using Huber's loss or Tukey's biweight loss, RMLC can be resistant to outliers in the responses or in both the responses and the covariates. A novel optimization algorithm that incorporates a 2-fold accelerated proximal gradient (APG) algorithm is developed to solve RMLC efficiently. We also demonstrate that our proposed RMLC enjoys the oracle property. Our simulation study shows that RMLC produces very reliable results for both the regression coefficients and the correlation structure of the responses, even if the data are contaminated. A real analysis on hyperspectral data further demonstrates the utility of RMLC. Supplementary materials for this article are available online.

关键词： Graphical lasso Lasso Multivariate regression Oracal property proximal gradient algorithm Robust estimation

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A varying coefficient model with matrix valued covariates

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JOURNAL OF NONPARAMETRIC STATISTICS 2024年第3期36卷 673-705页

作者： Zhang, Hong-Fan Southwest Jiaotong Univ Dept Stat Chengdu Peoples R China

Modern data are often collected in a matrix form. In this paper, we consider modelling the varying coefficient regression with matrix valued covariate X and scalar index variable U. The proposed model simultaneously makes principal component analysis for both the row and column dimensions of the matrix objects, maintaining the matrix structure while achieving substantial dimension reduction. We develop an iterative estimation method for the involved principal parameters and nonparametric functions. Under regularity conditions, the asymptotic distributions of the estimators are derived. In addition, by incorporating the estimation with the adaptive group Lasso and the group SCAD penalties, variables of X in entire rows or columns are selected. The proximal gradient algorithm is further utilised to solve the regularised optimisation problems. The asymptotic properties of the penalised estimators are also studied. Our model and estimation methods are demonstrated by simulated experiments. Real applications to the primary biliary cirrhosis (PBC) data reveal that the effects of the blood measurements to the survival time vary with levels of age.

关键词： Varying coefficient model matrix covariates principal component analysis variable selection proximal gradient algorithm

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EXTERNAL DIVISION OF TWO PROXIMITY OPERATORS: AN APPLICATION TO SIGNAL RECOVERY WITH STRUCTURED SPARSITY 49

EXTERNAL DIVISION OF TWO PROXIMITY OPERATORS: AN APPLICATION...

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49th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： Suzuki, Kyohei Yukawa, Masahiro Keio Univ Dept Elect & Elect Engn Minato City Tokyo Japan

ISBN: (纸本)9798350344868;9798350344851

This paper studies the external division operator, an external division (an affine combination with positive and negative weights) of two proximity operators. We show that the external division operator is cocoercive under some condition, and it can be expressed as the proximity operator of a certain weakly convex function. We then consider using the external division operator as an alternative to the proximity operator in the proximal gradient algorithm, which we show converges to a minimizer of the cost function penalized by the weakly convex function under some conditions. Our analysis covers the case when the fidelity function is convex but not strongly convex. In simulations, we employ the octagonal shrinkage and clustering algorithm for regression in the external division operator, and we show that significant improvements are attained in signal recovery with structured sparsity in both overdetermined and underdetermined cases.

关键词： Cocoercive operator proximity operator weakly convex function proximal gradient algorithm OSCAR

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