• Nonlinear Functional Analysis and Applications
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• 제27권4호
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• pp.869-879
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• 2022

In this paper, we consider matrix optimization problems. We investigate augmented Lagrangian methods of multipliers and alternating direction methods of multipliers for the problems. Following the proofs of Eckstein [3], and Eckstein and Yao [5], we prove convergence theorems for augmented Lagrangian methods of multipliers and alternating direction methods of multipliers for the problems.

• 대한수학회지
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• 제61권2호
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• pp.395-408
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• 2024

There have been numerous studies on the characteristics of the solutions of ordinary differential equations for optimization methods, including gradient descent methods and alternating direction methods of multipliers. To investigate computer simulation of ODE solutions, we need to trace pseudo-orbits by real orbits and it is called shadowing property in dynamics. In this paper, we demonstrate that the flow induced by the alternating direction methods of multipliers (ADMM) for a C² strongly convex objective function has the eventual shadowing property. For the converse, we partially answer that convexity with the eventual shadowing property guarantees a unique minimizer. In contrast, we show that the flow generated by a second-order ODE, which is related to the accelerated version of ADMM, does not have the eventual shadowing property.

• Communications for Statistical Applications and Methods
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• 제29권4호
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• pp.441-451
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• 2022

In the era of bigdata, scalability is a crucial issue in learning models. Among many others, the Alternating Direction of Multipliers (ADMM, Boyd et al., 2011) algorithm has gained great popularity in solving large-scale problems efficiently. In this article, we propose applying the ADMM algorithm to solve the least square problem penalized by the pairwise-difference penalty, frequently used to identify group structures among coefficients. ADMM algorithm enables us to solve the high-dimensional problem efficiently in a unified fashion and thus allows us to employ several different types of penalty functions such as LASSO, Elastic Net, SCAD, and MCP for the penalized problem. Additionally, the ADMM algorithm naturally extends the algorithm to distributed computation and real-time updates, both desirable when dealing with large amounts of data.

• Journal of Electrical Engineering and Technology
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• 제13권5호
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• pp.1778-1790
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• 2018

Based on the requirements and characteristics of multi-zone autonomous decision-making in modern power system, fully distributed computing methods are needed to optimize the economic dispatch (ED) problem coordination of multi-regional power system on the basis of constructing decomposition and interaction mechanism. In this paper, four fully distributed methods based on alternating direction method of multipliers (ADMM) are used for solving the ED problem in distributed manner. By duplicating variables, the 2-block classical ADMM can be directly used to solve ED problem fully distributed. The second method is employing ADMM to solve the dual problem of ED in fully distributed manner. N-block methods based on ADMM including Alternating Direction Method with Gaussian back substitution (ADM_G) and Exchange ADMM (E_ADMM) are employed also. These two methods all can solve ED problem in distributed manner. However, the former one cannot be carried out in parallel. In this paper, four fully distributed methods solve the ED problem in distributed collaborative manner. And we also discussed the difference of four algorithms from the aspects of algorithm convergence, calculation speed and parameter change. Some simulation results are reported to test the performance of these distributed algorithms in serial and parallel.

• 응용통계연구
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• 제35권2호
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• pp.265-283
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• 2022

사용자들의 영화정보를 기록한 MovieLens 데이터는 추천 시스템 연구에서 아이디어를 탐색하고 검증하는데 상당한 가치가 있는 데이터로, 기존 데이터 분할 및 군집화 알고리즘을 사용하여 사용자 평점 데이터를 기반으로 항목 집합을 분할하는 연구 등에 사용되는 데이터이다. 본 논문에서는 기존 연구에서 대표적으로 사용되었던 영화 평점 데이터와 영화 장르 데이터를 통해 사용자의 장르 선호도를 예측하여 선호도 패턴을 기반으로 사용자를 군집화(clustering)하고, 유의미한 정보를 얻는 연구를 진행하였다. MovieLens 데이터는 영화의 전체 개수에 비해 사용자별 평균 영화 평점 수가 낮아 결측 비율이 높다. 이러한 이유로 기존의 군집화 방법을 적용하는 데 한계가 존재한다. 본 논문에서는 MovieLens 데이터 특성에 모티브를 얻어 쌍별 규합 벌점함수(pairwise fused penalty)를 활용한 볼록 군집화(convex clustering) 기반의 방법을 제안한다. 특히 결측치 대체(missing imputation)도 동시에 해결하는 최적화 문제를 통해 기존의 군집화 분석과 차별화하였다. 군집화는 반복 알고리즘인 ADMM을 통해 제안하는 최적화 문제를 풀어 진행한다. 또한 시뮬레이션과 MovieLens 데이터 적용을 통해 제안하는 군집화 방법이 기존의 방법보다 노이즈 및 이상치에 상대적으로 민감하지 않은 것으로 보인다.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제14권9호
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• pp.3745-3761
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• 2020

Low-rank representation methods already achieve many applications in the image reconstruction. However, for high-gradient image patches with rich texture details and strong edge information, it is difficult to find sufficient similar patches. Existing low-rank representation methods usually destroy image critical details and fail to preserve edge structure. In order to promote the performance, a new representation-based image super-resolution reconstruction method is proposed, which combines gradient domain guided image filter with the structure-constrained low-rank representation so as to enhance image details as well as reveal the intrinsic structure of an input image. Firstly, we extract the gradient domain guided filter of each atom in high resolution dictionary in order to acquire high-frequency prior information. Secondly, this prior information is taken as a structure constraint and introduced into the low-rank representation framework to develop a new model so as to maintain the edges of reconstructed image. Thirdly, the approximate optimal solution of the model is solved through alternating direction method of multipliers. After that, experiments are performed and results show that the proposed algorithm has higher performances than conventional state-of-the-art algorithms in both quantitative and qualitative aspects.

• Structural Engineering and Mechanics
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• 제91권2호
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• pp.227-238
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• 2024

Sparse regularization methods have proven effective in addressing the ill-posed equations encountered in moving force identification (MFI). However, the complexity of vehicle loads is often ignored in existing studies aiming at enhancing MFI accuracy. To tackle this issue, a double 𝑙₁ regularization method is proposed for MFI based on a response spectrum-based weighted dictionary in this study. Firstly, the relationship between vehicle-induced responses and moving vehicle loads (MVL) is established. The structural responses are then expanded in the frequency domain to obtain the prior knowledge related to MVL and to further construct a response spectrum-based weighted dictionary for MFI with a higher accuracy. Secondly, with the utilization of this weighted dictionary, a double 𝑙₁ regularization framework is presented for identifying the static and dynamic components of MVL by the alternating direction method of multipliers (ADMM) method successively. To assess the performance of the proposed method, two different types of MVL, such as composed of trigonometric functions and driven from a 1/4 bridge-vehicle model, are adopted to conduct numerical simulations. Furthermore, a series of MFI experimental verifications are carried out in laboratory. The results shows that the proposed method's higher accuracy and strong robustness to noises compared with other traditional regularization methods.

• Journal of the Korean Data and Information Science Society
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• 제28권6호
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• pp.1229-1244
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• 2017

최근 여러 분야에서 데이터에 근거한 분석방법론에 대한 수요가 증대됨에 따라 이를 처리할 수 있는 최적화 방법이 발전되고 있다. 특히 통계학과 기계학습 분야의 문제들에서 요구되는 다양한 제약 조건은 볼록 최적화 (convex optimization) 방법으로 해결할 수 있다. 본 논문에서 리뷰하는 alternating direction method of multipliers (ADMM) 알고리즘은 선형 제약 조건을 효과적으로 처리할 수 있으며, 합의 방식을 통해 병렬연산을 수행할 수 있어서 범용적인 표준 최적화 툴로 자리매김 되고 있다. ADMM은 원래의 문제보다 최적화가 쉬운 부분문제로 분할하고 이를 취합함으로써 복잡한 원 문제를 해결하는 방식의 근사알고리즘이다. 부드럽지 않거나 복합적인 (composite) 목적 함수를 최적화할 때 유용하며, 쌍대이론과 proximal 작용소 이론을 토대로 체계적으로 알고리즘을 구성할 수 있기 때문에 통계 및 기계학습 분야에서 폭 넓게 활용되고 있다. 본 논문에서는 최근 통계와 관련된 여러 분야에서 ADMM알고리즘의 활용도를 살펴보고자 하며 주요한 두 가지 주제에 중점을 두고자 한다. (1) 목적식의 분할 전략과 증강 라그랑지안 방법 및 쌍대문제의 설명과 (2) proximal 작용소의 역할이다. 알고리즘이 적용된 사례로, 별점화 함수 추정 등의 조정화 (regularization)를 활용한 방법론들을 소개한다. 모의 자료를 활용하여 lasso 문제의 최적화에 대한 실증결과를 제시한다.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제10권10호
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• pp.5129-5152
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• 2016

Multi-view super-resolution (MVSR) aims to estimate a high-resolution (HR) image from a set of low-resolution (LR) images that are captured from different viewpoints (typically by different cameras). MVSR is usually applied in camera array imaging. Given that MVSR is an ill-posed problem and is typically computationally costly, we super-resolve multi-view LR images of the original scene via image fusion (IF) and blind deblurring (BD). First, we reformulate the MVSR problem into two easier problems: an IF problem and a BD problem. We further solve the IF problem on the premise of calculating the depth map of the desired image ahead, and then solve the BD problem, in which the optimization problems with respect to the desired image and with respect to the unknown blur are efficiently addressed by the alternating direction method of multipliers (ADMM). Our approach bridges the gap between MVSR and BD, taking advantages of existing BD methods to address MVSR. Thus, this approach is appropriate for camera array imaging because the blur kernel is typically unknown in practice. Corresponding experimental results using real and synthetic images demonstrate the effectiveness of the proposed method.