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NONNEGATIVE MATRIX FACTORIZATION BASED ON ALTERNATING NONNEGATIVITY CONSTRAINED LEAST SQUARES AND active set method

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2008年第2期30卷 713-730页

作者： Kim, Hyunsoo Park, Haesun Georgia Inst Technol Coll Comp Atlanta GA 30332 USA

Nonnegative matrix factorization (NMF) determines a lower rank approximation of a matrix A is an element of R-mxn approximate to WH where an integer k << min(m, n) is given and nonnegativity is imposed on all components of the factors W is an element of R-mxk and H is an element of R-kxn. NMF has attracted much attention for over a decade and has been successfully applied to numerous data analysis problems. In applications where the components of the data are necessarily nonnegative, such as chemical concentrations in experimental results or pixels in digital images, NMF provides a more relevant interpretation of the results since it gives nonsubtractive combinations of nonnegative basis vectors. In this paper, we introduce an algorithm for NMF based on alternating nonnegativity constrained least squares (NMF/ANLS) and the active set-based fast algorithm for nonnegativity constrained least squares with multiple right-hand side vectors, and we discuss its convergence properties and a rigorous convergence criterion based on the Karush-Kuhn-Tucker (KKT) conditions. In addition, we also describe algorithms for sparse NMFs and regularized NMF. We show how we impose a sparsity constraint on one of the factors by L-1-norm minimization and discuss its convergence properties. Our algorithms are compared to other commonly used NMF algorithms in the literature on several test data sets in terms of their convergence behavior.

关键词： nonnegative matrix factorization lower rank approximation two-block coordinate descent method Karush-Kuhn-Tucker (KKT) conditions nonnegativity constrained least squares active set method

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A DUAL active set method FOR?1-REGULARIZED PROBLEM

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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2023年第11期19卷 7826-7848页

作者： Hikima, Yuya Yamashita, Nobuo Univ Tokyo Tokyo Japan Kyoto Univ Kyoto Japan

We consider an l1-regularized problem, which has many applica-tions such as compressed sensing, MRI, and seismic images. Although there have been some methods for the problem, they cannot obtain a highly accurate solution. To overcome this drawback, we propose a dual active set method, which is an active set method for the Fenchel dual problem of the l1-regularized problem. Our method finds a highly accurate solution quickly because of the following features of the dual problem: (i) the dimension of the decision vari-ables is smaller;(ii) since the optimal solution of the primal problem is sparse, the active constraints of the optimal solution of the dual problem are few and can be identified quickly. Then, we show that the method outputs the solution with finite iterations when it solves certain subproblems exactly. Moreover, we give its efficient and concrete implementation for Basis Pursuit Denoising (BPDN), which is a typical l1-regularized problem. In numerical experiments for BPDN, our method provides more accurate solutions than all baselines in comparable computing time.

关键词： ?1-regularized problem active set method basis pursuit denoising compressed sensing

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Application of sequential quadratic programming based on active set method in cleaner production

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CLEAN TECHNOLOGIES AND ENVIRONMENTAL POLICY 2022年第1期24卷 413-422页

作者： Xia, Li Ling, Jianyang Xu, Zhen Bi, Rongshan Zhao, Wenying Xiang, Shuguang Qingdao Univ Sci & Technol Coll Chem Engn State Key Lab Base Ecochem Engn Zhengzhou Rd 53 Qingdao 266042 Peoples R China Qilu Normal Univ Chem & Chem Engn Fac Jinan 250013 Shandong Peoples R China

On the platform of general chemical process simulation software (it was named Optimization Engineer, OPEN), a general optimization algorithm for chemical process simulation is developed using C + + code. The algorithm is based on sequential quadratic programming (SQP). We adopt the activity set algorithm and the rotation axis algorithm to generate the activity set to solve the quadratic programming sub-problem. The active set method can simplify the number of constraints and speed up the calculation. At the same time, we used limited memory BFGS algorithm (L-BFGS) to simplify the solution of second derivative matrix. The special matrix storage mode of L-BFGS algorithm can save the storage space and speed up the computing efficiency. We use exact penalty function and traditional step-size rule in the algorithm. These two methods can ensure the convergence of the algorithm, a more correct search direction and suitable search step. The example shows that the advanced optimization function can meet the requirements of General Chemical Process Calculation. The number of iterations can reduce by about 6.0%. The computation time can reduce by about 6.5%. We combined this algorithm with chemical simulation technology to develop the optimization function of chemical engineering simulation. This optimization function can play an important role in the process optimization calculation aiming at energy saving and green production.

关键词： Sequential quadratic programming active set method Chemical process simulation Chemical optimization Green production

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An active set method for single-cone second-order cone programs

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SIAM JOURNAL ON OPTIMIZATION 2006年第2期17卷 459-484页

作者： Erdogan, E. Iyengar, G. Columbia Univ Dept Ind Engn & Operat Res New York NY 10027 USA

We develop an active set method for solving second-order cone programs that may have an arbitrary number of linear constraints but are restricted to having only one second-order cone constraint. Problems of this form arise in the context of robust optimization and trust region methods. The proposed active set method exploits the fact that a second-order cone program with only one second-order cone constraint and no inequality constraints can be solved in closed form.

关键词： second-order cone programs single-cone active set method robust optimization uncertain linear program duality

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A hybrid multilevel-active set method for large box-constrained linear discrete ill-posed problems

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CALCOLO 2011年第1期48卷 89-105页

作者： Morigi, S. Plemmons, R. Reichel, L. Sgallari, F. Univ Bologna Dept Math CIRAM I-40123 Bologna Italy Kent State Univ Dept Math Sci Kent OH 44242 USA Wake Forest Univ Dept Math Winston Salem NC 27106 USA Wake Forest Univ Dept Comp Sci Winston Salem NC 27106 USA Univ Bologna Dept Math CIRAM I-40126 Bologna Italy

Many questions in science and engineering give rise to linear ill-posed problems, whose solution is known to satisfy box constraints, such as nonnegativity. The solution of discretized versions of these problems is highly sensitive to perturbations in the data, discretization errors, and round-off errors introduced during the computations. It is therefore often beneficial to impose known constraints during the solution process. This paper describes a two-phase algorithm for the solution of large-scale box-constrained linear discrete ill-posed problems. The first phase applies a cascadic multilevel method and imposes the constraints on each level by orthogonal projection. The second phase improves the computed approximate solution on the finest level by an active set method. The latter allows several indices of the active set to be updated simultaneously. This reduces the computational effort significantly, when compared to standard active set methods that update one index at a time. Applications to image restoration are presented.

关键词： Constrained ill-posed problems Nonnegativity active set method Multilevel method Image restoration

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A RANDOM active set method FOR STRICTLY CONVEX QUADRATIC PROBLEM WITH SIMPLE BOUNDS

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MATHEMATICS OF COMPUTATION 2024年第350期93卷 2837-2860页

作者： Gu, Ran Gao, Bing Nankai Univ Sch Stat & Data Sci NITFID KLMDASRLEBPS Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China Nankai Univ Sch Math Sci Tianjin 300071 Peoples R China

. The active set method aims at finding the correct active set of the optimal solution and it is a powerful method for solving strictly convex quadratic problems with bound constraints. To guarantee the finite step convergence, existing active set methods all need strict conditions or some additional strategies, which can significantly impact the efficiency of the algorithm. In this paper, we propose a random active set method that introduces randomness in the active set's update process. We prove that the algorithm can converge in a finite number of iterations with probability one, without any extra conditions on the problem or any supplementary strategies. At last, numerical experiments show that the algorithm can obtain the correct active set within a few iterations, and it has better efficiency and robustness than the existing methods.

关键词： Convex quadratic problem bound constraint active set method

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Streamlining active set method in MPC using Cache Memory 7

Streamlining Active Set Method in MPC using Cache Memory

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7th IFAC Conference on Nonlinear Model Predictive Control (NMPC)

作者： Fedorova, K. Kohut, R. Kvasnica, M. Slovak Univ Technol Bratislava Inst Informat Engn Automat & Math Bratislava Slovakia

This paper investigates how various caching strategies can reduce the computational effort of the active set method (ASM) applied to solve constrained model predictive control problems with quadratic objective function and linear constraints. Specifically, we show that during closed-loop operation, the active set method often re-visits the same combination of active constraints while searching for optimal control inputs by factoring Karush-Kuhn-Tucker (KKT) systems. By storing the factors of the corresponding KKT system in a cache, these repetitive calculations can be simplified to a mere cache search and evaluation of the appropriate factors. Since the cache memory is typically fairly restricted, the efficiency of the scheme depends on how well the cache space can be utilized. In particular, when the cache is fully utilized, and a new element needs to be stored, the cache replacement policy needs to determine which element should be removed from the cache to make space for the new one. In the paper, we scrutinize various cache replacement policies and how well they work as a function of the cache size. The results show that by using a cache of modest size, the number of computational operations performed by the ASM can be reduced by up to 80%, thus significantly accelerating the implementation of model predictive control. Copyright (C) 2021 The Authors.

关键词： cache memory cache replacement policy active set method model predictive control

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Optimization of Overcurrent Relays in Microgrid Using Interior Point method and active set method 1

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5th International Conference on Frontiers in Intelligent Computing - Theory and Applications (FICTA)

作者： Swathika, O. V. Gnana Das, Arka Gupta, Yatharth Mukhopadhyay, Santanab Hemamalini, S. VIT Univ Sch Elect Engn Chennai Tamil Nadu India

ISBN: (数字)9789811031564

ISBN: (纸本)9789811031564;9789811031557

Microgrid is an aggregate of generating units and loads at the distribution level. It operates in two modes: grid connected mode and Islanded mode. Fault clearance in a microgrid is a key challenge for protection engineers. This paper aims to identify the optimized values of time of operation of overcurrent relays in a microgrid network. This paper solves the optimization problem using two methods-Interior Point method and active set method. Also three types of relays are used to determine which relay works best in similar constraint environment. These methods aid in providing suitable relay coordination in microgrid and clear the faulty portion of network quickly from the healthy portion of network.

关键词： Microgrid protection Optimization techniques Interior point method active set method

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Hidden Markov Model-based Warm-start of active set method in Model Predictive Control 23

Hidden Markov Model-based Warm-start of Active Set Method in...

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23rd International Conference on Process Control (PC)

作者： Kohut, Roman Galciova, Lenka Fedorova, Kristina Abelova, Tereza Bakosova, Monika Kvasnica, Michal Slovak Univ Technol Bratislava Inst Informat Engn Automat & Math Radlinskeho 9 Bratislava 81237 Slovakia

ISBN: (纸本)9781665403306

We propose a new idea of statistical warm-start of the active set method (ASM) utilized for solving the problem of model predictive control (MPC). The idea is based on the Hidden Markov Model (HMM). In our HMM, the emissions are defined as the system states. The hidden states of HMM are represented by the active sets of the MPC optimization problem. Given the measurement or estimation of system states, using the HMM we can determine which active set is the most likely optimal. This active set is then used for the initialization of the ASM. We show that our approach can streamline the iterative process of ASM compared to the cold-start or conventional warm-start method using the result from the previous control step.

关键词： model predictive control active set method hidden Markov model

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Energy storage system optimization strategy in the Distribution Network Based on active set method 6

Energy storage system optimization strategy in the Distribut...

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2016 IEEE International Conferences on BDCloud/SocialCom/SustainCom 2016

作者： Yan, Ren-Wu Xiao, Hong-min Fujian Univ Technol Coll Informat Sci & Engn Fuzhou Peoples R China Northwest Normal Univ Coll Math & Stat Lanzhou Peoples R China

ISBN: (纸本)9781509039364

On the basis of short-term power load data, an active set algorithm based energy storage system optimization strategy in the distribution network is presented in this paper. When energy storage system is connected to the power grid, the original load and output power of energy storage system is considered as the new network load. In order to improve the reliability and power supply capability of the distribution network, methods should be taken to make the new load curve as smooth as possible. Therefore the minimum variance in load curve is used as the objective function of optimization model of energy storage system. The active set method is adopted to solve the problem. The result shows that the active set method is effective in solving the optimization strategy in the distribution network.

关键词： energy storage system load data optimization strategy active set method

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