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Nonconvex optimization for third-order tensor completion under wavelet transform

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2023年第5期30卷 e2489-e2489页

作者： Yu, Quan Bai, Minru Hunan Univ Sch Math Changsha Peoples R China Hunan Univ Sch Math Changsha 410082 Peoples R China

The main aim of this paper is to develop a nonconvex optimization model for third-order tensor completion under wavelet transform. On the one hand, through wavelet transform of frontal slices, we divide a large tensor data into a main part tensor and three detail part tensors, and the elements of these four tensors are about a quarter of the original tensors. Solving these four small tensors can not only improve the operation efficiency, but also better restore the original tensor data. On the other hand, by using concave correction term, we are able to correct for low rank of tubal nuclear norm (TNN) data fidelity term and sparsity of l1$$ {l}_1 $$-norm data fidelity term. We prove that the proposed algorithm can converge to some critical point. Experimental results on image, magnetic resonance imaging and video inpainting tasks clearly demonstrate the superior performance and efficiency of our developed method over state-of-the-arts including the TNN and other methods.

关键词： difference of convex functions low rank sparse tensor completion wavelet transform

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Feasible Point Pursuit Based Sequential convex Approximation of Optimal Power Flow

Feasible Point Pursuit Based Sequential Convex Approximation...

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IEEE-Power-and-Energy-Society General Meeting (PESGM)

作者： Tiwari, Anamika Mohapatra, A. Sahoo, S. R. Indian Inst Technol Kanpur Dept Elect Engn Kanpur 208016 Uttar Pradesh India

ISBN: (纸本)9781665405072

Optimal Power Flow (OPF) is of paramount importance for the operation of the power network. However, the non-convexity of ACOPF poses convergence issues and limits its application in day to day operation. Therefore, a convex model is needed that gives guaranteed convergence and feasible solution. Among the existing convex OPF models, Second-Order Conic Programming (SOCP) is more efficient. However, the exact SOCP formulation of ACOPF has non-convex arctangent constraints. These constraints are generally either ignored or approximated to solve the relaxed and convex SOCP formulation of ACOPF. Consequently, retrieving a unique and feasible set of bus voltage phasors for ACOPF of meshed power networks is a challenge. Thus, this paper proposes a novel approach to recover a unique and feasible ACOPF solution from its convex formulation. The alternate difference of convex functions is used to reformulate the arctangent constraints. The proposed formulation is solved by a modified inner approximation technique such as Feasible Point Pursuit based Sequential convex Approximation (FPP-SCA) to preserve the feasibility of the solution.

关键词： Sequential convex Approximation difference of convex functions Second-Order Conic Programming convexity Feasibility Optimal Power Flow

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Oscillations in mixed-feedback systems

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SYSTEMS & CONTROL LETTERS 2022年 166卷

作者： Das, Amritam Chaffey, Thomas Sepulchre, Rodolphe Univ Cambridge Dept Engn Trumpington St Cambridge CB2 1PZ England

A new method is presented for the analysis of limit cycle oscillations in mixed-feedback systems. The calculation of the limit cycle is reformulated as the zero finding of a mixed-monotone relation, that is, of the difference of two maximally monotone relations. The problem can then be solved efficiently by borrowing existing algorithms that minimize the difference of two convex functions. The potential of the method is illustrated on the classical Van der Pol oscillator. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://***/licenses/by-nc-nd/4.0/).

关键词： Limit cycle analysis Describing function method Maximal monotonicity Mixed monotonicity difference of convex functions

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Nonconvex Optimization for Robust Tensor Completion from Grossly Sparse Observations

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JOURNAL OF SCIENTIFIC COMPUTING 2020年第2期85卷 1-32页

作者： Zhao, Xueying Bai, Minru Ng, Michael K. Hunan Univ Sch Math Changsha 410082 Hunan Peoples R China Univ Hong Kong Dept Math Pok Fu Lam Hong Kong Peoples R China

In this paper, we consider the robust tensor completion problem for recovering a low-rank tensor from limited samples and sparsely corrupted observations, especially by impulse noise. A convex relaxation of this problem is to minimize a weighted combination of tubal nuclear norm and the l(1)-norm data fidelity term. However, the l(1)-norm may yield biased estimators and fail to achieve the best estimation performance. To overcome this disadvantage, we propose and develop a nonconvex model, which minimizes a weighted combination of tubal nuclear norm, the l(1)-norm data fidelity term, and a concave smooth correction term. Further, we present a Gauss-Seidel difference of convex functions algorithm (GS-DCA) to solve the resulting optimization model by using a linearization technique. We prove that the iteration sequence generated by GS-DCA converges to the critical point of the proposed model. Furthermore, we propose an extrapolation technique of GS-DCA to improve the performance of the GS-DCA. Numerical experiments for color images, hyperspectral images, magnetic resonance imaging images and videos demonstrate that the effectiveness of the proposed method.

关键词： Robust tensor completion Low-rank Sparsity difference of convex functions Impulse noises

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Solving k-center problems involving sets based on optimization techniques

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JOURNAL OF GLOBAL OPTIMIZATION 2020年第1期76卷 189-209页

作者： Nguyen Thai An Nguyen Mau Nam Qin, Xiaolong Hangzhou Normal Univ Dept Math Hangzhou Zhejiang Peoples R China Univ Elect Sci & Technol China Inst Fundamental & Frontier Sci Chengdu 611731 Sichuan Peoples R China Duy Tan Univ Inst Res & Dev Da Nang Vietnam Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97207 USA Zhejiang Normal Univ Dept Math Jinhua Zhejiang Peoples R China

The continuous k-center problem aims at finding k balls with the smallest radius to cover a finite number of given points in R-n. In this paper, we propose and study the following generalized version of the k-center problem: Given a finite number of nonempty closed convex sets in R-n, find k balls with the smallest radius such that their union intersects all of the sets. Because of its nonsmoothness and nonconvexity, this problem is very challenging. Based on nonsmooth optimization techniques, we first derive some qualitative properties of the problem and then propose new algorithms to solve the problem. Numerical experiments are also provided to show the effectiveness of the proposed algorithms.

关键词： k-center problem Multifacility location problem Majorization-minimization principle difference of convex functions

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User Preference Aware Lossless Data Compression at the Edge

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IEEE TRANSACTIONS ON COMMUNICATIONS 2020年第6期68卷 3792-3807页

作者： Lu, Yawei Chen, Wei Poor, H. Vincent Tsinghua Univ Dept Elect Engn Beijing 100084 Peoples R China Princeton Univ Dept Elect Engn Princeton NJ 08544 USA

Data compression is an efficient technique for saving data storage and transmission costs in networks. Traditional data compression methods usually compress each content item according to its own statistical distribution of symbols and do not take into account user preferences on various content items. However, user preferences significantly impact the statistical distributions of symbols transmitted over communication links. This paper presents an edge source coding method to compress data at the network edge, in which codebooks are designed based on not only the statistical distributions of symbols in the content items but also the user preferences. In edge source coding, multiple content items might be compressed via the same codebook. For discrete user preferences, DCA (difference of convex functions programming algorithm) based and k-means++ based algorithms are proposed to derive codebook designs. For continuous user preferences, a sampling method is applied to yield codebook designs. In addition, edge source coding is extended to the two-user case and codebooks are designed to utilize multicasting opportunities. Simulation results demonstrate that edge source coding significantly reduces transmission costs for short content items.

关键词： Lossless data compression user preference edge source coding codebook design difference of convex functions

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Minimax D-optimal designs for multivariate regression models with multi-factors

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2020年 209卷 160-173页

作者： Gao, Lucy L. Zhou, Julie Univ Washington Dept Biostat Seattle WA 98195 USA Univ Victoria Dept Math & Stat Victoria BC V8W 2Y2 Canada

In multi-response regression models, the error covariance matrix is never known in practice. Thus, there is a need for optimal designs which are robust against possible misspecification of the error covariance matrix. In this paper, we approximate the error covariance matrix with a neighborhood of covariance matrices, in order to define minimax D-optimal designs that are robust against small departures from an assumed error covariance matrix. It is well known that the optimization problems associated with robust designs are non-convex, which makes it challenging to construct robust designs analytically or numerically, even for one-response regression models. We show that the objective function for the minimax D-optimal design is a difference of two convex functions, and develop a flexible algorithm for computing minimax D-optimal designs, which can be applied to any multi-response model with a discrete design space. We also derive several theoretical results for minimax D-optimal designs. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Robust regression design Minimax criterion Reflection symmetry Quantitative and qualitative factors convex optimization difference of convex functions

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Using Positive Spanning Sets to Achieve d-Stationarity with the Boosted DC Algorithm (vol 17, pg 853, 2020)

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VIETNAM JOURNAL OF MATHEMATICS 2020年第2期48卷 377-377页

作者： Artacho, F. J. Aragon Campoy, R. Vuong, P. T. Univ Alicante Dept Math Alicante Spain Univ Vienna Dept Math Vienna Austria Univ Southampton Sch Mathamt Sci Southampton SO17 1BJ Hants England

The difference of convex functions Algorithm (DCA) is widely used for minimizing the difference of two convex functions. A recently proposed accelerated version, termed BDCA for Boosted DC Algorithm, incorporates a line search step to achieve a larger decrease of the objective value at each iteration. Thanks to this step, BDCA usually converges much faster than DCA in practice. The solutions found by DCA are guaranteed to be critical points of the problem, but these may not be local minima. Although BDCA tends to improve the objective value of the solutions it finds, these are frequently just critical points as well. In this paper we combine BDCA with a simple Derivative-Free Optimization (DFO) algorithm to force the d-stationarity (lack of descent direction) at the point obtained. The potential of this approach is illustrated through some computational experiments on a Minimum-Sum-of-Squares clustering problem. Our numerical results demonstrate that the new method provides better solutions while still remains faster than DCA in the majority of test cases.

关键词： difference of convex functions Boosted difference of convex functions algorithm Positive spanning sets d-Stationary points Derivative-free optimization

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ON functions DETERMINED BY DENSE SETS

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Real Analysis Exchange 2021年第2期46卷 423-440页

作者： Nicholas P. M. Kayban Xianfu Wang

We show that a few basic classes of lower semicontinuous functions on ℝ n are densely recoverable. Specifically, we show that the sum of a convex and a continuous function, the difference of two convex and lower semicontinuous functions, a K-increasing function (where K is a cone of nonempty interior), and differences of K-increasing functions are all functions uniquely determined by their values on a dense set in ℝ n . Thus, sets of such functions of each type are densely recoverable sets. In general, the sum and difference of two densely recoverable sets of functions is shown to not be densely recoverable.

关键词： 26B05 26B25 90C26 cone-monotone function convex function dense set difference of cone-monotone functions difference of convex functions

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Solving global optimization problems using reformulations and signomial transformations

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COMPUTERS & CHEMICAL ENGINEERING 2018年 116卷 122-134页

作者： Lundell, A. Westerlund, T. Abo Akad Univ Math & Stat Tuomiokirkontori 1 FIN-20500 Turku Finland Abo Akad Univ Proc Design & Syst Engn Piispankatu 8 FIN-20500 Turku Finland

In this paper, a framework for reformulating nonconvex mixed-integer nonlinear programming (MINLP) problems containing twice-differentiable (C-2) functions to convex relaxed form is discussed. To provide flexibility and for utilizing more effective transformation strategies, the twice-differentiable functions can be partitioned into convex, signomial and general nonconvex functions. The latter two can then be convexified using lifting transformations in combination with approximations using piecewise linear functions (PLFs). However, since there are many degrees of freedom in how to select the set of transformations, an optimization-based method is proposed for finding an optimal set. The lifting transformations are based on single-variable power and exponential transformations for signomials. For nonconvex C-2-functions the alpha reformulation (alpha R) technique as well as more generally the method of difference of convex functions can be applied. In the alpha R, the alpha BB convex underestimator can be used. The framework is utilized in the alpha signomial global optimization (alpha SGO) algorithm to find the epsilon-global solution to a nonconvex problem by iteratively updating the approximations provided by the PLFs. The framework can also be used to directly obtain a convex relaxation of any nonconvex MINLP problem of the specified type to a determined accuracy. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Global optimization Nonconvex MINLP Reformulation techniques Signomial functions Twice-differentiable nonconvex functions Power transformations Exponential transformations alpha reformulation alpha BB convex underestimator difference of convex functions

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