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Globally and superlinearly convergent algorithm for minimizing a normal merit function

SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1998年第3期36卷 1005-1019页

作者： Polak, E Qi, LQ Univ Calif Berkeley Dept Elect Engn Berkeley CA 94720 USA Univ New S Wales Sch Math Sydney NSW 2052 Australia

In this paper we present two new concepts related to the solution of systems of nonsmooth equations (NE) and variational inequalities (VI). The first concept is that of a normal merit function, which summarizes the simple basic properties shared by various known merit functions. In general, normal merit functions are locally Lipschitz, but not differentiable. The second concept is that of a Newtonian operator, whose values generalize the concept of the Hessian for normal merit functions. These two concepts are then used to generalize the nonsmooth Newton method for solving the equation del f(x) = 0, where f is a normal merit function with f is an element of C-1, to the case where f is only locally Lipschitz and the set-valued inclusion 0 is an element of partial derivative f(x) needs to be solved. Combining the resulting generalized Newton method with certain first-order methods, we obtain a globally and superlinearly convergent algorithm for minimizing normal merit functions.

关键词： normal merit function generalized Newton method first-order algorithm global convergence superlinear convergence

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OPTIMAL METHODS FOR CONVEX RISK-AVERSE DISTRIBUTED OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2023年第3期33卷 1518-1557页

作者： Lan, Guanghui Zhang, Zhe Georgia Inst Technol H Milton Stewart Sch Ind & Syst Engn Atlanta GA 30332 USA

This paper studies the communication complexity of convex risk-averse optimization over a network. The problem generalizes the well-studied risk-neutral finite-sum distributed optimization problem, and its importance stems from the need to handle risk in an uncertain environment. For algorithms in the literature, a gap exists in communication complexities for solving risk-averse and risk-neutral problems. We propose two distributed algorithms, namely the distributed risk-averse optimization (DRAO) method and the distributed risk-averse optimization with sliding (DRAO-S) method, to close the gap. Specifically, the DRAO method achieves optimal communication complexity by assuming a certain saddle point subproblem can be easily solved in the server node. The DRAO-S method removes the strong assumption by introducing a novel saddle point sliding subroutine which only requires the projection over the ambiguity set P. We observe that the number of P-projections performed by DRAO-S is optimal. Moreover, we develop matching lower complexity bounds to show the communication complexities of both DRAO and DRAO-S to be unimprovable. Numerical experiments are conducted to demonstrate the encouraging empirical performance of the DRAO-S method.

关键词： risk-averse optimization distributed optimization first-order algorithm convex optimization lower complexity

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Cooperative Beamforming for Wireless Fronthaul and Access Links in Ultra-Dense C-RANs With SWIPT: A first-order Approach

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IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING 2021年第5期15卷 1242-1257页

作者： Tan, Fangqing Wu, Peiran Wu, Yik-Chung Xia, Minghua Sun Yat Sen Univ Sch Elect & Informat Technol Guangzhou 510006 Peoples R China Guilin Univ Elect Technol Key Lab Cognit Radio & Informat Proc Guilin 541004 Peoples R China Southern Marine Sci & Engn Guangdong Lab Zhuhai 519082 Peoples R China Univ Hong Kong Dept Elect & Elect Engn Hong Kong Peoples R China

This work studies multigroup multicasting transmission in cloud radio access networks (C-RANs) with simultaneous wireless information and power transfer where densely packed remote radio heads (RRHs) cooperatively provide information and energy services for information users (IUs) and energy users (EUs) respectively. To maximize the weighted sum rate (WSR) of information services while satisfying the energy harvesting levels at EUs an optimization of joint beamforming design for the fronthaul and access links is formulated which is however neither smooth nor convex and is indeed NP-hard. To tackle this difficulty the smooth and successive convex approximations are used to transform the original problem into a sequence of convex problems and two first-order algorithms are developed to find the initial feasible point and the nearly optimal solution respectively. Moreover an accelerated algorithm is designed to improve the convergence speed by exploiting both Nesterov and heavy-ball momentums. Numerical results demonstrate that the proposed first-order algorithms achieve almost the same WSR as that of traditional second-order approaches yet with much lower computational complexity and the proposed scheme outperforms state-of-the-art competing schemes in terms of WSR.

关键词： Signal processing algorithms Array signal processing Wireless communication Wireless sensor networks Clustering algorithms Heuristic algorithms Energy harvesting Beamforming cloud radio access networks first-order algorithm simultaneous wireless information and power transfer weighted sum rate wireless fronthauling

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Bregman primal-dual first-order method and application to sparse semidefinite programming

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2022年第1期81卷 127-159页

作者： Jiang, Xin Vandenberghe, Lieven Univ Calif Los Angeles Elect & Comp Engn Los Angeles CA 90095 USA

We present a new variant of the Chambolle-Pock primal-dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, we discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

关键词： Primal-dual algorithm first-order algorithm Semidefinite programming Bregman divergence

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A fixed-point augmented Lagrangian method for total variation minimization problems

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JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION 2013年第7期24卷 1168-1181页

作者： Duan, Yuping Huang, Weimin ASTAR Inst Infocomm Res Singapore Singapore

In this paper, we propose a fixed-point augmented Lagrangian method (FPALM) for general convex problems arising in image processing. We can easily obtain the alternating minimization algorithm (AMA) referred to [1] from the proposed FPALM. The proof for the convergence of the FPALM is provided under some mild assumptions. We present two kinds of first-order augmented Lagrangian schemes and show their connections to first-order primal-dual algorithms [2]. Furthermore, we apply an acceleration rule to both the FPALM and AMA to achieve better convergence rates. Numerical examples on different image denosing models including the ROF model, the vectorial TVmodel, high order models and the TV-L-1 model are provided to demonstrate the efficiency of the proposed algorithms. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Convex optimization Dual approaches Augmented Lagrangian method Fixed-point method Alternating minimization algorithm Total variation first-order algorithm Image denoising

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Super-resolution radar imaging using fast continuous compressed sensing

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ELECTRONICS LETTERS 2015年第24期51卷 2043-2044页

作者： Yang, Lei Zhou, Jianxiong Xiao, Huaitie Natl Univ Def Technol Coll Elect Sci & Engn Changsha 410073 Hunan Peoples R China

A continuous compressed sensing method for 2D radar imaging is adopted. Atomic norm minimisation is used for sparse signal recovery and the property of the dual optimal solution is utilised to generate a super-resolution radar image. A feasible first-order algorithm based on the alternating direction method of multipliers is presented for problem solving where the primal and dual optimal solution can be obtained simultaneously. A fast implementation is also developed by exploiting the low rank structure of the subproblem. Experimental results on real data validate the effectiveness of the proposed algorithm.

关键词： radar imaging compressed sensing super-resolution radar imaging fast continuous compressed sensing 2D radar imaging atomic norm minimisation sparse signal recovery dual optimal solution first-order algorithm multipliers alternating direction method

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Image denoising algorithm of compressed sensing based on alternating direction method of multipliers

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International Journal of Modeling, Simulation, and Scientific Computing 2022年第1期13卷 1-20页

作者： Changjie Fang Jingyu Chen Shenglan Chen College of Science Chongqing University of Posts and Telecommunications Chongqing 400065 P.R.China College of Automation Chongqing University of Posts and Telecommunications Chongqing 400065 P.R.China

In this paper,we propose an image denoising algorithm for compressed sensing based on alternating direction method of multipliers(ADMM).We prove that the objective func-tion of the iterates approaches the optimal *** also prove the O(1/N)convergence rate of our algorithm in the ergodic *** the same time,simulation results show that our algorithm is more efficient in image denoising compared with existing methods.

关键词： Compressed sensing alternating direction method of multipliers first-order algorithm convergence rate image denoising

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Design of experiments for locally weighted regression

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 1999年第2期81卷 363-382页

作者： Fedorov, VV Montepiedra, G Nachtsheim, CJ SmithKline Beecham Pharmaceut King Of Prussia PA 19406 USA Bowling Green State Univ Bowling Green OH 43403 USA Univ Minnesota Minneapolis MN 55455 USA

We consider the design of experiments when estimation is to be performed using locally weighted regression methods. We adopt criteria that consider both estimation error (variance) and error resulting from model misspecification (bias). Working with continuous designs, we use the ideas developed in convex design theory to analyze properties of the corresponding optimal designs. Numerical procedures for constructing optimal designs are developed and applied to a variety of design scenarios in one and two dimensions. Among the interesting properties of the constructed designs are the following: (1) Design points tend to be more spread throughout the design space than in the classical case. (2) The optimal designs appear to be less model and criterion dependent than their classical counterparts. (3) While the optimal designs are relatively insensitive to the specification of the design space boundaries, the allocation of supporting points is strongly governed by the points of interest and the selected weight function, if the latter is concentrated in areas significantly smaller than the design region. Some singular and unstable situations occur in the case of saturated designs. The corresponding phenomenon is discussed using a univariate linear regression example. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： mean squared error D-optimal linear-optimal equivalence theorem first-order algorithm local regression

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IMPROVING THE SOLUTION OF VARIATIONAL INEQUALITIES BASED ON THE OPTIMIZATION APPROACH

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CYBERNETICS AND SYSTEMS ANALYSIS 2013年第2期49卷 289-294页

作者： Aleksandrova, V. M. Shubenkova, I. A. Natl Tech Univ Ukraine Kyiv Polytech Inst Inst Appl Syst Anal Kiev Ukraine

An optimization approach to solving a convex finite-dimensional variational inequality with nonpotential operator is examined. It is shown how to construct an optimization problem equivalent to the variational inequality in the space of the original variables. A first-order algorithm to solve variational inequalitieis is formulated based on this optimization problem.

关键词： variational inequalities optimization approach monotone operator Kuhn-Tucker pair first-order algorithm

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OPTIMUM DESIGN IN PRESENCE OF PARAMETRIC UNCERTAINTY

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1976年第4期19卷 527-546页

作者： KWAK, BM HAUG, EJ UNIV IOWA COLL ENGNDEPT MECH & HYDRAULICSIOWA CITYIA 52240

A new class of optimal design problems that incorporates environmental uncertainty is formulated and related to worst-case design, minimax objective design, and game theory. A numerical solution technique is developed and applied to a weapon allocation problem, a structural design problem with an infinite family of load conditions, and a vibration isolator design problem with a band of excitation frequencies.

关键词： Environmental parameter parametric optimal design first-order algorithm gradient projection with constraint error compensation min-max problem

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