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quadratic semidefinite programming for Waveform-Constrained Joint Filter-Signal Design in STAP

IEEE TRANSACTIONS ON SIGNAL PROCESSING 2020年 68卷 1744-1759页

作者： O'Rourke, Sean M. Setlur, Pawan Rangaswamy, Muralidhar Swindlehurst, A. Lee Univ Calif Irvine Irvine CA 92697 USA US Air Force Res Lab Wright Patterson AFB OH 45433 USA Riverside Res Dayton OH 45431 USA

In waveform-adaptive space-time adaptive processing (STAP), signal-dependent interference can be mitigated through joint filter-signal design. For the common signal-to-interference-plus-noise ratio (SINR) optimality criterion, these designs result in biquadratic problems, which can be efficiently solved through joint relaxation. In this paper, we consider the addition of two realistic waveform constraints to this process, constant modulus (CM) and the similarity constraint (SC), and demonstrate that they are both amenable to a quadratic semidefinite program (QSDP) via relaxation. We examine the optimality conditions of the newly constrained problems and demonstrate, both analytically and numerically, improved performance over existing iterative techniques. To increase solution robustness, additional refinement strategies for rank-inflated relaxed solutions are also provided and analyzed which extend the aforementioned performance gain.

关键词： Radar fully adaptive radar convex optimization quadratic semidefinite programming semidefinite relaxation joint design beamformer design waveform design waveform constraints constant modulus similarity constraint space time adaptive radar

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Modified alternating direction method of multipliers for convex quadratic semidefinite programming

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NEUROCOMPUTING 2016年 214卷 575-586页

作者： Chang, Xiaokai Liu, Sanyang Li, Xu Xidian Univ Sch Math & Stat Xian 710071 Peoples R China Lanzhou Univ Technol Coll Sci Lanzhou 730050 Peoples R China

The direct extension of alternating direction method of multipliers (ADMM) is not necessarily convergent to the dual form of convex quadratic semidefinite programming (CQSDP) problems, though the convergence of ADMM is proved when two blocks of variables are alternatively updated. In this paper, we present a modified ADMM for solving the dual form of the CQSDP problems. At each iteration, the proposed method did not have to work out the sub-problem with the primal variable, compared with the existing ADMM. Assuming that the penalty parameter satisfies the condition related to the quadratic term of the primal objective function, the convergence is proved by using a fixed-point argument. Numerical results for the nearest correlation matrix problems and the random CQSDP problems demonstrate the efficiency of our proposed algorithm. (C) 2016 Elsevier B.V. All rights reserved.

关键词： quadratic semidefinite programming Alternating direction method of multipliers Convergence analysis Nearest correlation matrix problems

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QSDPNAL: a two-phase augmented Lagrangian method for convex quadratic semidefinite programming

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MATHEMATICAL programming COMPUTATION 2018年第4期10卷 703-743页

作者： Li, Xudong Sun, Defeng Toh, Kim-Chuan Princeton Univ Dept Operat Res & Financial Engn Sherrerd Hall Princeton NJ 08544 USA Hong Kong Polytech Univ Dept Appl Math Hong Kong Hong Kong Peoples R China Natl Univ Singapore Dept Math 10 Lower Kent Ridge Rd Singapore 119076 Singapore Natl Univ Singapore Inst Operat Res & Analyt 10 Lower Kent Ridge Rd Singapore 119076 Singapore

In this paper, we present a two-phase augmented Lagrangian method, called QSDPNAL, for solving convex quadratic semidefinite programming (QSDP) problems with constraints consisting of a large number of linear equality and inequality constraints, a simple convex polyhedral set constraint, and a positive semidefinite cone constraint. A first order algorithm which relies on the inexact Schur complement based decomposition technique is developed in QSDPNAL-Phase I with the aim of solving a QSDP problem to moderate accuracy or using it to generate a reasonably good initial point for the second phase. In QSDPNAL-Phase II, we design an augmented Lagrangian method (ALM) wherein the inner subproblem in each iteration is solved via inexact semismooth Newton based algorithms. Simple and implementable stopping criteria are designed for the ALM. Moreover, under mild conditions, we are able to establish the rate of convergence of the proposed algorithm and prove the R(super)linear convergence of the KKT residual. In the implementation of QSDPNAL, we also develop efficient techniques for solving large scale linear systems of equations under certain subspace constraints. More specifically, simpler and yet better conditioned linear systems are carefully designed to replace the original linear systems and novel shadow sequences are constructed to alleviate the numerical difficulties brought about by the crucial subspace constraints. Extensive numerical results for various large scale QSDPs show that our two-phase algorithm is highly efficient and robust in obtaining accurate solutions.

关键词： quadratic semidefinite programming Schur complement Augmented Lagrangian Inexact semismooth Newton method

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A sensitivity result for quadratic semidefinite programs with an application to a sequential quadratic semidefinite programming algorithm

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COMPUTATIONAL & APPLIED MATHEMATICS 2012年第1期31卷 205-218页

作者： Garces, Rodrigo Gomez, Walter Jarre, Florian Univ La Frontera Dept Engn Math Temuco 01145 Chile EuroAmerica SA Santiago Chile Univ Dusseldorf Inst Math D-40225 Dusseldorf Germany

In this short note a sensitivity result for quadratic semidefinite programming is presented under a weak form of second order sufficient condition. Based on this result, also the local convergence of a sequential quad... 详细信息

关键词： semidefinite programming second order sufficient condition sequential quadratic programming quadratic semidefinite programming sensitivity convergence

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Alternating direction method for bi-quadratic programming

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JOURNAL OF GLOBAL OPTIMIZATION 2011年第3期51卷 429-446页

作者： Hu, Sheng-Long Huang, Zheng-Hai Tianjin Univ Sch Sci Dept Math Tianjin 300072 Peoples R China

Bi-quadratic programming (Bi-QP for short) was studied systematically in Ling et al. (SIAM J. Optim. 20:1286-1320, 2009) due to its various applications in engineering as well as optimization. Several approximation methods were given in the same paper since it is NP-hard. In this paper, we introduce a quadratic SDP relaxation of Bi-QP and discuss the approximation ratio of the method. In particular, by exploiting the favorite structure of the quadratic SDP relaxation, we propose an alternating direction method for solving such a problem and show that the method is globally convergent without any assumption. Some preliminary numerical results are reported which show the effectiveness of the method proposed in this paper.

关键词： Alternating direction method Bi-quadratic programming quadratic semidefinite programming

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Convergent prediction-correction-based ADMM for multi-block separable convex programming

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2018年 335卷 270-288页

作者： Chang, Xiaokai Liu, Sanyang Zhao, Pengjun Li, Xu Xidian Univ Sch Math & Stat Xian 710071 Shaanxi Peoples R China Lanzhou Univ Technol Coll Sci Lanzhou 730050 Gansu Peoples R China Shanglue Univ Sch Math & Comp Applicat Shanglue 726000 Peoples R China

The direct extension of the classic alternating direction method with multipliers (ADMMe) to the multi-block separable convex optimization problem is not necessarily convergent, though it often performs very well in practice. In order to preserve the numerical advantages of ADMMe and obtain convergence, many modified ADMM were proposed by correcting the output of ADMMe or employing proximal terms to solve inexactly the subproblems in ADMMe. In this paper, we present an efficient Prediction-Correction-based ADMM (PCB-ADMM) to solve the multi-block separable convex minimization model. The prediction step takes a special block coordinate descent (BCD) cycle to update the variable blocks, then the correction step corrects the output slightly by computing a convex combination of two points from the prediction step and previous iteration. The convergence property is obtained by using the variational inequality. The numerical experiments illustrate effectiveness of the proposed PCB-ADMM to solve the quadratic semidefinite programming and image decomposition. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Alternating direction method of multipliers Prediction-correction Variational inequality Convergence analysis quadratic semidefinite programming Image decomposition

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A SEQUENTIAL SEMISMOOTH NEWTON METHOD FOR THE NEAREST LOW-RANK CORRELATION MATRIX PROBLEM

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SIAM JOURNAL ON OPTIMIZATION 2011年第4期21卷 1641-1666页

作者： Li, Qingna Qi, Hou-Duo Hunan Univ Coll Math & Econometr Changsha 410082 Hunan Peoples R China Univ Southampton Sch Math Southampton SO17 1BJ Hants England

Based on the well-known result that the sum of the largest eigenvalues of a symmetric matrix can be represented as a semidefinite programming problem (SDP), we formulate the nearest low-rank correlation matrix problem as a nonconvex SDP and propose a numerical method that solves a sequence of least-square problems. Each of the least-square problems can be solved by a specifically designed semismooth Newton method, which is shown to be quadratically convergent. The sequential method is guaranteed to produce a stationary point of the nonconvex SDP. Our numerical results demonstrate the high efficiency of the proposed method on large scale problems.

关键词： low-rank correlation matrix quadratic semidefinite programming semismooth Newton method quadratic convergence constraint nondegeneracy

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A primal majorized semismooth Newton-CG augmented Lagrangian method for large-scale linearly constrained convex programming

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2017年第3期68卷 503-532页

作者： Wang, Chengjing Tang, Peipei Southwest Jiaotong Univ Sch Math 999 Xian RdWest PkHightech Zone Chengdu 611756 Sichuan Peoples R China Zhejiang Univ City Coll Sch Comp Sci Hangzhou 310015 Zhejiang Peoples R China

In this paper, we propose a primal majorized semismooth Newton-CG augmented Lagrangian method for large-scale linearly constrained convex programming problems, especially for some difficult problems. The basic idea of this method is to apply the majorized semismooth Newton-CG augmented Lagrangian method to the primal convex problem. And we take two special nonlinear semidefinite programming problems as examples to illustrate the algorithm. Furthermore, we establish the global convergence and the iteration complexity of the algorithm. Numerical experiments demonstrate that our method works very well for the testing problems, especially for many ill-conditioned ones.

关键词： Majorized semismooth Newton-CG augmented Lagrangian method Iteration complexity quadratic semidefinite programming

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A projected semismooth Newton method for problems of calibrating least squares covariance matrix

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OPERATIONS RESEARCH LETTERS 2011年第2期39卷 103-108页

作者： Li, Qingna Li, Donghui S China Normal Univ Sch Math Sci Guangzhou 510631 Guangdong Peoples R China Chinese Acad Sci Inst Computat Math & Sci Engn Comp Beijing 100080 Peoples R China

In this paper we propose a projected semismooth Newton method to solve the problem of calibrating least squares covariance matrix with equality and inequality constraints. The method is globally and quadratically convergent with proper assumptions. The numerical results show that the proposed method is efficient and comparable with existing methods. (c) 2011 Elsevier B.V. All rights reserved.

关键词： Semismooth Newton method Active set Constraint nondegeneracy quadratic semidefinite programming

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A lower bound for the nearest correlation matrix problem based on the circulant mean

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COMPUTATIONAL & APPLIED MATHEMATICS 2014年第1期33卷 27-44页

作者： Travaglia, M. V. Univ Fed Piaui Ctr Ciencias Nat Dept Matemat BR-64049550 Teresina PI Brazil

This paper deals with the problem of finding the nearest correlation matrix to a given n x n symmetric matrix . We say that is a correlation matrix if it is symmetric positive semidefinite whose diagonal entries are equal to . The nearest correlation matrix to is the one that minimizes its distance to with relation to the Euclidean or Frobenius norm. The circulant mean of , denoted by , is the matrix obtained by averaging over the diagonals of , with the diagonals being extended to the length by a wrap-around. It turns out (due to a simple application of Jensen's trace inequality) that the distance from to its nearest correlation matrix will be a lower bound for the original distance. As the main result of this work, we compute the distance from to its nearest correlation matrix by relating it to the eigenvalues of . A motivation of the choice of the circulant mean is the following: we apply this lower bound and a related upper bound to the special case of the Laplacian matrix with Dirichlet boundary conditions (a three-band Toeplitz matrix, which is not circulant, that is, is not equal to its circulant mean). In this application we show that the distance to its nearest correlation matrix behaves asymptotically as with C-assymp approximate to 1.1165.

关键词： Matrix nearness problem Minimization problem involving matrices Convex programming quadratic semidefinite programming Circulant matrices Laplacian matrices

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