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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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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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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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A preconditioned iterative interior point approach to the conic bundle subproblem

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MATHEMATICAL programming 2024年第1-2期205卷 559-615页

作者： Helmberg, Christoph Tech Univ Chemnitz Dept Math D-09107 Chemnitz Germany

The conic bundle implementation of the spectral bundle method for large scale semidefinite programming solves in each iteration a semidefinite quadratic subproblem by an interior point approach. For larger cutting model sizes the limiting operation is collecting and factorizing a Schur complement of the primal-dual KKT system. We explore possibilities to improve on this by an iterative approach that exploits structural low rank properties. Two preconditioning approaches are proposed and analyzed. Both might be of interest for rank structured positive definite systems in general. The first employs projections onto random subspaces, the second projects onto a subspace that is chosen deterministically based on structural interior point properties. For both approaches theoretic bounds are derived for the associated condition number. In the instances tested the deterministic preconditioner provides surprisingly efficient control on the actual condition number. The results suggest that for large scale instances the iterative solver is usually the better choice if precision requirements are moderate or if the size of the Schur complemented system clearly exceeds the active dimension within the subspace giving rise to the cutting model of the bundle method.

关键词： Low rank preconditioner quadratic semidefinite programming Nonsmooth optimization Interior point method

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Convergence of a Weighted Barrier Algorithm for Stochastic Convex quadratic semidefinite Optimization

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2023年第2期196卷 490-515页

作者： Alzalg, Baha Gafour, Asma Univ Jordan Dept Math Amman 11942 Jordan Ohio State Univ Dept Comp Sci & Engn Columbus OH 43210 USA Univ Djillali Liabes Sidi Bel Abbes Dept Math Sidi Bel Abbes 22038 Algeria

Mehrotra and ozevin (SIAM J Optim 19:1846-1880, 2009) computationally found that a weighted barrier decomposition algorithm for two-stage stochastic conic programs achieves significantly superior performance when compared to standard barrier decomposition algorithms existing in the literature. Inspired by this motivation, Mehrotra and ozevin (SIAM J Optim 20:2474-2486, 2010) theoretically analyzed the iteration complexity for a decomposition algorithm based on the weighted logarithmic barrier function for two-stage stochastic linear optimization with discrete support. In this paper, we extend the aforementioned theoretical paper and its self-concordance analysis from the polyhedral case to the semidefinite case and analyze the iteration complexity for a weighted logarithmic barrier decomposition algorithm for two-stage stochastic convex quadratic SDP with discrete support.

关键词： quadratic semidefinite programming Two-stage stochastic programming Large-scale optimization Interior-point methods Decomposition

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A semidefinite programming approach for the projection onto the cone of negative semidefinite symmetric tensors with applications to solid mechanics

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CALCOLO 2022年第4期59卷 33-33页

作者： Padovani, Cristina Porcelli, Margherita ISTI CNR Inst Informat Sci & Technol A Faedo Via Moruzzi 1 Pisa Italy Univ Bologna AM2 Dipartimento Matemat Piazza Porta San Donato 5 I-40126 Bologna Italy

We propose an algorithm for computing the projection of a symmetric second-order tensor onto the cone of negative semidefinite symmetric tensors with respect to the inner product defined by an assigned positive definite symmetric fourth-order tensor C. The projection problem is written as a semidefinite programming problem and an algorithm based on a primal-dual path-following interior point method coupled with a Mehrotra's predictor-corrector approach is proposed. Implementations based on well-known symmetrization schemes and on direct methods are theoretically and numerically investigated taking into account tensors C arising in the modelling of masonry-like materials. For these special cases, indications on the preferable symmetrization scheme that take into account the conditioning of the arising linear systems are given.

关键词： Conic projection Negative semidefinite tensors quadratic semidefinite programming Interior point methods

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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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An Extension of Yuan's Lemma to Fourth-Order Tensor System

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第3期180卷 803-810页

作者： Yang, Qingzhi Zhou, Yang Yang, Yuning Kashgar Univ Sch Math & Stat Kashgar 844008 Peoples R China Nankai Univ Sch Math Sci Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China Guangxi Univ Sch Math Sci Nanning 530004 Peoples R China

Yuan's lemma is a basic proposition on homogeneous quadratic function systems. In this note, we extend Yuan's lemma to the fourth-order tensor system. We first give generalized definition of positive semidefinite of fourth-order tensor, and based on it, an extension of Yuan's lemma is proposed. As an application, we establish the strong duality result of a class of quadratic semidefinite programming problems using the extended Yuan's lemma.

关键词： Yuan's lemma S-lemma Fourth-order tensor Symmetry Positive semidefinite 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 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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