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Null broadening adaptive beamforming based on covariance matrix reconstruction and similarity constraint

EURASIP JOURNAL ON ADVANCES IN SIGNAL PROCESSING 2017年第1期2017卷 1页

作者： Qian, Junhui He, Zishu Xie, Julan Zhang, Yile Univ Elect Sci & Technol China Sch Elect Engn 2006 Xi Yuan Ave Chengdu Peoples R China

In this paper, a procedure for the null broadening algorithm design with respect to the nonstationary interference is proposed. In contrast to previous works, we first impose nulls toward the regions of the nonstationary interference based on the reconstruction of the interference-plus-noise covariance matrix. Additionally, in order to provide a restriction on the shape of the beam pattern, a similarity constraint is enforced at the design stage. Then, the adaptive weight vector can be computed via maximizing a new signal-to-interference-plus-noise ratio (SINR) criterion subject to similarity constraint. Mathematically, the design original problem is expressed as a nonconvex fractional quadratically constrained quadratic programming (QCQP) problem with additional constraint, which can be converted into a convex optimisation problem by semidefinite programming (SDP) techniques. Finally, an optimal solution can be found by using the Charnes-Cooper transformation and the rank-one matrix decomposition theorem. Several numerical examples are performed to validate the performance of the proposed algorithm.

关键词： Robust adaptive beamforming Similarity constraint quadratically constrained quadratic programming Semidefinite programming

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Optimal fractionation in radiotherapy with multiple normal tissues

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MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2016年第2期33卷 211-252页

作者： Saberian, Fatemeh Ghate, Archis Kim, Minsun Univ Washington Ind & Syst Engn Box 352650 Seattle WA 98195 USA Univ Washington Radiat Oncol Seattle WA 98195 USA

The goal in radiotherapy is to maximize the biological effect (BE) of radiation on the tumour while limiting its toxic effects on healthy anatomies. Treatment is administered over several sessions to give the normal tissue time to recover as it has better damage-repair capabilities than tumour cells. This is termed fractionation. A key problem in radiotherapy involves finding an optimal number of treatment sessions (fractions) and the corresponding dosing schedule. A major limitation of existing mathematically rigorous work on this problem is that it includes only a single normal tissue. Since essentially no anatomical region of interest includes only one normal tissue, these models may incorrectly identify the optimal number of fractions and the corresponding dosing schedule. We present a formulation of the optimal fractionation problem that includes multiple normal tissues. Our model can tackle any combination of maximum dose, mean dose and dose-volume type constraints for serial and parallel normal tissues as this is characteristic of most treatment protocols. We also allow for a spatially heterogeneous dose distribution within each normal tissue. Furthermore, we do not a priori assume that the doses are invariant across fractions. Finally, our model uses a spatially optimized treatment plan as input and hence can be seamlessly combined with any treatment planning system. Our formulation is a mixed-integer, non-convex, quadratically constrained quadratic programming problem. In order to simplify this computationally challenging problem without loss of optimality, we establish sufficient conditions under which equal-dosage or single-dosage fractionation is optimal. Based on the prevalent estimates of tumour and normal tissue model parameters, these conditions are expected to hold in many types of commonly studied tumours, such as those similar to head-and-neck and prostate cancers. This motivates a simple reformulation of our problem that leads to a closed-form fo

关键词： intensity modulated radiation therapy quadratically constrained quadratic programming linear-quadratic cell survival model

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NONCONVEX PHASE SYNCHRONIZATION

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SIAM JOURNAL ON OPTIMIZATION 2016年第4期26卷 2355-2377页

作者： Boumal, Nicolas Princeton Univ Dept Math Princeton NJ 08544 USA

We estimate n phases (angles) from noisy pairwise relative phase measurements. The task is modeled as a nonconvex least-squares optimization problem. It was recently shown that this problem can be solved in polynomial time via convex relaxation, under some conditions on the noise. In this paper, under similar but more restrictive conditions, we show that a modified version of the power method converges to the global optimum. This is simpler and (empirically) faster than convex approaches. Empirically, they both succeed in the same regime. Further analysis shows that, in the same noise regime as previously studied, second-order necessary optimality conditions for this quadratically constrained quadratic program are also sufficient, despite nonconvexity.

关键词： angular synchronization nonconvex optimization generalized power method projected power method sufficient optimality conditions quadratically constrained quadratic programming optimization on manifolds

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Maximizing the sum of a generalized Rayleigh quotient and another Rayleigh quotient on the unit sphere via semidefinite programming

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JOURNAL OF GLOBAL OPTIMIZATION 2016年第2期64卷 399-416页

作者： Van-Bong Nguyen Sheu, Ruey-Lin Xia, Yong Natl Cheng Kung Univ Dept Math Tainan 701 Taiwan Beihang Univ Sch Math & Syst Sci State Key Lab Software Dev Environm LMIB Minist Educ Beijing 100191 Peoples R China

The problem is a type of "sum-of-ratios" fractional programming and is known to be NP-hard. Due to many local maxima, finding the global maximizer is in general difficult. The best attempt so far is a critical point approach based on a necessary optimality condition. The problem therefore has not been completely solved. Our novel idea is to replace the generalized Rayleigh quotient by a parameter and generate a family of quadratic subproblems subject to two quadratic constraints. Each , if the problem dimension , can be solved in polynomial time by incorporating a version of S-lemma;a tight SDP relaxation;and a matrix rank-one decomposition procedure. Then, the difficulty of the problem is largely reduced to become a one-dimensional maximization problem over an interval of parameters . We propose a two-stage scheme incorporating the quadratic fit line search algorithm to find numerically. Computational experiments show that our method solves the problem correctly and efficiently.

关键词： Fractional programming (Generalized) Rayleigh quotient quadratically constrained quadratic programming S-Lemma Semidefinite programming quadratic fit line search

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SOLVING GENERALIZED CDT PROBLEMS VIA TWO-PARAMETER EIGENVALUES

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SIAM JOURNAL ON OPTIMIZATION 2016年第3期26卷 1669-1694页

作者： Sakaue, Shinsaku Nakatsukasa, Yuji Takeda, Akiko Iwata, Satoru Univ Tokyo Dept Math Informat Tokyo 1138656 Japan

We consider solving a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. This problem is a generalization of the Celis-Denis-Tapia (CDT) problem and thus we refer to it as GCDT (Generalized CDT). The CDT problem has been widely studied, but no polynomial-time algorithm was known until Bienstock's recent work. His algorithm solves the CDT problem in polynomial time with respect to the number of bits in data and log is an element of(-1) by admitting an is an element of error in the constraints. The algorithm, however, appears to be difficult to implement. In this paper, we present another algorithm for GCDT, which is guaranteed to find a global solution for almost all GCDT instances (and slightly perturbed ones in some exceptionally rare cases), in exact arithmetic (including eigenvalue computation). Our algorithm is based on the approach proposed by Iwata, Nakatsukasa, and Takeda (2015) for computing the signed distance between overlapping ellipsoids. Our algorithm computes all the Lagrange multipliers of GCDT by solving a two-parameter linear eigenvalue problem, obtains the corresponding KKT points, and finds a global solution as the KKT point with the smallest objective value. In practice, in finite precision arithmetic, our algorithm requires O(n(6) log log u(-1)) computational time, where n is the number of variables and u is the unit roundoff. Although we derive our algorithm under the unrealistic assumption that exact eigenvalues can be computed, numerical experiments illustrate that our algorithm performs well in finite precision arithmetic.

关键词： quadratically constrained quadratic programming nonconvex optimization Celis-Dennis-Tapia problem two-parameter eigenvalue problem

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A decomposition method for nonconvex quadratically constrained quadratic programs

A decomposition method for nonconvex quadratically constrain...

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American Control Conference

作者： Chuangchuang Sun Ran Dai Aerospace Engineering Department Iowa State University Ames United States of America

ISBN: (纸本)9781509045839

This paper examines the nonconvex quadratically constrained quadratic programming (QCQP) problems using a decomposition method. It is well known that a QCQP can be transformed into a rank-one constrained optimization problem. Finding a rank-one matrix is computationally complicated, especially for large scale QCQPs. A decomposition method is applied to decompose the single rank-one constraint on original unknown matrix into multiple rank-one constraints on small scale submatrices. An iterative rank minimization (IRM) algorithm is then proposed to gradually approach all of the rank-one constraints. To satisfy each rank-one constraint in the decomposed formulation, linear matrix inequalities (LMIs) are introduced in IRM with local convergence analysis. The decomposition method reduces the overall computational cost by decreasing size of LMIs, especially when the problem is sparse. Simulation examples with comparative results obtained from an alternative method are presented to demonstrate advantages of the proposed method.

关键词： quadratically constrained quadratic programming Rank Constraint Optimization Semidefinite programming Matrix Decomposition Sparse Matrix

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Null Broadening Adaptive Beamforming Based on Semidefinite programming 13

Null Broadening Adaptive Beamforming Based on Semidefinite P...

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13th IEEE International Conference on Signal Processing (ICSP)

作者： Qian, Junhui He, Zishu Zhang, Yile Wu, Bo Univ Elect Sci & Technol China Dept EE Chengdu 611731 Sichuan Peoples R China Georgia Inst Technol Atlanta GA 30332 USA

ISBN: (纸本)9781509013456

In this paper, we introduce a procedure for the null broadening algorithm design with respect to the perturbation of the interference location. This method is based on maximizing the array output signal-to-interference-plus-noise-ratio (SINR) subject to quadratic constraints. The design problem can be cast as a fractional quadratically constrained quadratic programming (QCQP) problem that can be solved efficiently using the semidefinite programming (SDP) techniques, the semidefinite relaxation can be used to obtain a lower bound on the optimal objective function. This proposed approach imposes broadened nulls towards the interference region while possesses well-maintained pattern. Theoretical analysis and numerical results demonstrate that the performance of proposed adaptive beamformer is almost always close to optimal.

关键词： Robust adaptive beamforming beampattern shaping constraints quadratically constrained quadratic programming semidefinite programming

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Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming

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JOURNAL OF GLOBAL OPTIMIZATION 2009年第2-3期43卷 471-484页

作者： Anstreicher, Kurt M. Univ Iowa Dept Management Sci Tippie Coll Business Iowa City IA 52242 USA

We consider relaxations for nonconvex quadratically constrained quadratic programming (QCQP) based on semidefinite programming (SDP) and the reformulation-linearization technique (RLT). From a theoretical standpoint we show that the addition of a semidefiniteness condition removes a substantial portion of the feasible region corresponding to product terms in the RLT relaxation. On test problems we show that the use of SDP and RLT constraints together can produce bounds that are substantially better than either technique used alone. For highly symmetric problems we also consider the effect of symmetry-breaking based on tightened bounds on variables and/or order constraints.

关键词： Semidefinite programming Reformulation-linearization technique quadratically constrained quadratic programming

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Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

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OPTIMIZATION METHODS & SOFTWARE 2015年第1期30卷 215-249页

作者： Misener, Ruth Smadbeck, James B. Floudas, Christodoulos A. Princeton Univ Dept Chem & Biol Engn Princeton NJ 08544 USA Univ London Imperial Coll Sci Technol & Med Dept Chem Engn London SW7 2AZ England

The global mixed-integer quadratic optimizer, GloMIQO, addresses mixed-integer quadratically constrained quadratic programs (MIQCQP) to epsilon-global optimality. This paper documents the branch-and-cut framework integrated into GloMIQO 2. Cutting planes are derived from reformulation-linearization technique equations, convex multivariable terms, BB convexifications, and low- and high-dimensional edge-concave aggregations. Cuts are based on both individual equations and collections of nonlinear terms in MIQCQP. Novel contributions of this paper include: development of a corollary to Crama's [Concave extensions for nonlinear 0-1 maximization problems, Math. Program. 61 (1993), pp. 53-60] necessary and sufficient condition for the existence of a cut dominating the termwise relaxation of a bilinear expression;algorithmic descriptions for deriving each class of cut;presentation of a branch-and-cut framework integrating the cuts. Computational results are presented along with comparison of the GloMIQO 2 performance to several state-of-the-art solvers.

关键词： quadratically constrained quadratic programming cutting planes global optimization

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Response Vector constrained Robust LCMV Beamforming Based on Semidefinite programming

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2015年第21期63卷 5720-5732页

作者： Xu, Jingwei Liao, Guisheng Zhu, Shengqi Huang, Lei Xidian Univ Natl Lab Radar Signal Proc Xian 710071 Shaanxi Peoples R China Shenzhen Univ Coll Informat Engn Shenzhen 518060 Guangdong Peoples R China

Although linearly constrained minimum variance (LCMV) beamforming is robust against imprecise target information, it usually leads to relatively high sidelobe and distorted mainlobe which would induce a high false alarm probability. To circumvent this problem, this work devises a novel robust LCMV beamforming approach by utilizing response vector optimization. It intends to find the optimal response vector in lieu of the all-one response vector in traditional LCMV beamformer. The proposed robust beamformer is first formulated as a nonconvex quadratically constrained quadratic programming problem, and then transformed into a semidefinite programming problem which can be efficiently and exactly solved. The proposed beamformer not only improves the performance in terms of signal-to-interference-plus-noise ratio substantially, but also possesses low sidelobe and well-maintained mainlobe. Moreover, since the response vector is quite small in size, the complexity of calculating the optimal response vector is negligible. Additionally, the proposed beamformer is also extended to two-dimensional space-time adaptive processing. Simulation results are presented to demonstrate the superiority of the proposed approach.

关键词： Robust adaptive beamforming linearly constrained minimum variance beamformer response vector optimization quadratically constrained quadratic programming semidefinite programming

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