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A hierarchy of semidefinite relaxations for completely positive tensor optimization problems

JOURNAL OF GLOBAL OPTIMIZATION 2019年第2期75卷 417-437页

作者： Zhou, Anwa Fan, Jinyan Shanghai Univ Dept Math Shanghai 200444 Peoples R China Shanghai Jiao Tong Univ Sch Math Sci Shanghai 200240 Peoples R China Shanghai Jiao Tong Univ MOE LSC Shanghai 200240 Peoples R China

In this paper, we study the completely positive (CP) tensor program, which is a linear optimization problem with the cone of CP tensors and some linear constraints. We reformulate it as a linear program over the cone of moments, then construct a hierarchy of semidefinite relaxations for solving it. We also discuss how to find a best CP approximation of a given tensor. Numerical experiments are presented to show the efficiency of the proposed methods.

关键词： CP tensor program Best CP tensor approximation Linear optimization with moments Nonnegative decomposition semidefinite program

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Doubly Autoparallel Structure on Positive Definite Matrices and Its Applications 4th

Doubly Autoparallel Structure on Positive Definite Matrices ...

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4th International SEE Conference on Geometric Science of Information (GSI)

作者： Ohara, Atsumi Univ Fukui 3-9-1 Bunkyo Fukui Fukui 9108507 Japan

ISBN: (纸本)9783030269807;9783030269791

In a statistical manifold, we can naturally define submanifolds that are simultaneously autoparallel with respect to both the primal and the dual affine connections of the statistical manifold. We call them doubly autoparallel submanifolds. The aim of this paper is to mainly introduce doubly autoparallelism on positive definite matrices in linear algebraic way and show its applicability to two related topics.

关键词： semidefinite program Structured covariance estimation Doubly autoparallelism Jordan subalgebra

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Spatio-Temporal Waveform Design in Active Sensing Systems with Multilayer Targets 27

Spatio-Temporal Waveform Design in Active Sensing Systems wi...

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27th European Signal Processing Conference (EUSIPCO)

作者： Kariminezhad, Ali Sezgin, Aydin Ruhr Univ Bochum RUB Inst Digital Commun Syst Bochum Germany

ISBN: (纸本)9789082797039

In this paper, we study the optimal spatio-temporal waveform design for active sensing applications. For this purpose a multi-antenna radar is exploited. The targets in the radar vision are naturally composed of multiple layers of different materials. Therefore, the interaction of these layers with the incident wave effects targets detection and classification. In order to enhance the quality of detection, we propose to exploit space-time waveforms which adapt with the targets multilayer response. We consider the backscattered signal power as the utility function to be maximized The backscattered signal power maximization under transmit signal power constraint is formulated as a semidefinite program (SDP). First, we assume a single-target scenario, where the resulting SDP yields an analytical solution. Second, we study the optimal waveform which considers the angle uncertainties of a target in the presence of a clutter. Third, having multiple targets and multiple clutters, the weighted sum of the backscattered signals power from the targets is maximized to deliver the backscattered power region outermost boundary. We observe that, when the targets material is given, the backscattered signal power can be significantly increased by optimal spatio-temporal waveform design. Moreover, we observe that by utilizing multiple temporal dimensions in the waveform design process, the number of exploited antennas can be significantly decreased.

关键词： Spatio-temporal waveform target material response semidefinite program Pareto boundary uncertainty region

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Optimal liquidation of financial derivatives

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FINANCE RESEARCH LETTERS 2020年 34卷 101233-101233页

作者： Chen, Jingnan Beihang Univ Sch Econ & Management 37 Xueyuan Rd Beijing 100191 Peoples R China

We propose a two-period robust optimization model for portfolio liquidation under a cash requirement that finds the least costly liquidation strategy. The basic asset return is assumed to belong to a scaled ellipsoid while the derivative return is modeled as a quadratic function of the underlying asset return via delta-gamma approximation. We show that the robust liquidation model is equivalent to a computationally tractable semidefinite program. We obtain analytical properties regarding how derivative Greek letters affect the optimal liquidation strategy.

关键词： Robust portfolio liquidation Financial derivatives Delta-gamma approximation Greek letters semidefinite program

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SOS-Convex Semialgebraic programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization

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SET-VALUED AND VARIATIONAL ANALYSIS 2018年第2期26卷 305-326页

作者： Chieu, N. H. Feng, J. W. Gao, W. Li, G. Wu, D. Univ New South Wales Sch Math Stat Sydney NSW 2052 Australia Vinh Univ Dept Math Vinh 42118 Nghe An Vietnam Univ New South Wales Sch Civil & Environm Engn Sydney NSW 2052 Australia

In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with a"" (1)-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semialgebraic programs can be found by solving a single semidefinite programming problem (SDP). We achieve the results by using tools from semialgebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we show that robust SOS-convex optimization proble ms under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers an open question in the literature on how to recover a robust solution of uncertain SOS-convex polynomial programs from its semidefinite programming relaxation in this broader setting.

关键词： Nonsmooth optimization Convex optimization SOS-convex polynomial semidefinite program Robust optimization

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Completely positive tensor recovery with minimal nuclear value

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2018年第2期70卷 419-441页

作者： Zhou, Anwa Fan, Jinyan Shanghai Univ Dept Math Shanghai 200444 Peoples R China Shanghai Jiao Tong Univ Sch Math Sci Shanghai 200240 Peoples R China

In this paper, we introduce the CP-nuclear value of a completely positive (CP) tensor and study its properties. A semidefinite relaxation algorithm is proposed for solving the minimal CP-nuclear-value tensor recovery. If a partial tensor is CP-recoverable, the algorithm can give a CP tensor recovery with the minimal CP-nuclear value, as well as a CP-nuclear decomposition of the recovered CP tensor. If it is not CP-recoverable, the algorithm can always give a certificate for that, when it is regular. Some numerical experiments are also presented.

关键词： Completely positive tensor Tensor recovery The CP-nuclear values Moment problem semidefinite program

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Tensor maximal correlation problems

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JOURNAL OF GLOBAL OPTIMIZATION 2018年第4期70卷 843-858页

作者： Zhou, Anwa Zhao, Xin Fan, Jinyan Bai, Yanqin Shanghai Univ Dept Math Shanghai 200444 Peoples R China Shanghai Jiao Tong Univ Sch Math Sci Shanghai 200240 Peoples R China Shanghai Jiao Tong Univ Sch Math Sci Shanghai 200240 Peoples R China Shanghai Jiao Tong Univ MOE LSC Shanghai 200240 Peoples R China

This paper studies the tensor maximal correlation problem, which aims at optimizing correlations between sets of variables in many statistical applications. We reformulate the problem as an equivalent polynomial optimization problem, by adding the first order optimality condition to the constraints, then construct a hierarchy of semidefinite relaxations for solving it. The global maximizers of the problem can be detected by solving a finite number of such semidefinite relaxations. Numerical experiments show the efficiency of the proposed method.

关键词： Tensor maximal correlation problems Polynomial optimization Lasserre relaxation semidefinite program

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Tensor eigenvalue complementarity problems

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MATHEMATICAL programMING 2018年第2期170卷 507-539页

作者： Fan, Jinyan Nie, Jiawang Zhou, Anwa Shanghai Jiao Tong Univ Sch Math Sci Shanghai 200240 Peoples R China Shanghai Jiao Tong Univ MOE LSC Shanghai 200240 Peoples R China Univ Calif San Diego Dept Math 9500 Gilman Dr La Jolla CA 92093 USA Shanghai Univ Dept Math Shanghai 200444 Peoples R China

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre's hierarchy of semidefinite relaxations. We show that it has finite convergence for generic tensors. Numerical experiments are presented to show the efficiency of proposed methods.

关键词： Tensor eigenvalues Eigenvalue complementarity Polynomial optimization Lasserre relaxation semidefinite program

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Security-Constrained Unit Commitment for AC-DC Grids With Generation and Load Uncertainty

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IEEE TRANSACTIONS ON POWER SYSTEMS 2018年第3期33卷 2717-2732页

作者： Bahrami, Shahab Wong, Vincent W. S. Univ British Columbia Dept Elect & Comp Engn Vancouver BC V6T 1Z4 Canada

The uncertainties in renewable generators and load demand make it a challenge for system operators to execute the security-constrained unit commitment (SCUC) program in an ac-dc grid. The SCUC is a nonlinear mixed-integer optimization problem due to the power flow equations, constraints imposed by the ac-dc converters, and the binary variables associated with the generators' on/off state. In this paper, we study the SCUC problem in ac-dc grids with generation and load uncertainty. We introduce the concept of conditional value-at-risk to limit the risk of deviations in the load demand and renewable generation. We relax the binary variables and introduce a l(1)-norm regularization term to the objective function, and then use convex relaxation techniques to transform the problem into a semidefinite program (SDP). We develop an algorithm based on the iterative reweighted l(1)-norm approximation that involves solving a sequence of SDPs. Simulations are performed on an IEEE 30-bus test system. Results show that the proposed algorithm returns a solution within 2% gap from the global optimal solution for the underlying test system. When compared with the multi-stage algorithm in the literature, our algorithm has a lower running time and returns a solution with a smaller gap from the global optimal solution.

关键词： Conditional value-at-risk l(1)-norm regularization security-constrained unit commitment semidefinite program

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Secure transmit optimization based on QoS for MISOMEs channel with artificial noise aided

Secure transmit optimization based on QoS for MISOMEs channe...

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Signal, Information and Data Processing (ICSIDP), IEEE International Conference on

作者： Xiang Gao Yong Li Wei Cheng Ge Shi Penglu Liu Yi Zheng School of Electronics and Information Northwestern Polytechnical University Xi'an Shaanxi China

ISBN: (数字)9781728123455

ISBN: (纸本)9781728123462

Consider the multi-input single-output multi-eavesdroppers (MISOMEs) wireless communication system. Our goal is to design an artificial noise (AN)-aided secure transmit strategy to safeguard this system from the perspective of quality-of-service (QoS). Specifically, we propose two design formulas: (1) total power minimization design under signal-to-interference-plus-noise ratio (SINR) constraints on receiver and eavesdroppers; (2) receiver's SINR maximization design under constraints of transmitted total power and eavesdroppers' SINRs. The proposed design problems jointly optimizing the information covariance and AN covariance are solved by semidefinite program (SDP) optimization approach. Through exploring the Karush-Kuhn-Tucker conditions of SDP problems, it is proved that transmit beamforming is the optimal strategy for confidential information transmission in the two designs. Finally, simulation results demonstrate that the proposed AN-aided approach in the two designs provides the best performance compared to isotropic AN and no-AN approach.

关键词： artificial noise power minimization SINR maximization semidefinite program transmit beamforming

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