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SDO and LDO relaxation approaches to complex fractional quadratic optimization

RAIRO-OPERATIONS RESEARCH 2021年第1-Sup期55卷 S2241-S2258页

作者： Ashrafi, Ali Zare, Arezu Semnan Univ Dept Math Stat & Comp Sci Semnan Iran

This paper examines a complex fractional quadratic optimization problem subject to two quadratic constraints. The original problem is transformed into a parametric quadratic programming problem by the well-known classical Dinkelbach method. Then a semidefinite and Lagrangian dual optimization approaches are presented to solve the nonconvex parametric problem at each iteration of the bisection and generalized Newton algorithms. Finally, the numerical results demonstrate the effectiveness of the proposed approaches.

关键词： Fractional quadratic optimization nonconvex problem semidefinite programming Lagrangian dual optimization

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TDOA-Based Source Collaborative Localization via semidefinite Relaxation in Sensor Networks

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INTERNATIONAL JOURNAL OF DISTRIBUTED SENSOR NETWORKS 2015年第9期11卷 1-16页

作者： Yan, Yongsheng Wang, Haiyan Shen, Xiaohong He, Ke Zhong, Xionghu Northwestern Polytech Univ Sch Marine Sci & Technol Xian 710072 Shaanxi Peoples R China Nanyang Technol Univ Sch Elect & Elect Engn Singapore 639798 Singapore

The time delay of arrival- (TDOA-) based source localization using a wireless sensor network has been considered in this paper. The maximum likelihood estimate (MLE) is formulated by taking the correlated TDOA noise into account, which is caused by the difference with the TOA of the reference sensor. The global optimal solution is difficult to obtain due to the nonconvex nature of the ML function. We propose an alternative semidefinite programming method, which transforms the original ML problem into a convex one by relaxing nonconvex equalities into convex matrix inequalities. In addition, the source localization algorithm in the presence of sensor location errors and non-line-of-sight (NLOS) observations is developed. Our simulation results demonstrate the potential advantages of the proposed method. Furthermore, the proposed source localization algorithm by taking the NLOS TOA measurements as the constraints of the convex problem can provide a good estimate.

关键词： WIRELESS sensor networks RELAXATION methods (Mathematics) semidefinite programming TIME delay systems MAXIMUM likelihood statistics INEQUALITIES (Mathematics)

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Undrained stability of unsupported rectangular excavations in non-homogeneous clays

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COMPUTERS AND GEOTECHNICS 2020年 117卷 103281-000页

作者： Ukritchon, Boonchai Yoang, Sothoan Keawsawasvong, Suraparb Chulalongkorn Univ Fac Engn Ctr Excellence Geotech & Geoenvironm Engn Dept Civil Engn Bangkok 10330 Thailand Thammasat Univ Thammasat Sch Engn Dept Civil Engn Pathum Thani 12120 Thailand

Solutions for the undrained stability of unsupported excavations are important in practice as they can be used for assessing the safety of temporary excavations associated with various civil works. Even though the previous stability solutions of unsupported excavations namely infinitely long trenches and circular excavations are available in the literature, there is a lack of the stability solution of unsupported rectangular ones. In this paper, the lower bound solutions for the undrained stability of unsupported rectangular excavations in non-homogeneous clays are presented for the first time. A three-dimensional lower bound finite element limit analysis is developed in order to investigate the stability of this problem. The undrained shear strength of non-homogeneous clays are considered as a linearly increasing one with depth. The effects of aspect ratios of rectangular excavations, excavated depth ratios, and normalized strength gradients on the stability number of the problem and its associated failure mechanisms are examined parametrically. A new design equation of the problem is also firstly presented for practical use by practising engineers.

关键词： Stability Excavations Non-homogeneous clays Lower bound Finite element limit analysis semidefinite programming

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Time-Varying semidefinite Programs

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MATHEMATICS OF OPERATIONS RESEARCH 2021年第3期46卷 1054-1080页

作者： Ahmadi, Amir Ali El Khadir, Bachir Princeton Univ Dept Operat Res & Financial Engn Princeton NJ 08544 USA

We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data vary polynomially with time. We show that under a strict feasibility assumption, restricting the solutions to also be polynomial functions of time does not change the optimal value of the TV-SDP. Moreover, by using a Positivstellensatz (positive locus theorem) on univariate polynomial matrices, we show that the best polynomial solution of a given degree to a TV-SDP can be found by solving a semidefinite program of tractable size. We also provide a sequence of dual problems that can be cast as SDPs and that give upper bounds on the optimal value of a TV-SDP (in maximization form). We prove that under a boundedness assumption, this sequence of upper bounds converges to the optimal value of the TV-SDP. Under the same assumption, we also show that the optimal value of the TV-SDP is attained. We demonstrate the efficacy of our algorithms on a maximum-flow problem with time-varying edge capacities, a wireless coverage problem with time-varying coverage requirements, and on biobjective semidefinite optimization where the goal is to approximate the Pareto curve in one shot.

关键词： semidefinite programming time-varying convex optimization univariate polynomial matrices Positivstellensatz continuous linear programs biobjective optimization

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Inverse linear-quadratic discrete-time finite-horizon optimal control for indistinguishable homogeneous agents: A convex optimization approach

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AUTOMATICA 2023年 148卷

作者： Zhang, Han Ringh, Axel Shanghai Jiao Tong Univ Sch Elect Informat & Elect Engn Dept Automat Shanghai Peoples R China Minist Educ China Key Lab Syst Control & Informat Proc Shanghai 200240 Peoples R China Shanghai Engn Res Ctr Intelligent Control & Manag Shanghai 200240 Peoples R China Chalmers Univ Technol Dept Math Sci S-41296 Gothenburg Sweden Univ Gothenburg S-41296 Gothenburg Sweden Hong Kong Univ Sci & Technol Dept Elect & Comp Engn Kowloon Clear Water Bay Hong Kong Peoples R China

The inverse linear-quadratic optimal control problem is a system identification problem whose aim is to recover the quadratic cost function and hence the closed-loop system matrices based on observations of optimal trajectories. In this paper, the discrete-time, finite-horizon case is considered, where the agents are also assumed to be homogeneous and indistinguishable. The latter means that the agents all have the same dynamics and objective functions and the observations are in terms of "snap shots" of all agents at different time instants, but what is not known is "which agent moved where" for consecutive observations. This absence of linked optimal trajectories makes the problem challenging. We first show that this problem is globally identifiable. Then, for the case of noiseless observations, we show that the true cost matrix, and hence the closed-loop system matrices, can be recovered as the unique global optimal solution to a convex optimization problem. Next, for the case of noisy observations, we formulate an estimator as the unique global optimal solution to a modified convex optimization problem. Moreover, the statistical consistency of this estimator is shown. Finally, the performance of the proposed method is demonstrated by a number of numerical examples. (c) 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://***/licenses/by-nc-nd/4.0/).

关键词： Inverse optimal control Linear quadratic regulator System identification Closed-loop identification Time-varying system matrices Convex optimization semidefinite programming

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Extended non-local games and monogamy-of-entanglement games

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PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 2016年第2189期472卷 20160003-20160003页

作者： Johnston, Nathaniel Mittal, Rajat Russo, Vincent Watrous, John Mt Allison Univ Dept Math & Comp Sci Sackville NB E4L 1E4 Canada Univ Waterloo Inst Quantum Comp Dept Combinator & Optimizat Dept Combinator & Optimizat Waterloo ON N2L 3G1 Canada Univ Waterloo Sch Comp Sci Inst Quantum Comp Waterloo ON N2L 3G1 Canada IIT Kanpur Dept Comp Sci & Engn Kanpur 208016 Uttar Pradesh India Canadian Inst Adv Res Toronto ON M5G 1Z8 Canada

We study a generalization of non-local games-which we call extended non-local games-in which the players, Alice and Bob, initially share a tripartite quantum state with the referee. In such games, the winning conditions for Alice and Bob may depend on the outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to randomly selected questions. Our study of this class of games was inspired by the monogamy-of-entanglement games introduced by Tomamichel, Fehr, Kaniewski and Wehner, which they also generalize. We prove that a natural extension of the Navascues-Pironio-Acin hierarchy of semidefinite programmes converges to the optimal commuting measurement value of extended non-local games, and we prove two extensions of results of Tomamichel et al. concerning monogamy-of-entanglement games.

关键词： non-local games semidefinite programming monogamy-of-entanglement quantum steering extended non-local games

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Optimal Cardinality Constrained Portfolio Selection

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OPERATIONS RESEARCH 2013年第3期61卷 745-761页

作者： Gao, Jianjun Li, Duan Shanghai Jiao Tong Univ Dept Automat Shanghai 200030 Peoples R China Chinese Univ Hong Kong Dept Syst Engn & Engn Shatin Hong Kong Peoples R China

One long-standing challenge in both the optimization and investment communities is to devise an efficient algorithm to select a small number of assets from an asset pool such that a portfolio objective is optimized. This cardinality constrained investment situation naturally arises due to the presence of various forms of market friction, such as transaction costs and management fees, or even due to the consideration of mental cost. Unfortunately, the combinatorial nature of such a portfolio selection problem formulation makes the exact solution process NP-hard in general. We focus in this paper on the cardinality constrained mean-variance portfolio selection problem. Instead of tailoring such a difficult problem into the general solution framework of mixed-integer programming formulation, we explore the special structures and rich geometric properties behind the mathematical formulation. Applying the Lagrangian relaxation to the primal problem results in a pure cardinality constrained portfolio selection problem, which possesses a symmetric property, and to which geometric approaches can be developed. Different from the existing literature that has primarily focused on some direct relaxations of the cardinality constraint, we consider modifying the objective function to some separable relaxations, which are immune to the hard cardinality constraint. More specifically, we develop efficient lower bounding schemes by using the circumscribed box, the circumscribed ball, and the circumscribed axis-aligned ellipsoid to approximate the objective contour of the problem. In particular, all these cardinality constrained relaxation problems can be solved analytically. Furthermore, we derive efficient polynomial-time algorithms for the corresponding dual search problems. Most promisingly, the lower bounding scheme using the circumscribed axis-aligned ellipsoid leads to a semidefinite programming (SDP) formulation and offers a sharp bound and high-quality feasible solution. By i

关键词： cardinality constrained quadratic programming cardinality constrained portfolio selection mean-variance formulation semidefinite programming relaxation branch-and-bound

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Applications of the alternating direction method of multipliers to the semidefinite inverse quadratic eigenvalue problem with a partial eigenstructure

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INVERSE PROBLEMS 2013年第7期29卷 075011-75011页

作者： Bai, Zhengjian Chen, Meixiang Yuan, Xiaoming Xiamen Univ Sch Math Sci Xiamen 361005 Peoples R China Huaqiao Univ Sch Math Sci Quanzhou 362021 Peoples R China Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China

This paper shows that the alternating direction method of multipliers (ADMM) is efficient for solving the semidefinite inverse quadratic eigenvalue problem (SDIQEP) with a partial eigenstructure. We derive several ADM... 详细信息

关键词： MULTIPLIERS (Mathematical analysis) RESEARCH semidefinite programming EIGENFUNCTIONS INVERSE problems (Differential equations) ITERATIVE methods (Mathematics)

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Strong SDP based bounds on the cutwidth of a graph

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COMPUTERS & OPERATIONS RESEARCH 2024年 161卷

作者： Gaar, Elisabeth Puges, Diane Wiegele, Angelika Univ Augsburg Univ Str 14 D-86159 Augsburg Germany Johannes Kepler Univ Linz Altenberger Str 69 A-4040 Linz Austria Alpen Adria Univ Klagenfurt Unv Str 65-67 A-9020 Klagenfurt Austria Univ Cologne Albertus Magnus Pl D-50923 Cologne Germany

Given a linear ordering of the vertices of a graph, the cutwidth of a vertex nu with respect to this ordering is the number of edges from any vertex before nu (including nu) to any vertex after nu in this ordering. The cutwidth of an ordering is the maximum cutwidth of any vertex with respect to this ordering. We are interested in finding the cutwidth of a graph, that is, the minimum cutwidth over all orderings, which is an NP-hard problem. In order to approximate the cutwidth of a given graph, we present a semidefinite relaxation. We identify several classes of valid inequalities and equalities that we use to strengthen the semidefinite relaxation. These classes are on the one hand the well-known 3-dicycle equations and the triangle inequalities and on the other hand we obtain inequalities from the squared linear ordering polytope and via lifting the linear ordering polytope. The solution of the semidefinite program serves to obtain a lower bound and also to construct a feasible solution and thereby having an upper bound on the cutwidth. In order to evaluate the quality of our bounds, we perform numerical experiments on graphs of different sizes and densities. It turns out that we produce high quality bounds for graphs of medium size independent of their density in reasonable time. Compared to that, obtaining bounds for dense instances of the same quality is out of reach for solvers using integer linear programming techniques.

关键词： Cutwidth Linear ordering semidefinite programming Combinatorial optimization

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Comparison of Two Structure-Exploiting Optimization Algorithms for Integral Quadratic Constraints

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IFAC Proceedings Volumes 2003年第11期36卷 199-204页

作者： Ragnar Wallin Anders Hansson Lieven Vandenberghe Department of Electrical Engineering Division of Automatic Control Linköping University SE-581 83 Linköping Sweden Electrical Engineering Department University of California Los Angeles Los Angeles CA 90095-1594 USA

As the semidefinite programs that result from integral quadratic constraints are usually large it is important to implement efficient algorithms. The interior-point algorithms in this paper are primal-dual potential reduction methods and handle multiple constraints. Two approaches are made. For the first approach the computational cost is dominated by a least-squares problem that has to be solved in each iteration. The least-squares problem is solved using an iterative method, namely the conjugate gradient method. The computational effort for the second approach is dominated by forming a linear system of equations. This system of equations is used to compute exact search direction in each iteration. If the number of variables are reduced by solving a small subproblem the resulting system has a very nice structure and can be solved efficiently. The first approach is more efficient for larger problems but is not as numerically stable.

关键词： semidefinite programming Robustness analysis

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