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Polynomial Level-Set Method for Polynomial System Reachable Set Estimation

IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2013年第10期58卷 2508-2521页

作者： Wang, Ta-Chung Lall, Sanjay West, Matthew Natl Cheng Kung Univ Inst Civil Aviat Tainan 708 Taiwan Stanford Univ Dept Aeronaut & Astronaut Stanford CA 94305 USA Univ Illinois Dept Mech Sci & Engn Urbana IL 61801 USA

In this paper, we present a polynomial level-set method for advecting a semi-algebraic set for polynomial systems. This method uses the sub-level representation of sets. The problem of flowing these sets under the advection map of a dynamical system is converted to a semi-definite program, which is then used to compute the coefficients of the polynomials. The method presented in this paper does not require either the sets being positively invariant or star-shaped. Hence, the proposed algorithm can describe the behavior of system states both inside and outside the domain of attraction and can also be used to describe more complex shapes of sets. We further address the related problems of constraining the degree of the polynomials. Various numerical examples are presented to show the effectiveness of advection approach.

关键词： Algebraic/geometric methods level-set methods nonlinear systems semi-definite programming

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Decode-and-Forward Relay Beamforming for Secrecy with Finite-Alphabet Input

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IEEE COMMUNICATIONS LETTERS 2013年第5期17卷 912-915页

作者： Vishwakarma, Sanjay Chockalingam, A. Indian Inst Sci Dept Elect Commun Engn Bangalore 560012 Karnataka India

In this letter, we compute the secrecy rate of decode-and-forward (DF) relay beamforming with finite input alphabet of size M. Source and relays operate under a total power constraint. First, we observe that the secrecy rate with finite-alphabet input can go to zero as the total power increases, when we use the source power and the relay weights obtained assuming Gaussian input. This is because the capacity of an eavesdropper can approach the finite-alphabet capacity of 1/2 log(2) M with increasing total power, due to the inability to completely null in the direction of the eavesdropper. We then propose a transmit power control scheme where the optimum source power and relay weights are obtained by carrying out transmit power (source power plus relay power) control on DF with Gaussian input using semi-definite programming, and then obtaining the corresponding source power and relay weights which maximize the secrecy rate for DF with finite-alphabet input. The proposed power control scheme is shown to achieve increasing secrecy rates with increasing total power with a saturation behavior at high total powers.

关键词： Cooperative relay beamforming physical layer security multiple eavesdroppers finite-alphabet input semi-definite programming

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Classification of companies using maximal margin ellipsoidal surfaces

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2013年第2期55卷 469-480页

作者： Konno, Hiroshi Saito, Masato Chuo Univ Dept Ind & Syst Engn Bunkyo Ku Tokyo 1128551 Japan

We recently proposed a data mining approach for classifying companies into several groups using ellipsoidal surfaces. This problem can be formulated as a semi-definite programming problem, which can be solved within a practical amount of computation time by using a state-of-the-art semi-definite programming software. It turned out that this method performs better for this application than earlier methods based on linear and general quadratic surfaces. In this paper we will improve the performance of ellipsoidal separation by incorporating the idea of maximal margin hyperplane developed in the field of support vector machine. It will be demonstrated that the new method can very well simulate the rating of a leading rating company of Japan by using up to 18 financial attributes of 363 companies. This paper is expected to provide another evidence of the importance of ellipsoidal separation approach in credit risk analysis.

关键词： Credit risk Classification of companies Rating Ellipsoidal surface Maximal margin hyperplane semi-definite programming

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A new Branch and Bound method for a discrete truss topology design problem

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2013年第1期54卷 163-187页

作者： Cerveira, Adelaide Agra, Agostinho Bastos, Fernando Gromicho, Joaquim Univ Tras Os Montes & Alto Douro DM & CIO Vila Real Portugal Univ Aveiro Dmat & CIDMA P-3800 Aveiro Portugal Univ Lisbon DEIO FC P-1699 Lisbon Portugal Univ Lisbon CIO P-1699 Lisbon Portugal Vrije Univ Amsterdam Amsterdam Netherlands

Our paper considers a classic problem in the field of Truss Topology Design, the goal of which is to determine the stiffest truss, under a given load, with a bound on the total volume and discrete requirements in the cross-sectional areas of the bars. To solve this problem we propose a new two-stage Branch and Bound algorithm. In the first stage we perform a Branch and Bound algorithm on the nodes of the structure. This is based on the following dichotomy study: either a node is in the final structure or not. In the second stage, a Branch and Bound on the bar areas is conducted. The existence or otherwise of a node in this structure is ensured by adding constraints on the cross-sectional areas of its incident bars. In practice, for reasons of stability, free bars linked at free nodes should be avoided. Therefore, if a node exists in the structure, then there must be at least two incident bars on it, unless it is a supported node. Thus, a new constraint is added, which lower bounds the sum of the cross-sectional areas of bars incident to the node. Otherwise, if a free node does not belong to the final structure, then all the bar area variables corresponding to bars incident to this node may be set to zero. These constraints are added during the first stage and lead to a tight model. We report the computational experiments conducted to test the effectiveness of this two-stage approach, enhanced by the rule to prevent free bars, as compared to a classical Branch and Bound algorithm, where branching is only performed on the bar areas.

关键词： Truss topology design Stiffness semi-definite programming Quadratic programming Global optimization Branch and Bound

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A direct optimization method for low group delay FIR filter design

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SIGNAL PROCESSING 2013年第7期93卷 1764-1772页

作者： Wu, Changzhi Gao, David Teo, Kok Lay Univ Ballarat Sch Sci Informat Technol & Engn Ballarat Vic 3353 Australia Curtin Univ Dept Math & Stat Perth WA Australia

This paper studies the design of FIR filter with low group delay, where the desired phase response is not being approximated. It is formulated as a constrained optimization problem, which is then solved globally. Numerical experiments show that our design method can produce a filter with smaller group delay than that obtained by the existing convex optimization method used in conjunction with a minimum phase spectral factorization method under the same design criteria. Furthermore, our formulation offers us the flexibility for the trade-off between the group delay and the magnitude response directly. It also allows the feasibility of imposing constraints on the group delay. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Low group delay Duality semi-definite programming

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Moment matrices, border bases and real radical computation

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JOURNAL OF SYMBOLIC COMPUTATION 2013年 51卷 63-85页

作者： Lasserre, Jean-Bernard Laurent, Monique Mourrain, Bernard Rostalski, Philipp Trebuchet, Philippe CNRS LAAS F-31077 Toulouse 4 France CWI NL-1098 SJ Amsterdam Netherlands Tilburg Univ NL-5000 LE Tilburg Netherlands INRIA Mediterranee GALAAD F-06902 Sophia Antipolis France Univ Calif Berkeley Dept Math Berkeley CA 94720 USA LIP6 APR F-75005 Paris France

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming its complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of Mourrain and Trebuchet (2005) are efficient and numerically stable for computing complex roots, algorithms based on moment matrices (Lasserre et al., 2008) allow the incorporation of additional polynomials, e.g., to restrict the computation to real roots or to eliminate multiple solutions. The proposed algorithm can be used to compute a border basis of the input ideal and, as opposed to other approaches, it can also compute the quotient structure of the (real) radical ideal directly, i.e., without prior algebraic techniques such as Grobner bases. It thus combines the strength of existing algorithms and provides a unified treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Solving polynomial equations Border basis Real root Real radical Moment matrix Hankel operator Duality Relaxation semi-definite programming

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semi-definite programming based pulse waveform design and its further analysis

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Journal of Harbin Institute of Technology(New Series) 2008年第4期15卷 578-581页

作者：王利利沙学军吴宣利 Communication Research Center Harbin Institute of Technology

In order to provide a judicious pulse waveform design required for ultra-wideband(UWB)communication to enable the UWB spectral mask compatible and coexistent with other existing wireless communication systems,a semi-definite programming(SDP)based pulse waveform design method for UWB radios is introduced and a further analysis is given in this *** using Sedumi and Yalmip toolboxes of Matlab,the procedure of solving the SDP problem is *** results show that this SDP based pulse waveform design method can be used to design pulses that fulfill the Federal Communications Commission(FCC)spectral mask strictly and optimize the power efficiency at the same *** paper also analyzes the influences of the power efficiency duing to the changes of sampling interval and the number of combined pulses,and then the optimal sampling interval that maximizes the transmission power can be found.

关键词： ultra-wideband pulse waveform design FCC spectral mask semi-definite programming

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Constructing Uniquely Realizable Graphs

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DISCRETE & COMPUTATIONAL GEOMETRY 2013年第4期50卷 1051-1071页

作者： Pak, Igor Vilenchik, Dan Univ Calif Los Angeles Dept Math Los Angeles CA 90095 USA Weizmann Inst Sci Dept Comp Sci & Appl Math IL-76100 Rehovot Israel

In the Graph Realization Problem (GRP), one is given a graph , a set of non-negative edge-weights, and an integer . The goal is to find, if possible, a realization of in the Euclidian space , such that the distance between any two vertices is the assigned edge weight. The problem has many applications in mathematics and computer science, but is NP-hard when the dimension is fixed. Characterizing tractable instances of GRP is a classical problem, first studied by Menger in 1931. We construct two new infinite families of GRP instances which can be solved in polynomial time. Both constructions are based on the blow-up of fixed small graphs with large expanders. Our main tool is the Connelly's condition in Rigidity Theory, combined with an explicit construction and algebraic calculations of the rigidity (stress) matrix. As an application of our results, we describe a general framework to construct uniquely k-colorable graphs. These graphs have the extra property of being uniquely vector k-colorable. We give a deterministic explicit construction of such a family using Cayley expander graphs.

关键词： Rigidity Graph colorability semi-definite programming Graph realization problem

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Deterministic Discrepancy Minimization

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ALGORITHMICA 2013年第4期67卷 451-471页

作者： Bansal, Nikhil Spencer, Joel Eindhoven Univ Technol NL-5600 MB Eindhoven Netherlands NYU New York NY USA

We derandomize a recent algorithmic approach due to Bansal (Foundations of Computer Science, FOCS, pp. 3-10, 2010) to efficiently compute low discrepancy colorings for several problems, for which only existential results were previously known. In particular, we give an efficient deterministic algorithm for Spencer's six standard deviations result (Spencer in Trans. Am. Math. Soc. 289:679-706, 1985), and to find a low discrepancy coloring for a set system with low hereditary discrepancy. The main new idea is to add certain extra constraints to the natural semidefinite programming formulation for discrepancy, which allow us to argue about the existence of a good deterministic move at each step of the algorithm. The non-constructive entropy method is used to argue the feasibility of this enhanced SDP.

关键词： Discrepancy Derandomization Algorithms semi-definite programming

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Amplify-and-Forward Relay Beamforming for Secrecy with Cooperative Jamming and Imperfect CSI

Amplify-and-Forward Relay Beamforming for Secrecy with Coope...

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IEEE International Conference on Communications (ICC)

作者： Vishwakarma, Sanjay Chockalingam, A. Indian Inst Sci Dept ECE Bangalore 560012 Karnataka India

ISBN: (纸本)9781467331227

In this paper, we compute worst case secrecy rates in amplify-and-forward (AF) relay beamforming with cooperative jamming (CJ) in the presence of imperfect channel state information (CSI). A source-destination pair aided by M relays is considered. Number of eavesdroppers J can be more than the number of relays. Out of the M relays, k(1) relays (1 <= k(1) <= M) act as data relays and the remaining k(2) = M - k(1) relays act as jamming relays. Data relays aid the communication by relaying data in AF mode, and jamming relays cooperate by transmitting jamming signals (artificial noise). The jamming signals are created such that they degrade the eavesdroppers' channels but do not significantly affect the intended receiver's channel, thereby improving secrecy rate. Imperfection in the CSI is modeled using a norm-bounded error model. We solve for the optimum (k(1), k(2)) and the weights of data relays and jamming relays that maximize the secrecy rate subject to a total relay power constraint. We relax the rank-one constraint on the complex semi-definite data relays and jamming relays weight matrices and reformulate the optimization problem into a form that can be solved using convex semi-definite programming. Numerical results on the secrecy rate that illustrate the effect of cooperative jamming, imperfect CSI, and number of eavesdroppers are presented.

关键词： Cooperative relay beamforming physical layer security secrecy rate amplify-and-forward cooperative jamming imperfect CSI multiple eavesdroppers semi-definite programming

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