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## Fast linear iterations for distributed averaging. (2004)

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Venue: | Systems & Control Letters, |

Citations: | 432 - 12 self |

### Citations

7718 |
Matrix Analysis
- Horn, Johnson
- 1990
(Show Context)
Citation Context ...rivial, because the spectral radius condition (3) is not convex. As a special case, if one further restricts W by three linear constraints: (i) every entry of W is non-negative; (ii) the corresponding (directed) graph is strongly connected (see page 358 in (Horn & Johnson, 1985) for the notion of strongly connectedness); (iii) one of the diagonal entries ofW is positive; then one can drop the spectral radius condition in (3) and thus obtain at least a desired W (satisfying (3)) by solving, for example, Ps (to be introduced shortly) with the additional three linear constraints (see page 522 in Horn and Johnson (1985)).minimize the spectral abscissa α(·) (the largest real part of the eigenvalues). The gradient sampling method is a variation of the reliable steepest descent method. It uses gradient information on the neighbourhood of each iteration point yk, not just the gradient at the single point yk. As a result, the gradient sampling method becomesparticularly usefulwhen it is applied tominimizing anonsmooth (non-differentiable) function such as ρ(·) (see Burke et al. (2002) and Section 2.2 for details). In this paper, we further develop the foregoing two ideas for finding W ∗. In the following Section ... |

7448 | Convex Optimization
- Boyd, Vanderberghe
- 2004
(Show Context)
Citation Context ...− 11 T /n W T − 11 T /n sI W ∈ S, 1 T W = 1 T , W 1 = 1. Here the symbol � denotes matrix inequality, i.e., X � Y means that X − Y is positive semidefinite. For background on SDP and LMIs, see, e.g., =-=[1, 2, 11, 12, 16, 41, 42, 43]-=-. Related background on eigenvalue optimization can be found in, e.g., [10, 22, 36]. Similarly, the symmetric FDLA problem (15) can be expressed as the SDP with variables s ∈ R and W ∈ R n×n . minimiz... |

5400 | Convex Analysis
- Rockafellar
- 1972
(Show Context)
Citation Context ...ient; but when r is not differentiable at w, it can have multiple subgradients. Subgradients play a key role in convex analysis, and are used in several algorithms for convex optimization (see, e.g., =-=[36, 38, 19, 4, 7]-=-). We can compute a subgradient of r at w as follows. If r(w) = 2(W) and u is the associated unit eigenvector, then a subgradient g is given by g!=-(ui-uj) 2, l{i,j}, l=l,...,m. Similarly, if r(w) = A... |

1984 |
Distributed Algorithms
- Lynch
- 1997
(Show Context)
Citation Context ...s of autonomous agents, in particular, the consensus or agreement problem among the agents. Distributed consensus problems have been studied extensively in the computer science literature (see, e.g., =-=[22]-=-). Recently it has found a wide range of applications, in areas such as formation flight of unmanned air vehicles and clustered satellites, and coordination of mobile robots. The recent paper [31] stu... |

1288 | Coordination of groups of mobile autonomous agents using nearest neighbor rules
- Jadbabaie, Lin, et al.
- 2003
(Show Context)
Citation Context ...cent paper [31] studies linear and nonlinear consensus protocols in these new applications with fixed network topology. Related coordination problems with time-varying topologies have been studied in =-=[20]-=- using a switched linear system model. In these previous works, the edge weights used in the linear consensus protocols are either constant or only dependent on the degrees of their incident nodes. Wi... |

1222 |
Nonlinear Programming. Athena Scientific, 2 edition
- Bertsekas
- 1999
(Show Context)
Citation Context ...em (19), repeated here as minimize r(w) = ‖I − Adiag(w)AT − 11T/n‖2. Here r represents rasym or rstep, which are the same for symmetric weight matrices. The objective function r(w) is a nonsmooth convex function. A subgradient of r at w is a vector g ∈ Rm that satisfies the inequality r(w) ≥ r(w) + gT (w − w) for any vector w ∈ Rm. If r is differentiable at w, then g = ∇r(w) is the only subgradient; but when r is not differentiable at w, it can have multiple subgradients. Subgradients play a key role in convex analysis, and are used in several algorithms for convex optimization (see, e.g., [4, 7, 20, 38, 40]). We can compute a subgradient of r at w as follows. If r(w) = λ2(W ) and u is the associated unit eigenvector, then a subgradient g is given by gl = −(ui − uj)2, l ∼ {i, j}, l = 1, . . . ,m. Similarly, if r(w) = λn(W ) and v is a unit eigenvector associated with λn(W ), then gl = (vi − vj)2, l ∼ {i, j}, l = 1, . . . ,m. 13 A more detailed derivation of these formulas can be found in [9]. For large sparse symmetric matrices W , we can compute a few extreme eigenvalues and their corresponding eigenvectors very efficiently using Lanczos methods (see, e.g., [37, 39]). Thus, we can compute a subg... |

1180 |
Nonlinear Programming
- Bertsekas
- 1999
(Show Context)
Citation Context ...ient; but when r is not differentiable at w, it can have multiple subgradients. Subgradients play a key role in convex analysis, and are used in several algorithms for convex optimization (see, e.g., =-=[36, 38, 19, 4, 7]-=-). We can compute a subgradient of r at w as follows. If r(w) = 2(W) and u is the associated unit eigenvector, then a subgradient g is given by g!=-(ui-uj) 2, l{i,j}, l=l,...,m. Similarly, if r(w) = A... |

1101 | Semidefinite programming
- Vandenberghe, Boyd
- 1994
(Show Context)
Citation Context ... subject to W T - 11T/r sI - W$, 1TW--I T, W1--1. o (16) Here the symbol _ denotes matrix inequality, i.e., X _ Y means that X - Y is positive semidefinite. For background on SDP and LMIs, see, e.g., =-=[10, 1, 40, 15, 41, 2, 39, 11]-=-. Related background on eigenvalue optimization can be found in, e.g., [34, 9, 21]. Similarly, the symmetric FDLA problem (15) can be expressed as the SDP minimize s subject to-sI _ W - 11T/r _ sI (17... |

983 |
The symmetric eigenvalue problem
- Parlett
- 1980
(Show Context)
Citation Context ...e formulas can be found in [8]. For large sparse symmetric matrices W, we can compute a few extreme eigenvalues and their corresponding eigenvectots very efficiently using Lanczos methods (see, e.g., =-=[35, 37]-=-). Thus, we can compute a subgradient of r very efficiently. The subgradient method is very simple: given a feasible w (1) (e.g., from the maximum-degree or local-degree heuristics) k:--1 repeat 1. Co... |

891 | Algebraic graph theory
- Godsil, Royle
- 2001
(Show Context)
Citation Context ...lites, and coordination of mobile robots. The recent paper [33] studies linear and nonlinear consensus protocols in these new applications with fixed network topology. Related coordination problems with time-varying topologies have been studied in [21] using a switched linear system model, and in [27] using set-valued Lyapunov theory. In previous work, the edge weights used in the linear consensus protocols are either constant or only dependent on the degrees of their incident nodes. With these simple methods of choosing edge weights, many concepts and tools from algebraic graph theory (e.g., [5, 17]), in particular the Laplacian matrix of the associated graph, appear to be very useful in the convergence analysis of consensus protocols (see, e.g., [33] and §4 of this paper). The graph Laplacian has also been used in control of distributed dynamic systems (e.g., [13, 14, 25]). This paper is concerned with general conditions on the weight matrix W for the linear iteration (1) to converge to the average at each node, and how we choose W to make the convergence as fast as possible. 1.1 Fastest distributed linear averaging problem The linear iteration (1) implies that x(t) = W tx(0) for t = 0,... |

834 |
Algebraic Graph Theory
- Biggs
- 1993
(Show Context)
Citation Context ... protocols are either constant or only dependent on the degrees of their incident nodes. With these simple methods of choosing edge weights, many concepts and tools from algebraic graph theory (e.g., =-=[5, 16]-=-), in particular the Laplacian matrix of the associated graph, appear to be very usetiff in the convergence analysis of consensus protocols [31]. The graph Laplacian has also been used in control of d... |

622 |
Numerical Methods for Large Eigenvalue Problems
- Saad
- 1992
(Show Context)
Citation Context ...e formulas can be found in [8]. For large sparse symmetric matrices W, we can compute a few extreme eigenvalues and their corresponding eigenvectots very efficiently using Lanczos methods (see, e.g., =-=[35, 37]-=-). Thus, we can compute a subgradient of r very efficiently. The subgradient method is very simple: given a feasible w (1) (e.g., from the maximum-degree or local-degree heuristics) k:--1 repeat 1. Co... |

613 |
Convex Analysis and Minimization Algorithms
- Hiriart-Urruty, Lemarechal
- 1993
(Show Context)
Citation Context ...ient; but when r is not differentiable at w, it can have multiple subgradients. Subgradients play a key role in convex analysis, and are used in several algorithms for convex optimization (see, e.g., =-=[36, 38, 19, 4, 7]-=-). We can compute a subgradient of r at w as follows. If r(w) = 2(W) and u is the associated unit eigenvector, then a subgradient g is given by g!=-(ui-uj) 2, l{i,j}, l=l,...,m. Similarly, if r(w) = A... |

604 | Primal-Dual Interior-Point Methods.
- Wright
- 1997
(Show Context)
Citation Context ...d through numerical simulations in Xiao and Boyd (2004). The second part of the statement indicates that W (1)s is a local minimizer of the function ρ(W ), and thus q-SNM may not be better than 1-SNMwhen E is symmetric. This is just because of the linearization aroundW (1)s introduced to approximately solve (4). Note that the computational complexity ofAs is dominated by solving P (q)s in Step 2 at each iteration. As P (q) s involves order n2 variables, it costs order n6L flops if one uses an interior-point method (Newton’s method), where L is the length of a binary coding of the input data (Potra & Wright, 2000). 2.2. The constrained gradient sampling method As briefly introduced earlier, the gradient sampling method (CGSM) seems quite attractive for optimization problems with non-smooth functions, e.g. Burke, Henrion, Lewis, and Overton (2006). This method subsumes and generalizes the classical steepest descent method by collecting more gradient information at each iterate. Once the iterates jam near the manifold on which theminimized functional is not differentiable, themethod samples a bundle of gradients nearby the jamming point and finds a wayout, as opposed to the classical steepest descentmeth... |

549 | Information flow and cooperative control of vehicle formations
- Fax, Murray
- 2004
(Show Context)
Citation Context ...f the associated graph, appear to be very usetiff in the convergence analysis of consensus protocols [31]. The graph Laplacian has also been used in control of distributed dynamic systems (see, e.g., =-=[12, 13, 24]-=-). The FDLA problem (4) is closely related to the problem of finding the fastest mixing Markov chain on a graph [8]; the only difference is that in the FDLA problem, the weights can be (and the optima... |

547 |
Interior-Point Polynomial Algorithms in Convex Programming, volume 13
- Nesterov, Nemirovski
- 1994
(Show Context)
Citation Context ...ee weights local-degree weights best constant edge weight optimal symmetric weights Figure 3: Distribution of the eigenvalues of W with four different strategies. The dashed lines indicate ±ρ(W − 11T/n). −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 2 4 6 8 10 12 PSfrag replacements Figure 4: Distribution of the optimal symmetric edge and node weights, found by solving the SDP (17). Note that many weights are negative. 11 6 Computational methods 6.1 Interior-point method Standard interior-point algorithms for solving SDPs work well for problems with up to a thousand or so edges (see, e.g., [1, 12, 31, 41, 42, 43, 44]). The particular structure of the SDPs encountered in FDLA problems can be exploited for some gain in efficiency, but problems with more than a few thousand edges are probably beyond the capabilities of current interior-point SDP solvers. We consider a simple primal barrier method, with the standard log-determinant barrier function (see, e.g., [12, chapter 11]); the same techniques can be used to compute the search directions in other interior-point methods (e.g., primal-dual). In the primal barrier method (for solving the SDP (17)), at each step we need to compute a Newton step for the probl... |

543 | Interior point methods in semidefinite programming with applications to combinatorial optimization
- Alizadeh
- 1995
(Show Context)
Citation Context ... subject to W T - 11T/r sI - W$, 1TW--I T, W1--1. o (16) Here the symbol _ denotes matrix inequality, i.e., X _ Y means that X - Y is positive semidefinite. For background on SDP and LMIs, see, e.g., =-=[10, 1, 40, 15, 41, 2, 39, 11]-=-. Related background on eigenvalue optimization can be found in, e.g., [34, 9, 21]. Similarly, the symmetric FDLA problem (15) can be expressed as the SDP minimize s subject to-sI _ W - 11T/r _ sI (17... |

477 |
Stability of multiagent systems with time-dependent communication links
- Moreau
- 2005
(Show Context)
Citation Context ...nsus protocols in these new applications with fixed network topology. Related coordination problems with time-varying topologies have been studied in [21] using a switched linear system model, and in =-=[27]-=- using set-valued Lyapunov theory. In previous work, the edge weights used in the linear consensus protocols are either constant or only dependent on the degrees of their incident nodes. With these si... |

398 |
Minimization Methods for Nondifferentiable Functions
- Shor
- 1985
(Show Context)
Citation Context |

287 | Interior Point Algorithms, Theory and Analysis
- Ye
- 1997
(Show Context)
Citation Context ...any weights are negative. 11 6 Computational methods 6.1 Interior-point method Standard interior-point algorithms for solving SDPs work well for problems with up to a thousand or so edges (see, e.g., =-=[29, 1, 40, 42, 41, 39, 11]-=-). The particular structure of the SDPs encountered in FDLA problems can be exploited for some gain in efficiency, but problems with more than a few thousand edges are probably beyond the capabilities... |

274 | A rank minimization heuristic with application to minimum order system approximation.
- Fazel, Hindi, et al.
- 2001
(Show Context)
Citation Context ...is also possible to assign weights to the edges, to achieve (hopefully) some desired sparsity pattern. More sophisticated heuristics for sparse design and minimum rank problems can be found in, e.g., =-=[14]-=-. 15 To demonstrate this idea, we applied thesheuristic (29) to the example described in 5.1. We set the guaranteed convergence factor r TM z 0.910, which is only slightly larger than the minimum fact... |

274 |
A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications
- Ben-Tal, Nemirovsky
- 2001
(Show Context)
Citation Context ...s a convex problem. We refer to this problem as the symmetric FDLA problem. The problem of minimizing the per-step convergence factor, i.e., the spectral norm minimization problem (14), can be expressed as an SDP, by introducing a scalar variable s to bound the spectral norm ‖W − 11T/n‖2, and expressing the norm bound constraint as a linear matrix inequality (LMI): minimize s subject to [ sI W − 11T/n W T − 11T/n sI ] º 0 W ∈ S, 1TW = 1T , W1 = 1. (16) Here the symbol º denotes matrix inequality, i.e., X º Y means that X − Y is positive semidefinite. For background on SDP and LMIs, see, e.g., [1, 2, 11, 12, 16, 41, 42, 43]. Related background on eigenvalue optimization can be found in, e.g., [10, 22, 36]. Similarly, the symmetric FDLA problem (15) can be expressed as the SDP minimize s subject to −sI W − 11T/n sI W ∈ S, W = W T , W1 = 1, (17) with variables s ∈ R and W ∈ Rn×n. 6 4 Heuristics based on the Laplacian There are some simple heuristics for choosing W ∈ S that guarantee convergence of the distributed linear averaging iteration, and sometimes give reasonably fast convergence. These heuristics are based on the Laplacian matrix of the associated graph and assign symmetric edge weights. To describe th... |

236 | Convex analysis and nonlinear optimization: theory and examples
- Borwein, Lewis
- 2010
(Show Context)
Citation Context |

170 | A spectral bundle method for semidefinite programming
- Helmberg, Rendl
(Show Context)
Citation Context ... lim_/3 -- 0 and oos/3 z oo. The convergence of this algorithm is proved in [38, 2.2]. Some closely related methods for solving large-scale SDPs and eigenvalue problems are the spectral bundle method =-=[18]-=- and a prox-method [28]; see also [32]. To demonstrate the subgradient method, we apply it to a large-scale network with 10000 nodes and 100000 edges. The graph is generated as follows. First we gener... |

155 | Fastest mixing markov chain on a graph,” - Boyd, Diaconis, et al. - 2004 |

151 | Semidefinite optimization.
- Todd
- 2001
(Show Context)
Citation Context ... subject to W T - 11T/r sI - W$, 1TW--I T, W1--1. o (16) Here the symbol _ denotes matrix inequality, i.e., X _ Y means that X - Y is positive semidefinite. For background on SDP and LMIs, see, e.g., =-=[10, 1, 40, 15, 41, 2, 39, 11]-=-. Related background on eigenvalue optimization can be found in, e.g., [34, 9, 21]. Similarly, the symmetric FDLA problem (15) can be expressed as the SDP minimize s subject to-sI _ W - 11T/r _ sI (17... |

146 |
Laplacian Matrices of Graphs: A Survey,” Linear Algebra Appl
- Merris
- 1994
(Show Context)
Citation Context ...s not depend on the choice of reference directions). The Laplacian matrix is a useful tool in algebraic graph theory, and its eigenstructure reveals many important properties of the graph (see, e.g., =-=[17, 24]-=-). We note for future use that L is positive semidefinite, and since our graph is assumed connected, L has a simple eigenvalue zero, with corresponding eigenvector 1. We can use the incidence matrix t... |

132 | Algebraic Graph Theory, Graduate Texts - Godsil, Royle - 2000 |

129 | Consensus filters for sensor networks and distributed sensor fusion.
- Olfati-Saber, Shamma
- 2005
(Show Context)
Citation Context ...adjacent UAVs. Therefore, one may easily conclude that the mission heavily hinges on the ramifications of the limited information exchange pattern. Such a decentralized tracking problem that requires each agent (processor) to do iterative weighted average operations in a decentralized manner is called the average consensus problem, and has been studied for numerous applications, e.g.mobile ad-hoc and wireless sensor networks. These applications include consensus with statically or dynamically changing information-exchange topologies (Olfati-Saber & Murray, 2004), high-frequency channel noise (Olfati-Saber & Shamma, 2005), corrupted measurement data (Ren, Beard, & Kingston, 2005), network link failures (Cortes, Martinez, & Bullo, 2006), or state-dependent graph settings (Kim & Mesbahi, 2006). In this paper, we are particularly interested in the optimal matrixW ∈ Rn×n (denoted byW ∗) such that the following rule x(k+ 1) = Wx(k), (1) 1380 Y. Kim et al. / Automatica 45 (2009) 1379–1386allows xi(k) to converge to 1Tx(0)/n with minimum k∗ within a prescribed tolerance for every i (∈ {1, 2, . . . , n}). Here, x(k) = [x1(k), . . . , xn(k)]T, xi(k) is the value possessed by the ith agent at time step k on a network (g... |

119 | Solving large-scale sparse semidefinite programs for combinatorial optimization.
- Benson, Ye, et al.
- 2000
(Show Context)
Citation Context ...ese formulas are derived using equation (27). The structure exploited here is similar to the methods used in the dual-scaling algorithm for large-scale combinatorial optimization problems, studied in =-=[3]-=-. The total costs of this step (number of flops) is on the order of m 2 (negligible compared with step I and 3). 3. Compute the Newton step-H-lg by Cholesky factorization and back substitution. The co... |

115 | Eigenvalue optimization.
- Lewis, Overton
- 1996
(Show Context)
Citation Context ... i.e., X _ Y means that X - Y is positive semidefinite. For background on SDP and LMIs, see, e.g., [10, 1, 40, 15, 41, 2, 39, 11]. Related background on eigenvalue optimization can be found in, e.g., =-=[34, 9, 21]-=-. Similarly, the symmetric FDLA problem (15) can be expressed as the SDP minimize s subject to-sI _ W - 11T/r _ sI (17) W$, W--W T, W1--1, with variables s G R and W G R nXn. 4 Heuristics based on the... |

113 | Consensus protocol for networks of dynamic agents
- Saber, Murray
- 2003
(Show Context)
Citation Context ...g., [22]). Recently it has found a wide range of applications, in areas such as formation flight of unmanned air vehicles and clustered satellites, and coordination of mobile robots. The recent paper =-=[31]-=- studies linear and nonlinear consensus protocols in these new applications with fixed network topology. Related coordination problems with time-varying topologies have been studied in [20] using a sw... |

101 |
Algebraic graph theory, volume 207 of Graduate Texts in Mathematics
- Godsil, Royle
- 2001
(Show Context)
Citation Context ... protocols are either constant or only dependent on the degrees of their incident nodes. With these simple methods of choosing edge weights, many concepts and tools from algebraic graph theory (e.g., =-=[5, 16]-=-), in particular the Laplacian matrix of the associated graph, appear to be very usetiff in the convergence analysis of consensus protocols [31]. The graph Laplacian has also been used in control of d... |

100 |
On maximizing the second smallest eigenvalue of a state-dependent Laplacian.
- Kim, Mesbahi
- 2006
(Show Context)
Citation Context ...blem that requires each agent (processor) to do iterative weighted average operations in a decentralized manner is called the average consensus problem, and has been studied for numerous applications, e.g.mobile ad-hoc and wireless sensor networks. These applications include consensus with statically or dynamically changing information-exchange topologies (Olfati-Saber & Murray, 2004), high-frequency channel noise (Olfati-Saber & Shamma, 2005), corrupted measurement data (Ren, Beard, & Kingston, 2005), network link failures (Cortes, Martinez, & Bullo, 2006), or state-dependent graph settings (Kim & Mesbahi, 2006). In this paper, we are particularly interested in the optimal matrixW ∈ Rn×n (denoted byW ∗) such that the following rule x(k+ 1) = Wx(k), (1) 1380 Y. Kim et al. / Automatica 45 (2009) 1379–1386allows xi(k) to converge to 1Tx(0)/n with minimum k∗ within a prescribed tolerance for every i (∈ {1, 2, . . . , n}). Here, x(k) = [x1(k), . . . , xn(k)]T, xi(k) is the value possessed by the ith agent at time step k on a network (graph) G with a proper1 information exchange pattern E , and k (∈ {0, 1, 2, . . .}) is the discrete-time step index. Anetwork (graph)G consists of a setV of nodes (agents) vi... |

89 | Nonlinear Programming: 2nd Edition, Athena Scientific - Bertsekas - 1999 |

84 | Tsitsiklis, “NP-hardness of some linear control design problems
- Blondel, N
- 1997
(Show Context)
Citation Context ....e., the spectral radius of a matrix, is in general not a convex function; indeed it is not even Lipschitz continuous (see, e.g., [33]). Some related spectral radius minimization problems are NP-hard =-=[6, 27]-=-. We can also formulate the FDLA problem, with per-step convergence factor, as the following spectral norm minimization problem: minimize IlW- 11T/II subject to W,S, lrW--1 r, W1--1. In contrast to th... |

84 | Large-scale optimization of eigenvalues.
- Overton
- 1992
(Show Context)
Citation Context ...rgence of this algorithm is proved in [38, 2.2]. Some closely related methods for solving large-scale SDPs and eigenvalue problems are the spectral bundle method [18] and a prox-method [28]; see also =-=[32]-=-. To demonstrate the subgradient method, we apply it to a large-scale network with 10000 nodes and 100000 edges. The graph is generated as follows. First we generate a 10000 by 10000 symmetric matrix ... |

79 |
Laplacian matrices of graphs: a survey
- Merris
- 1994
(Show Context)
Citation Context ...s not depend on the choice of reference directions). The Laplacian matrix is a useful tool in algebraic graph theory, and its eigenstructure reveals many important properties of the graph (see, e.g., =-=[23, 16]-=-). We note for future use that L is positive semidefinite, and since our graph is assumed connected, L has a simple eigenvalue zero, with corresponding eigenvector 1. We can use the incidence matrix t... |

78 | Ghaoui, “Method of centers for minimizing generalized eigenvalues
- Boyd, E
- 1993
(Show Context)
Citation Context ... i.e., X � Y means that X − Y is positive semidefinite. For background on SDP and LMIs, see, e.g., [1, 2, 11, 12, 16, 41, 42, 43]. Related background on eigenvalue optimization can be found in, e.g., =-=[10, 22, 36]-=-. Similarly, the symmetric FDLA problem (15) can be expressed as the SDP with variables s ∈ R and W ∈ R n×n . minimize s subject to −sI � W − 11 T /n � sI W ∈ S, W = W T , W 1 = 1, 6 � � 0 (13) (14) (... |

69 | Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices,
- Overton, Womersley
- 1993
(Show Context)
Citation Context ... i.e., X _ Y means that X - Y is positive semidefinite. For background on SDP and LMIs, see, e.g., [10, 1, 40, 15, 41, 2, 39, 11]. Related background on eigenvalue optimization can be found in, e.g., =-=[34, 9, 21]-=-. Similarly, the symmetric FDLA problem (15) can be expressed as the SDP minimize s subject to-sI _ W - 11T/r _ sI (17) W$, W--W T, W1--1, with variables s G R and W G R nXn. 4 Heuristics based on the... |

64 | The Symmetric Eigenvalue Problem, Prentice-Hall, - Parlett - 1980 |

62 | Algebraic Graph Theory, 2nd edition, - Biggs - 1993 |

58 | Multi-agent kalman consensus with relative uncertainty,”
- Ren, Beard, et al.
- 2005
(Show Context)
Citation Context ... mission heavily hinges on the ramifications of the limited information exchange pattern. Such a decentralized tracking problem that requires each agent (processor) to do iterative weighted average operations in a decentralized manner is called the average consensus problem, and has been studied for numerous applications, e.g.mobile ad-hoc and wireless sensor networks. These applications include consensus with statically or dynamically changing information-exchange topologies (Olfati-Saber & Murray, 2004), high-frequency channel noise (Olfati-Saber & Shamma, 2005), corrupted measurement data (Ren, Beard, & Kingston, 2005), network link failures (Cortes, Martinez, & Bullo, 2006), or state-dependent graph settings (Kim & Mesbahi, 2006). In this paper, we are particularly interested in the optimal matrixW ∈ Rn×n (denoted byW ∗) such that the following rule x(k+ 1) = Wx(k), (1) 1380 Y. Kim et al. / Automatica 45 (2009) 1379–1386allows xi(k) to converge to 1Tx(0)/n with minimum k∗ within a prescribed tolerance for every i (∈ {1, 2, . . . , n}). Here, x(k) = [x1(k), . . . , xn(k)]T, xi(k) is the value possessed by the ith agent at time step k on a network (graph) G with a proper1 information exchange pattern E , an... |

51 |
Several np-hard problems arising in robust stability analysis
- Nemirovskii
- 1993
(Show Context)
Citation Context ....e., the spectral radius of a matrix, is in general not a convex function; indeed it is not even Lipschitz continuous (see, e.g., [33]). Some related spectral radius minimization problems are NP-hard =-=[6, 27]-=-. We can also formulate the FDLA problem, with per-step convergence factor, as the following spectral norm minimization problem: minimize IlW- 11T/II subject to W,S, lrW--1 r, W1--1. In contrast to th... |

51 | Bayesian analysis for reversible Markov chains.
- Diaconis, Rolles
- 2006
(Show Context)
Citation Context ...d = 1 , (25) provided the graph is not bipartite. Compared with the optimal weights, the maximum-degree weights often lead to much slower convergence when there are bottle-neck links in the graph. In =-=[8]-=-, we give an example of two complete graphs connected by a bridge, where the optimal weight matrix W ⋆ can perform arbitrarily better than the maximum-degree weights, in the sense that the ratio (1 − ... |

48 | Static output feedback - a survey. - Syrmos, Abdallah, et al. - 1997 |

36 | Semide nite Programming - Vandenberghe, Boyd - 1996 |

36 | Two numerical methods for optimizing matrix stability. Linear Algebra and its Applications,
- Burke, Lewis, et al.
- 2002
(Show Context)
Citation Context ...ying (3)) by solving, for example, Ps (to be introduced shortly) with the additional three linear constraints (see page 522 in Horn and Johnson (1985)).minimize the spectral abscissa α(·) (the largest real part of the eigenvalues). The gradient sampling method is a variation of the reliable steepest descent method. It uses gradient information on the neighbourhood of each iteration point yk, not just the gradient at the single point yk. As a result, the gradient sampling method becomesparticularly usefulwhen it is applied tominimizing anonsmooth (non-differentiable) function such as ρ(·) (see Burke et al. (2002) and Section 2.2 for details). In this paper, we further develop the foregoing two ideas for finding W ∗. In the following Section 2.1, we first introduce the qth order spectral norm (2-norm) minimization method (q-SNM) to improve the solutionWs toPs. We then show that 1-SNM can yield a symmetric solution W (1)s such that σ (W (1) s ) = ρ(W (1) s ) if E is symmetric, and, as a consequence,W (1)s is a local minimizer of the objective functional in (4) with any positive integer q, and thus of the function ρ(W ). In Section 2.2, we propose the constrained gradient sampling method (CGSM) to ac... |

33 | Stabilization via nonsmooth, nonconvex optimization. - Burke, Henrion, et al. - 2006 |

31 | Low-authority controller design via convex optimization,” in
- Hassibi, How, et al.
- 1998
(Show Context)
Citation Context ...he convergence factor. This is a difficult combinatorial problem, but one very effective heuristic to achieve this goal is to minimize the 1 norm of the vector of edge weights; see, e.g., [11, 6] and =-=[17]-=-. For example, given the maximum allowed asymptotic convergence factor r TM, the 1 heuristic for the sparse graph design problem (with symmetric edge weights) can be posed as the convex problem minimi... |

31 | Minimization Methods for Non-Dierentiable Functions - Shor - 1985 |

29 | Approximating subdifferentials by random sampling of gradients.Mathematics of
- Burke, Lewis, et al.
- 2002
(Show Context)
Citation Context ...ying (3)) by solving, for example, Ps (to be introduced shortly) with the additional three linear constraints (see page 522 in Horn and Johnson (1985)).minimize the spectral abscissa α(·) (the largest real part of the eigenvalues). The gradient sampling method is a variation of the reliable steepest descent method. It uses gradient information on the neighbourhood of each iteration point yk, not just the gradient at the single point yk. As a result, the gradient sampling method becomesparticularly usefulwhen it is applied tominimizing anonsmooth (non-differentiable) function such as ρ(·) (see Burke et al. (2002) and Section 2.2 for details). In this paper, we further develop the foregoing two ideas for finding W ∗. In the following Section 2.1, we first introduce the qth order spectral norm (2-norm) minimization method (q-SNM) to improve the solutionWs toPs. We then show that 1-SNM can yield a symmetric solution W (1)s such that σ (W (1) s ) = ρ(W (1) s ) if E is symmetric, and, as a consequence,W (1)s is a local minimizer of the objective functional in (4) with any positive integer q, and thus of the function ρ(W ). In Section 2.2, we propose the constrained gradient sampling method (CGSM) to ac... |

29 | An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems - Leibfritz, Mostafa |

27 | Variational analysis of non-Lipschitz spectral functions.Mathematical Programming
- Burke, Overton
- 2001
(Show Context)
Citation Context ...nts Y1 = 1 and1TY = 1T, weneed2n linear constraints hj(xi) = 0 (j = 1, 2, . . . , 2n). Recalling the constrained steepest descent method, we therefore choose a direction d for each sampled Y such that ∇ρ(Y )Td ≤ 0, (7) where ∇ρ(Y ) is the gradient of ρ(·) at Y − 11T/n, and ∇hj(x1, x2, . . . , xm)Td = 0 ∀ j = 1, 2, . . . , 2n. (8) If there are multiple feasible directions d, we choose the one such that ∇ρ(Y )Td is minimized. We now present the gradient formula of ρ(·). The proof is motivated by that of Theorem 6.3.12 in Horn and Johnson (1985), and could be deduced from existing works, e.g. Burke and Overton (2001). Proposition 2. Suppose ρ(·) is differentiable at X, and λ = Re(λ)+ iIm(λ) is the largest in magnitude eigenvalue of X = ∑ i xiXi and it (and its conjugate pair) is algebraically simple, i.e. has multiplicityone. Then, for k ∈ {1, 2, . . . ,m} ∂ρ(X) ∂xk = Re(λ) |λ| (uTXkv + uTXkv)+ Im(λ) |λ| (uTXkv − uTXkv), where u+ iu and v+ iv are the left and right eigenvectors associated with λ, respectively. Proof. The conditions given in the statement guarantee that (u− iu)T(v + iv) 6= 0 (see Lemma 6.3.10 in Horn and Johnson (1985)), which allows the eigenvectors to be normalized such that (u −... |

24 |
Robust Control with Structured Perturbations,"
- Keel, Bhattacharyya, et al.
- 1988
(Show Context)
Citation Context ...ion to P (1) s . P (q) s is similarly defined as we did in Section 2.1, i.e. the linearized version of P (q)s , and (W (q)s , K (q) s ) is the solution to P (q) s . If K = Φ after running the algorithm, then the considered system may have no stabilizing static output feedback controllers. As opposed to the previous version ofAs in Section 2.1, the current version can be terminated before q reaches itsmaximumvalue of 7, i.e. as soon as a stabilizing controller is found. We now proceed with the following benchmark systems found in the literature (Blondel et al., 1999; Leibfritz & Mostafa, 2002; Keel, Bhattacharyya, & Howze, 1988; Mesbahi, 2008; Nesterov & Nemirovskii, 1994): Case 1: A = [ 1 1.05 −1.05 0 ] , B = [ 0 1 ] , C = [ 0 1 ] ; Case 2: A = [ 1 1.05 0 −1.05 0 0 0 0 0 ] , B = [ 0 0 0 1 1 0 ] , C = [ 0 0 1 0 1 0 ] ; Case 3: A = 1 1.05 0 0−1.05 0 0 00 0 0 0 0 0 0 0 , B = 0 0 00 0 11 0 0 0 1 0 , C = [0 0 1 00 0 0 1 0 1 0 0 ] ; Case 4: A = −0.0366 0.0271 0.0188 −0.45550.0482 −1.0100 0.0024 −4.02080.1002 0.3681 −0.7070 1.4200 0.0000 0.0000 1.0000 0.0000 , B = 0.4422 0.17613.5446 −7.5922 −5.5200 4.4900 0.0000 0.0000 , C = [0 1 0 0] . Case 5: A = 1.3800 −0.2077 6.7150 −5.6760−0.5814 −4... |

23 |
Minimum Entropy H∞ Control.
- Mustafa, Glover
- 1990
(Show Context)
Citation Context ... those that result in asymptotic convergence and minimize the logarithmic barrier function log det(I - 11T/n + W) - + log det(I + 11,/ - W) -. (28) (The terminology follows control theory; see, e.g., =-=[26]-=-.) In terms of the eigenvalues hi of W, the central weights minimize the objective 1og 1 14 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0 ' A' W 1 O0 150 200 250 300 350 400 iteration number Figure 5: Progress of... |

23 | Fast linear iterations for distributed averaging,” Sys - Xiao, Boyd - 2004 |

19 |
On miminizing the spectral radius of a nonsymmetric matrix function– optimality conditions and duality theory.
- Overton, Womersley
- 1988
(Show Context)
Citation Context ... in general is very hard. The reason is that the objective function, i.e., the spectral radius of a matrix, is in general not a convex function; indeed it is not even Lipschitz continuous (see, e.g., =-=[33]-=-). Some related spectral radius minimization problems are NP-hard [6, 27]. We can also formulate the FDLA problem, with per-step convergence factor, as the following spectral norm minimization problem... |

17 |
Convergence powers of a matrix with applications to iterative methods for singular linear systems.
- Meyer, Plemmons
- 1977
(Show Context)
Citation Context ...r matrix T such that W = T � Iκ 0 0 Z where Iκ is the κ-dimensional identity matrix (0 ≤ κ ≤ n) and Z is a convergent matrix, i.e., ρ(Z) < 1. (This can be derived using the Jordan canonical form; see =-=[32, 26]-=-.) Let u1, . . . , un be the columns of T and vT 1 , . . . , vT n be the rows of T −1 . Then we have lim t→∞ W t � Iκ 0 = lim T t→∞ 0 Zt � T −1 � � Iκ 0 = T T 0 0 −1 κ� = uiv i=1 T i . (12) Since each... |

16 |
Consensus problems in networks of agentswith switching topology and time-delays.
- Olfati-Saber, Murray
- 2004
(Show Context)
Citation Context ...V is only capable of communicating with a limited number of adjacent UAVs. Therefore, one may easily conclude that the mission heavily hinges on the ramifications of the limited information exchange pattern. Such a decentralized tracking problem that requires each agent (processor) to do iterative weighted average operations in a decentralized manner is called the average consensus problem, and has been studied for numerous applications, e.g.mobile ad-hoc and wireless sensor networks. These applications include consensus with statically or dynamically changing information-exchange topologies (Olfati-Saber & Murray, 2004), high-frequency channel noise (Olfati-Saber & Shamma, 2005), corrupted measurement data (Ren, Beard, & Kingston, 2005), network link failures (Cortes, Martinez, & Bullo, 2006), or state-dependent graph settings (Kim & Mesbahi, 2006). In this paper, we are particularly interested in the optimal matrixW ∈ Rn×n (denoted byW ∗) such that the following rule x(k+ 1) = Wx(k), (1) 1380 Y. Kim et al. / Automatica 45 (2009) 1379–1386allows xi(k) to converge to 1Tx(0)/n with minimum k∗ within a prescribed tolerance for every i (∈ {1, 2, . . . , n}). Here, x(k) = [x1(k), . . . , xn(k)]T, xi(k) is the val... |

14 |
On a dynamic extension of the theory of graphs.
- Mesbahi
- 2002
(Show Context)
Citation Context ...f the associated graph, appear to be very usetiff in the convergence analysis of consensus protocols [31]. The graph Laplacian has also been used in control of distributed dynamic systems (see, e.g., =-=[12, 13, 24]-=-). The FDLA problem (4) is closely related to the problem of finding the fastest mixing Markov chain on a graph [8]; the only difference is that in the FDLA problem, the weights can be (and the optima... |

14 | Interior point methods in semide nite programming with applications to combinatorial optimization - Alizadeh - 1995 |

12 |
Graph Laplacians and vehicle formation stabilization.
- Fax, Murray
- 2002
(Show Context)
Citation Context ...f the associated graph, appear to be very usetiff in the convergence analysis of consensus protocols [31]. The graph Laplacian has also been used in control of distributed dynamic systems (see, e.g., =-=[12, 13, 24]-=-). The FDLA problem (4) is closely related to the problem of finding the fastest mixing Markov chain on a graph [8]; the only difference is that in the FDLA problem, the weights can be (and the optima... |

11 |
Infinite powers of matrices and characteristic roots.
- Oldenburger
- 1940
(Show Context)
Citation Context ...r matrix T such that W = T � Iκ 0 0 Z where Iκ is the κ-dimensional identity matrix (0 ≤ κ ≤ n) and Z is a convergent matrix, i.e., ρ(Z) < 1. (This can be derived using the Jordan canonical form; see =-=[32, 26]-=-.) Let u1, . . . , un be the columns of T and vT 1 , . . . , vT n be the rows of T −1 . Then we have lim t→∞ W t � Iκ 0 = lim T t→∞ 0 Zt � T −1 � � Iκ 0 = T T 0 0 −1 κ� = uiv i=1 T i . (12) Since each... |

9 | Minimum Entropy H - Mustafa, Glover - 1990 |

8 |
Open problems in mathematical systems and control theory.
- Blondel, Sontag, et al.
- 1999
(Show Context)
Citation Context ...=A+BKC ‖W q‖ and (W (1)s , K (1) s ) is the solution to P (1) s . P (q) s is similarly defined as we did in Section 2.1, i.e. the linearized version of P (q)s , and (W (q)s , K (q) s ) is the solution to P (q) s . If K = Φ after running the algorithm, then the considered system may have no stabilizing static output feedback controllers. As opposed to the previous version ofAs in Section 2.1, the current version can be terminated before q reaches itsmaximumvalue of 7, i.e. as soon as a stabilizing controller is found. We now proceed with the following benchmark systems found in the literature (Blondel et al., 1999; Leibfritz & Mostafa, 2002; Keel, Bhattacharyya, & Howze, 1988; Mesbahi, 2008; Nesterov & Nemirovskii, 1994): Case 1: A = [ 1 1.05 −1.05 0 ] , B = [ 0 1 ] , C = [ 0 1 ] ; Case 2: A = [ 1 1.05 0 −1.05 0 0 0 0 0 ] , B = [ 0 0 0 1 1 0 ] , C = [ 0 0 1 0 1 0 ] ; Case 3: A = 1 1.05 0 0−1.05 0 0 00 0 0 0 0 0 0 0 , B = 0 0 00 0 11 0 0 0 1 0 , C = [0 0 1 00 0 0 1 0 1 0 0 ] ; Case 4: A = −0.0366 0.0271 0.0188 −0.45550.0482 −1.0100 0.0024 −4.02080.1002 0.3681 −0.7070 1.4200 0.0000 0.0000 1.0000 0.0000 , B = 0.4422 0.17613.5446 −7.5922 −5.5200 4.4900 0.0000 0.0000 , C = [0 1... |

7 | Information °ow and cooperative control of vehicle formations - Fax, Murray - 2002 |

4 |
Fastest mixing Markov chain on a graph. Submitted to SIAM Review, problems and techniques section, February 2003. Available at http://www.stanford.edu/~boyd/fmmc.html
- Boyd, Diaconis, et al.
- 1993
(Show Context)
Citation Context |

4 | Solving large-scale sparse semide nite programs for combinatorial optimization - BENSON, YE, et al. - 1997 |

2 |
Convex Optimization. forthcoming book, 2003. Draft available at h;;p://www. s;anord. edu/-boyd/cvxbook. h;m
- Boyd, Vandenberghe
(Show Context)
Citation Context |

2 |
Fastest mixing Markov chain on a graph. Accepted for publication in SIAM Review, problems and techniques section,
- Boyd, Diaconis, et al.
- 2004
(Show Context)
Citation Context ...constraints on the weight matrix) the fastest distributed linear averaging (FDLA) problem. The FDLA problem (4) is closely related to the problem of finding the fastest mixing Markov chain on a graph =-=[9]-=-; the only difference in the two problem formulations is that in the FDLA problem, the weights can be (and the optimal ones often are) negative, hence faster convergence could be achieved compared wit... |

2 | A spectral bundle method for semide nite programming - Helmberg, Rendl - 2000 |

2 |
Robust Rendezvous formobile autonomous agents via proximity graphs in arbitrary dimensions.
- Cortes, Martinez, et al.
- 2006
(Show Context)
Citation Context ...mited information exchange pattern. Such a decentralized tracking problem that requires each agent (processor) to do iterative weighted average operations in a decentralized manner is called the average consensus problem, and has been studied for numerous applications, e.g.mobile ad-hoc and wireless sensor networks. These applications include consensus with statically or dynamically changing information-exchange topologies (Olfati-Saber & Murray, 2004), high-frequency channel noise (Olfati-Saber & Shamma, 2005), corrupted measurement data (Ren, Beard, & Kingston, 2005), network link failures (Cortes, Martinez, & Bullo, 2006), or state-dependent graph settings (Kim & Mesbahi, 2006). In this paper, we are particularly interested in the optimal matrixW ∈ Rn×n (denoted byW ∗) such that the following rule x(k+ 1) = Wx(k), (1) 1380 Y. Kim et al. / Automatica 45 (2009) 1379–1386allows xi(k) to converge to 1Tx(0)/n with minimum k∗ within a prescribed tolerance for every i (∈ {1, 2, . . . , n}). Here, x(k) = [x1(k), . . . , xn(k)]T, xi(k) is the value possessed by the ith agent at time step k on a network (graph) G with a proper1 information exchange pattern E , and k (∈ {0, 1, 2, . . .}) is the discrete-time step index.... |

1 |
Prox-method with rate of convergence O(1/t) for Lipschitz continuous variational inequalities and smooth convex-concave saddle point problems.
- Nemirovskii
- 2003
(Show Context)
Citation Context ...z oo. The convergence of this algorithm is proved in [38, 2.2]. Some closely related methods for solving large-scale SDPs and eigenvalue problems are the spectral bundle method [18] and a prox-method =-=[28]-=-; see also [32]. To demonstrate the subgradient method, we apply it to a large-scale network with 10000 nodes and 100000 edges. The graph is generated as follows. First we generate a 10000 by 10000 sy... |

1 | Semide nite optimization - Todd - 2001 |

1 | Vandengerghe (Eds.), Handbook of Semide nite - Wolkowicz, Saigal, et al. - 2000 |

1 |
On miminizing the spectral radius of a nonsymmetricmatrix function –Optimality conditions andduality theory.
- Overton, Womersley
- 1988
(Show Context)
Citation Context ...e optimal W ∗ is obtained when ρ(W ) or q is minimized. FindingW ∗ is an old problem, although the structure embedded in W may not be old. It can be translated into finding the most stable discrete-time linear system, or finding the fastest mixing Markov chain (discrete-time stochastic process) when W is non-negative (entry-wise). These areas have been popular research topics in the control community. However, findingW ∗ or minimizing the spectral radius matrix function ρ(·) is known as a very hard problem in general. This is becauseρ(·) is continuous but neither convex nor locally Lipschtz (Overton & Womersley, 1988). For this reason, there are few works in the literature that directly address the problem in question. In Xiao and Boyd (2004), the authors approach the problem by solving the following program: Ps : min W ‖W‖ s.t. W ∈ S(E), W1 = 1, 1TW = 1T for a given G. The programPs minimizes the largest singular value of W , σ (W ), instead of ρ(W ). Thus, the solution Ws to Ps only guarantees the well-known bound ρ(W ∗) ≤ ρ(Ws) ≤ σ (Ws), where the gap between ρ(W ∗) and σ (Ws) can be unacceptably large in general. In Burke, Lewis, and Overton (2002), the authors propose the so-called (uncons... |

1 | was born in Goheung, Republic of Korea in 1974. He received his B.Sc., M.Sc. and Ph.D. - Kim - 1999 |

1 | He graduated from Department of Mathematics, Fudan University, - Shanghai, China - 1979 |