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## Convex optimization of graph Laplacian eigenvalues (2006)

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Citations: | 48 - 0 self |

### Citations

7184 | Convex Optimization
- Boyd, Vandenberghe
- 2004
(Show Context)
Citation Context ...he problem (2) is a convex optimization problem. Roughly speaking, this means that the analysis of the problem is fairly straightforward, and that the problem is easily solved numerically; see, e.g., =-=[6]-=-. In the cases we will consider, (2)sConvex Optimization of Graph Laplacian Eigenvalues 3 the problem (2) can be formulated even more specifically as a semidefinite program (SDP), which has the form m... |

5222 | Convex Analysis
- Rockafellar
- 1970
(Show Context)
Citation Context ...ds a sharp lower bound on the optimal SLEM. (5)sConvex Optimization of Graph Laplacian Eigenvalues 5 Both of these conclusions follow from general results for convex optimization problems (see, e.g., =-=[10, 1, 6]-=-). We can conclude strong duality using (a refined form of) Slater’s condition (see, e.g., [1, §3.3] and [6, §5.2]), since the constraints are all linear equalities and inequalities. 3. Fastest mixing... |

1151 |
Nonlinear Programming
- Bertsekas
- 1995
(Show Context)
Citation Context ...ds a sharp lower bound on the optimal SLEM. (5)sConvex Optimization of Graph Laplacian Eigenvalues 5 Both of these conclusions follow from general results for convex optimization problems (see, e.g., =-=[10, 1, 6]-=-). We can conclude strong duality using (a refined form of) Slater’s condition (see, e.g., [1, §3.3] and [6, §5.2]), since the constraints are all linear equalities and inequalities. 3. Fastest mixing... |

416 | Fast linear iterations for distributed averaging
- Xiao, Boyd
- 2004
(Show Context)
Citation Context ...ables w ∈ R m and the (slack) matrix Y = Y T ∈ R n×n (see [8]). 5. Fast averaging Here we describe the problem of choosing edge weights that give fastest averaging, using a classical linear iteration =-=[13]-=-. The nodes start with value x(0) ∈ R n , and at each iteration we update the node values as x(t + 1) = (I − L)x(t). The goal is to choose the edge weights so that xi(t) converges, as rapidly as possi... |

265 | Unsupervised Learning of Image Manifolds by Semidefinite Programming
- Weinberger, Saul
- 2006
(Show Context)
Citation Context ...izes the total variance, given by � i�=j �xi − xj�2 . This problem was recently formulated in the machine learning literature as a method for identifying low dimensional structure in data; see, e.g., =-=[12]-=-.s6 Stephen Boyd 4. Minimum total effective resistance Here we describe the problem of choosing the edge weights to minimize the total effective resistance of a graph, subject to some given total weig... |

228 | Convex Analysis and NonLinear Optimization. Theory and Exmaples - Borwein, Lewis - 2000 |

197 | Distributed average consensus with least-mean-square deviation,” Submitted to
- Xiao, Boyd, et al.
- 2006
(Show Context)
Citation Context ...ghts are negative [13]. 6. Minimum RMS consensus error Here we describe a variation on the fastest linear averaging problem described above, in which an additive random noise perturbs the node values =-=[15]-=-. The iteration is x(t + 1) = (I − L)x(t) + v(t), where v(t) ∈ R n are uncorrelated zero mean unit variance random variables, i.e., E v(t) = 0, E v(t)v(t) T = I, E v(t)v(s) T = 0, t �= s. This iterati... |

150 | L.: Fastest mixing Markov chain on a graph
- Boyd, Diaconis, et al.
- 2004
(Show Context)
Citation Context ...ixing Markov chain In this section we briefly describe the problem of finding the fastest mixing symmetric Markov chain on a given graph. Many more details (and additional references) can be found in =-=[4, 5]-=-. We consider a symmetric Markov chain on the graph G, with transition matrix P ∈ R n×n , where Pij = Pji is the probability of a transition from vertex i to vertex j. Since P is symmetric, the unifor... |

63 | The fastest mixing markov process on a graph and a connection to a maximum variance unfolding problem
- SUN, BOYD, et al.
(Show Context)
Citation Context ...e all linear equalities and inequalities. 3. Fastest mixing Markov process Here we briefly describe the problem of finding the fastest mixing continuous-time symmetric Markov process on a given graph =-=[11]-=-. Consider a continuous-time Markov process on the graph G, with transition rate (or intensity) wl across edge l. The probability density π(t) ∈ R 1×n at time t ≥ 0 is given by π(t) = π(0)e −tL . It f... |

63 | Minimizing effective resistance of a graph
- Ghosh, Boyd, et al.
(Show Context)
Citation Context ... Stephen Boyd 4. Minimum total effective resistance Here we describe the problem of choosing the edge weights to minimize the total effective resistance of a graph, subject to some given total weight =-=[8]-=-. We consider the graph as an electrical circuit or network, with the edge weight representing the conductance (inverse of resistance) of the associated electrical branch. We define Rij as the resista... |

51 | Symmetry analysis of reversible markov chains, 2003. URL citeseer.ist.psu.edu/boyd03symmetry. html - Boyd, Diaconis, et al. |

24 | Optimal scaling of a gradient method for distributed resource allocation - Xiao, Boyd - 2006 |

20 | Nonlinear Programming. Second edition, Athena Scientific - Bertsekas - 1999 |

17 | L.: Fastest mixing Markov chain on a path
- Boyd, Diaconis, et al.
- 2006
(Show Context)
Citation Context ...ixing Markov chain In this section we briefly describe the problem of finding the fastest mixing symmetric Markov chain on a given graph. Many more details (and additional references) can be found in =-=[4, 5]-=-. We consider a symmetric Markov chain on the graph G, with transition matrix P ∈ R n×n , where Pij = Pji is the probability of a transition from vertex i to vertex j. Since P is symmetric, the unifor... |

8 | Upper bounds on algebraic connectivity via convex optimization - Ghosh, Boyd |

7 |
Absolute algebraic connectivity of trees
- Fiedler
- 1990
(Show Context)
Citation Context ... algebraic connectivity of the graph. Fiedler refers to the maximum value of λ2 that can be obtained by allocating a fixed total weight to the edges of a graph, as its absolute algebraic connectivity =-=[7]-=-. The dual of the fastest mixing Markov process problem can be given a very interesting interpretation. It is equivalent to the following problem. We are given some distances d1, . . . , dm on the gra... |

3 | Absolute algebraic connectivity of trees. Linear and Multilinear Algebra 26 - Fiedler - 1990 |

1 |
Minimizing effective resistance of a graph. Submitted to
- Ghosh, Boyd
- 2005
(Show Context)
Citation Context ... Stephen Boyd 4. Minimum total effective resistance Here we describe the problem of choosing the edge weights to minimize the total effective resistance of a graph, subject to some given total weight =-=[8]-=-. We consider the graph as an electrical circuit or network, with the edge weight representing the conductance (inverse of resistance) of the associated electrical branch. We define Rij as the resista... |