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65
Convex optimization of graph Laplacian eigenvalues
 IN INTERNATIONAL CONGRESS OF MATHEMATICIANS
"... We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this p ..."
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Cited by 51 (0 self)
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We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this problem is convex, i.e., it involves minimizing a convex function (or maximizing a concave function) over a convex set. This allows us to give simple necessary and sufficient optimality conditions, derive interesting dual problems, find analytical solutions in some cases, and efficiently compute numerical solutions in all cases. In this overview we briefly describe some more specific cases of this general problem, which have been addressed in a series of recent papers. • Fastest mixing Markov chain. Find edge transition probabilities that give the fastest mixing (symmetric, discretetime) Markov chain on the graph. • Fastest mixing Markov process. Find the edge transition rates that give the fastest mixing (symmetric, continuoustime) Markov process on the graph. • Absolute algebraic connectivity. Find edge weights that maximize the algebraic
Kron Reduction of Graphs with Applications to Electrical Networks
"... Consider a weighted undirected graph and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to selfloops. The Kron reduction of this graph is again a graph whose Laplacian matrix is obtained by the Schur complement of the original Laplacian mat ..."
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Cited by 39 (16 self)
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Consider a weighted undirected graph and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to selfloops. The Kron reduction of this graph is again a graph whose Laplacian matrix is obtained by the Schur complement of the original Laplacian matrix with respect to a specified subset of nodes. The Kron reduction process is ubiquitous in classic circuit theory and in related disciplines such as electrical impedance tomography, smart grid monitoring, transient stability assessment, and analysis of power electronics. Kron reduction is also relevant in other physical domains, in computational applications, and in the reduction of Markov chains. Related concepts have also been studied as purely theoretic problems in the literature on linear algebra. In this paper we analyze the Kron reduction process from the viewpoint of algebraic graph theory. Specifically, we provide a comprehensive and detailed graphtheoretic analysis of Kron reduction encompassing topological, algebraic, spectral, resistive, and sensitivity analyses. Throughout our theoretic elaborations we especially emphasize the practical applicability of our results to various problem setups arising in engineering, computation, and linear algebra. Our analysis of Kron reduction leads to novel insights both on the mathematical and the physical side.
Design of optimal sparse feedback gains via the alternating direction method of multipliers
 IEEE Trans. Automat. Control
"... Abstract—We design sparse and block sparse feedback gains that minimize the variance amplification (i.e., the norm) of distributed systems. Our approach consists of two steps. First, we identify sparsity patterns of feedback gains by incorporating sparsitypromoting penalty functions into the optim ..."
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Cited by 33 (8 self)
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Abstract—We design sparse and block sparse feedback gains that minimize the variance amplification (i.e., the norm) of distributed systems. Our approach consists of two steps. First, we identify sparsity patterns of feedback gains by incorporating sparsitypromoting penalty functions into the optimal control problem, where the added terms penalize the number of communication links in the distributed controller. Second, we optimize feedback gains subject to structural constraints determined by the identified sparsity patterns. In the first step, the sparsity structure of feedback gains is identified using the alternating direction method of multipliers, which is a powerful algorithm wellsuited to large optimization problems. This method alternates between promoting the sparsity of the controller and optimizing the closedloop performance, which allows us to exploit the structure of the corresponding objective functions. In particular, we take advantage of the separability of the sparsitypromoting penalty functions to decompose the minimization problem into subproblems that can be solved analytically. Several examples are provided to illustrate the effectiveness of the developed approach. Index Terms—Alternating direction method of multipliers (ADMM), communication architectures, continuation methods, minimization, optimization, separable penalty functions, sparsitypromoting optimal control, structured distributed design. I.
Robustness of noisy consensus dynamics with directed communication
 in Proceedings of the American Control Conference
, 2010
"... Abstract — In this paper we study robustness of consensus in networks of coupled single integrators driven by white noise. Robustness is quantified as the H2 norm of the closedloop system. In particular we investigate how robustness depends on the properties of the underlying (directed) communicati ..."
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Cited by 25 (7 self)
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Abstract — In this paper we study robustness of consensus in networks of coupled single integrators driven by white noise. Robustness is quantified as the H2 norm of the closedloop system. In particular we investigate how robustness depends on the properties of the underlying (directed) communication graph. To this end several classes of directed and undirected communication topologies are analyzed and compared. The tradeoff between speed of convergence and robustness to noise is also investigated. I.
Algorithms for leader selection in large dynamical networks: Noisefree leaders
 in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference
, 2011
"... Abstract — We examine the leader selection problem in multiagent dynamical networks where leaders, in addition to relative information from their neighbors, also have access to their own states. We are interested in selecting an a priori specified number of agents as leaders in order to minimize th ..."
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Cited by 22 (8 self)
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Abstract — We examine the leader selection problem in multiagent dynamical networks where leaders, in addition to relative information from their neighbors, also have access to their own states. We are interested in selecting an a priori specified number of agents as leaders in order to minimize the total variance of the stochastically forced network. Combinatorial nature of this optimal control problem makes computation of the global minimum difficult. We propose a convex relaxation to obtain a lower bound on the global optimal value, and use simple but efficient greedy algorithms to obtain an upper bound. Furthermore, we employ the alternating direction method of multipliers to search for a local minimum. Two examples are provided to illustrate the effectiveness of the developed methods. Index Terms — Alternating direction method of multipliers, consensus, convex optimization/relaxation, greedy algorithm, leader selection, performance bounds, variance amplification. I.
Subgraph Sparsification and Nearly Optimal Ultrasparsifiers
, 2010
"... We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs G and W and an integer k, we are asked to find a kedge weighted graph Wk such that G + Wk is a good spectral sparsifer of G + ..."
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Cited by 17 (3 self)
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We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs G and W and an integer k, we are asked to find a kedge weighted graph Wk such that G + Wk is a good spectral sparsifer of G + W. We will refer to this problem as the subgraph (spectral) sparsification. We present a nontrivial condition on G and W such that a good sparsifier exists and give a polynomialtime algorithm to find the sparsifer. As a significant application of our technique, we show that for each positive integer k, every nvertex weighted graph has an (n − 1 + k)edge spectral sparsifier with relative condition number at most n log n Õ(log log n) where Õ() hides k lower order terms. Our bound nearly settles a question left open by Spielman and Teng about ultrasparsifiers, which is a key component in their nearly lineartime algorithms for solving diagonally dominant symmetric linear systems. We also present another application of our technique to spectral optimization in which the goal is to maximize the algebraic connectivity of a graph (e.g. turn it into an expander) with a limited number of edges.
Algorithms for leader selection in stochastically forced consensus networks
 IEEE Trans. Automat. Control
"... Abstract—We are interested in assigning a prespecified number of nodes as leaders in order to minimize the meansquare deviation from consensus in stochastically forced networks. This problem arises in several applications including control of vehicular formations and localization in sensor networ ..."
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Cited by 12 (3 self)
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Abstract—We are interested in assigning a prespecified number of nodes as leaders in order to minimize the meansquare deviation from consensus in stochastically forced networks. This problem arises in several applications including control of vehicular formations and localization in sensor networks. For networks with leaders subject to noise, we show that the Boolean constraints (which indicate whether a node is a leader) are the only source of nonconvexity. By relaxing these constraints to their convex hull we obtain a lower bound on the global optimal value. We also use a simple but efficient greedy algorithm to identify leaders and to compute an upper bound. For networks with leaders that perfectly follow their desired trajectories, we identify an additional source of nonconvexity in the form of a rank constraint. Removal of the rank constraint and relaxation of the Boolean constraints yields a semidefinite program for which we develop a customized algorithm wellsuited for large networks. Several examples ranging from regular lattices to random graphs are provided to illustrate the effectiveness of the developed algorithms. Index Terms—Alternating direction method of multipliers (ADMMs), consensus networks, convex optimization, convex relaxations, greedy algorithm, leader selection, performance bounds, semidefinite programming (SDP), sensor selection, variance amplification. I.
Design of Optimal Sparse Interconnection Graphs for Synchronization of Oscillator Networks
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2014
"... We study the optimal design of a conductance network as a means for synchronizing a given set of oscillators. Synchronization is achieved when all oscillator voltages reach consensus, and performance is quantified by the meansquare deviation from the consensus value. We formulate optimization probl ..."
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Cited by 10 (3 self)
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We study the optimal design of a conductance network as a means for synchronizing a given set of oscillators. Synchronization is achieved when all oscillator voltages reach consensus, and performance is quantified by the meansquare deviation from the consensus value. We formulate optimization problems that address the tradeoff between synchronization performance and the number and strength of oscillator couplings. We promote the sparsity of the coupling network by penalizing the number of interconnection links. For identical oscillators, we establish convexity of the optimization problem and demonstrate that the design problem can be formulated as a semidefinite program. Finally, for special classes of oscillator networks we derive explicit analytical expressions for the optimal conductance values.
Design is as easy as optimization
 In 33rd International Colloquium on Automata, Languages and Programming (ICALP
, 2006
"... We consider the class of maxmin and minmax optimization problems subject to a global budget (or weight) constraint and we undertake a systematic algorithmic and complexitytheoretic study of such problems, which we call problems design problems. Every optimization problem leads to a natural design ..."
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Cited by 8 (0 self)
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We consider the class of maxmin and minmax optimization problems subject to a global budget (or weight) constraint and we undertake a systematic algorithmic and complexitytheoretic study of such problems, which we call problems design problems. Every optimization problem leads to a natural design problem. Our main result uses techniques of FreundSchapire [FS99] from learning theory, and its generalizations, to show that for a large class of optimization problems, the design version is as easy as the optimization version. We also observe a close relationship between design problems and packing problems; this yields relationships between fractional packing of spanning and Steiner trees in a graph, the strength of the graph, and the integrality gap of the bidirected cut relaxation for the graph. 1