We propose a method for optimizing the lift-and-project relaxations of binary integer programs introduced by Lovasz and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a re...
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We propose a method for optimizing the lift-and-project relaxations of binary integer programs introduced by Lovasz and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constraints and allows for a Lagrangian approach. We detail an enhanced subgradient method and discuss its efficient implementation. Computational results illustrate that our algorithm produces tight bounds more quickly than state-of-the-art linear and semidefinite solvers.
We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determi...
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We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue lambda(2) Of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize lambda(2) subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, i.e., the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for "unfolding" high-dimensional data that lies on a low-dimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions.
In content-based image retrieval (CBIR), relevant images are identified based on their similarities to query images. Most CBIR algorithms are hindered by the semantic gap between the low-level image features used for ...
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In content-based image retrieval (CBIR), relevant images are identified based on their similarities to query images. Most CBIR algorithms are hindered by the semantic gap between the low-level image features used for computing image similarity and the high-level semantic concepts conveyed in images. One way to reduce the semantic gap is to utilize the log data of users' feedback that has been collected by CBIR systems in history, which is also called "collaborative image retrieval." In this paper, we present a novel metric learning approach, named "regularized metric learning," for collaborative image retrieval, which learns a distance metric by exploring the correlation between low-level image features and the log data of users' relevance judgments. Compared to the previous research, a regularization mechanism is used in our algorithm to effectively prevent overfitting. Meanwhile, we formulate the proposed learning algorithm into a semidefinite programming problem, which can be solved very efficiently by existing software packages and is scalable to the size of log data. An extensive set of experiments has been conducted to show that the new algorithm can substantially improve the retrieval accuracy of a baseline CBIR system using Euclidean distance metric, even with a modest amount of log data. The experiment also indicates that the new algorithm is more effective and more efficient than two alternative algorithms, which exploit log data for image retrieval.
We establish polynomial-time convergence of infeasible-interior-point methods for conic programs over symmetric cones using a wide neighborhood of the central path. The convergence is shown for a commutative family of...
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We establish polynomial-time convergence of infeasible-interior-point methods for conic programs over symmetric cones using a wide neighborhood of the central path. The convergence is shown for a commutative family of search directions used in Schmieta and Alizadeh [ Math. Program. 96 ( 2003), pp. 409 - 438]. Monteiro and Zhang [ Math. Program., 81 ( 1998), pp. 281 - 299] introduced this family of directions when analyzing semidefinite programs. These conic programs include linear and semidefinite programs. This extends the work of Rangarajan and Todd [ Tech. rep. 1388, School of OR & IE, Cornell University, Ithaca, NY, 2003], which established convergence of infeasible-interior-point methods for self-scaled conic programs using the NT direction. Our work is built on earlier analyses by Faybusovich [ J. Comput. Appl. Math., 86 ( 1997), pp. 149 - 175] and Schmieta and Alizadeh [ Math. Program. 96 ( 2003), pp. 409 - 438]. Of independent interest, we provide a constructive proof of Lyapunov lemma in the Jordan algebraic setting.
An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitab...
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An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequence of semidefinite programs. Similar ideas apply to conic optimization over the cone of copositive matrices and to certain optimization problems involving random variables with some known moment information. We bring together several of these approximation results by studying the approximability of cones of positive semidefinite forms ( homogeneous polynomials). Our approach enables us to extend the existing methodology to new approximation schemes. In particular, we derive a novel approximation to the cone of copositive forms, that is, the cone of forms that are positive semidefinite over the nonnegative orthant. The format of our construction can be extended to forms that are positive semidefinite over more general conic domains. We also construct polyhedral approximations to cones of positive semidefinite forms over a polyhedral domain. This opens the possibility of using linear programming technology in optimization problems over these cones.
Multi-way partitioning of an undirected weighted graph where pairwise similarities are assigned as edge weights, provides an important tool for data clustering, but is an NP-hard problem. Spectral relaxation is a popu...
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Multi-way partitioning of an undirected weighted graph where pairwise similarities are assigned as edge weights, provides an important tool for data clustering, but is an NP-hard problem. Spectral relaxation is a popular way of relaxation, leading to spectral clustering where the clustering is peformed by the eigen-decomposition of the (normalized) graph Laplacian. On the other hand, semidefinite relaxation, is an alternative way of relaxing a combinatorial optimization, leading to a convex optimization. In this paper we employ a semidefinite programming (SDP) approach to the graph equipartitioning for clustering, where sufficient conditions for strong duality hold. The method is referred to as semidefinite spectral clustering, where the clustering is based on the eigen-decomposition of the optimal feasible matrix computed by SDR Numerical experiments with several data sets, demonstrate the useful behavior of our semidefinite spectral clustering, compared to existing spectral clustering methods. (c) 2006 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved.
A new stability test for d-dimensional discrete-time systems is presented. It consists of maximizing the minimum eigen-value of a positive definite Gram matrix associated with a polynomial positive on the unit d-circl...
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A new stability test for d-dimensional discrete-time systems is presented. It consists of maximizing the minimum eigen-value of a positive definite Gram matrix associated with a polynomial positive on the unit d-circle. This formulation is based on expressing the polynomial as a sum-of-squares and leads to a semidefinite programming (SDP) problem. Several heuristics are introduced for reducing the complexity of the problem in the case of sparse polynomials. Although in its practical form the test is based on a sufficient condition, the experimental results show that correct stability decisions are given. Comparisons with previous methods are favorable.
Motivated by applications to sensor, peer-to-peer, and ad hoc networks, we study distributed algorithms, also known as gossip algorithms, for exchanging information and for computing in an arbitrarily connected networ...
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Motivated by applications to sensor, peer-to-peer, and ad hoc networks, we study distributed algorithms, also known as gossip algorithms, for exchanging information and for computing in an arbitrarily connected network of nodes. The topology of such networks changes continuously as new nodes join and old nodes leave the network. Algorithms for such networks need to be robust against changes in topology. Additionally, nodes in sensor networks operate under limited computational, communication, and energy resources. These constraints have motivated the design of "gossip" algorithms: schemes which distribute the computational burden and in which a node communicates with a randomly chosen neighbor. We analyze the averaging problem under the gossip constraint for an arbitrary network graph, and find that the averaging time of a gossip algorithm depends on the second largest eigenvalue of a doubly stochastic matrix characterizing the algorithm. Designing the fastest gossip algorithm corresponds to minimizing this eigenvalue, which is a semidefinite program (SDP). In general, SDPs cannot be solved in a distributed fashion;however, exploiting problem structure, we propose a distributed subgradient method that solves the optimization problem over the network. The relation of averaging time to the second largest eigenvalue naturally relates it to the mixing time of a random walk with transition probabilities derived from the gossip algorithm. We use this connection to study the performance and scaling of gossip algorithms on two popular networks: Wireless Sensor Networks, which are modeled as Geometric Random Graphs, and the Internet graph under the so-called Preferential Connectivity (PC) model.
We investigate the augmented Lagrangian penalty function approach to solve semidefinite programs. It turns out that this method generates iterates which lie on the boundary of the cone of semidefinite matrices which a...
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We investigate the augmented Lagrangian penalty function approach to solve semidefinite programs. It turns out that this method generates iterates which lie on the boundary of the cone of semidefinite matrices which are driven to the affine subspace described by the linear equations defining the semidefinite program. We provide some computational experience with this method and show in particular, that it allows to compute the theta number of a graph to reasonably high accuracy for instances which are beyond reach by other methods.
We study the properties of the augmented Lagrangian function for nonlinear semidefinite programming. It is shown that, under a set of sufficient conditions, the augmented Lagrangian algorithm is locally convergent whe...
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We study the properties of the augmented Lagrangian function for nonlinear semidefinite programming. It is shown that, under a set of sufficient conditions, the augmented Lagrangian algorithm is locally convergent when the penalty parameter is larger than a certain threshold. An error estimate of the solution, depending on the penalty parameter, is also established.
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