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Theory of semidefinite programming for sensor network localization
 IN SODA05
, 2005
"... We analyze the semidefinite programming (SDP) based model and method for the position estimation problem in sensor network localization and other Euclidean distance geometry applications. We use SDP duality and interior–point algorithm theories to prove that the SDP localizes any network or graph th ..."
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Cited by 83 (5 self)
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We analyze the semidefinite programming (SDP) based model and method for the position estimation problem in sensor network localization and other Euclidean distance geometry applications. We use SDP duality and interior–point algorithm theories to prove that the SDP localizes any network or graph that has unique sensor positions to fit given distance measures. Therefore, we show, for the first time, that these networks can be localized in polynomial time. We also give a simple and efficient criterion for checking whether a given instance of the localization problem has a unique realization in R 2 using graph rigidity theory. Finally, we introduce a notion called strong localizability and show that the SDP model will identify all strongly localizable sub–networks in the input network.
Initialization in Semidefinite Programming Via a SelfDual SkewSymmetric Embedding
, 1996
"... The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for large scale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, bu ..."
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Cited by 32 (10 self)
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The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for large scale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, but the initialization problem remained unsolved. In this paper we show that the initialization strategy of embedding the problem in a selfdual skewsymmetric problem can also be extended to the semidefinite case. This way the initialization problem of semidefinite problems is solved. This method also provides solution for the initialization of quadratic programs and it is applicable to more general convex problems with conic formulation. Key words: Semidefinite programming, complementarity, skewsymmetric embedding, initialization, selfdual problems, central path. iii 1 Introduction The extension of interior point algorithms from linear programming (LP) to semidefinite programmi...
A Truncated PrimalInfeasible DualFeasible Network Interior Point Method
, 1994
"... . In this paper we introduce the truncated primalinfeasible dualfeasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum cost network flow problem. In each iteration, the linear system that determines the search direction is ..."
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Cited by 29 (3 self)
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. In this paper we introduce the truncated primalinfeasible dualfeasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum cost network flow problem. In each iteration, the linear system that determines the search direction is computed inexactly, and the norm of the resulting residual vector is used in the stopping criteria of the iterative solver employed for the solution of the system. In the implementation, a preconditioned conjugate gradient method is used as the iterative solver. The details of the implementation are described and the code, pdnet, is tested on a large set of standard minimum cost network flow test problems. Computational results indicate that the implementation is competitive with stateoftheart network flow codes. Key Words. Interior point method, linear programming, network flows, primalinfeasible dualfeasible, truncated Newton method, conjugate gradient, maximum flow, experimental test...
Further relaxation of the semidefinite programming approach to sensor network localization
 SIAM Journal on Optimization
, 2008
"... Abstract. Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating the sensor locations, the speed of the SDP approach is not satisfactory for practical applications. In this pa ..."
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Cited by 27 (2 self)
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Abstract. Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating the sensor locations, the speed of the SDP approach is not satisfactory for practical applications. In this paper we propose methods to further relax the SDP relaxation, more precisely, to relax the single semidefinite matrix cone into a set of smallsize semidefinite submatrix cones, which we call a subSDP (SSDP) approach. We present two such relaxations. Although they are weaker than the original SDP relaxation, they retain the key theoretical property, and numerical experiments show that they are both efficient and accurate. The speed of the SSDP is even faster than that of other approaches based on weaker relaxations. The SSDP approach may also pave a way to efficiently solving general SDP problems without sacrificing the solution quality.
On a Homogeneous Algorithm for the Monotone Complementarity Problem
 Mathematical Programming
, 1995
"... We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and compleme ..."
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Cited by 24 (3 self)
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We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interiorpoint and infeasiblestarting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and selfdual, infeasiblestarting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...
Further relaxations of the SDP approach to sensor network localization
, 2006
"... Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating sensor’s locations, the speed of the SDP approach is not satisfactory for practical applications. In this paper we prop ..."
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Cited by 23 (0 self)
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Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating sensor’s locations, the speed of the SDP approach is not satisfactory for practical applications. In this paper we propose methods to further relax the SDP relaxation; more precisely, to relax the single semidefinite matrix cone into a set of smallsize semidefinite matrix cones, which we call the smaller SDP (SSDP) approach. We present two such relaxations; and they are, although weaker than the original SDP relaxation, retaining the key theoretical property and tested to be both efficient and accurate in computation. The speed of the SSDP is even faster than that of other further weaker approaches. The SSDP approach may also pave a way to efficiently solve general SDP relaxations without sacrificing their solution quality.
Duality And SelfDuality For Conic Convex Programming
, 1996
"... This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we i ..."
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Cited by 20 (7 self)
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This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we introduce the notions of weak/strong feasibility or infeasibility for a general primaldual pair of conic convex programs, and then establish various relations between these notions and the duality properties of the problem. In the second half of the paper, we propose a selfdual embedding with the following properties: Any weakly centered sequence converging to a complementary pair either induces a sequence converging to a certificate of strong infeasibility, or induces a sequence of primaldual pairs for which the amount of constraint violation converges to zero, and the corresponding objective values are in the limit not worse than the optimal objective value(s). In case of strong duality, ...
Conic Convex Programming And SelfDual Embedding
 Optim. Methods Softw
, 1998
"... How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the twophase approach or using the socalled big M technique. In the interior point method, there is a more ..."
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Cited by 18 (2 self)
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How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the twophase approach or using the socalled big M technique. In the interior point method, there is a more elegant way to deal with the initialization problem, viz. the selfdual embedding technique proposed by Ye, Todd and Mizuno [30]. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving its embedded selfdual problem. The embedded selfdual problem has a trivial initial solution and has the same structure as the original problem. Hence, it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programming. Since a nonlinear conic convex programming problem may lack the socalled stri...
A Distributed SDP approach for Largescale Noisy Anchorfree Graph Realization with Applications to Molecular Conformation
, 2007
"... We propose a distributed algorithm for solving Euclidean metric realization problems arising from large 3D graphs, using only noisy distance information, and without any prior knowledge of the positions of any of the vertices. In our distributed algorithm, the graph is first subdivided into smaller ..."
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Cited by 16 (0 self)
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We propose a distributed algorithm for solving Euclidean metric realization problems arising from large 3D graphs, using only noisy distance information, and without any prior knowledge of the positions of any of the vertices. In our distributed algorithm, the graph is first subdivided into smaller subgraphs using intelligent clustering methods. Then a semidefinite programming relaxation and gradient search method is used to localize each subgraph. Finally, a stitching algorithm is used to find affine maps between adjacent clusters and the positions of all points in a global coordinate system are then derived. In particular, we apply our method to the problem of finding the 3D molecular configurations of proteins based on a limited number of given pairwise distances between atoms. The protein molecules, all with known molecular configurations, are taken from the Protein Data Bank. Our algorithm is able to reconstruct reliably and efficiently the configurations of large protein molecules from a limited number of pairwise distances corrupted by noise, without incorporating domain knowledge such as the minimum separation distance constraints derived from van der Waals interactions. 1
Complementarity Constraint Qualifications and Simplified BStationarity Conditions for Mathematical Programs with Equilibrium Constraints
, 1998
"... With the aid of some novel complementarity constraint qualifications, we derive some simplied primaldual characterizations of a Bstationary point for a mathematical program with complementarity constraints (MPEC). The approach is based on a locally equivalent piecewise formulation of such a prog ..."
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Cited by 15 (6 self)
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With the aid of some novel complementarity constraint qualifications, we derive some simplied primaldual characterizations of a Bstationary point for a mathematical program with complementarity constraints (MPEC). The approach is based on a locally equivalent piecewise formulation of such a program near a feasible point. The simplied results, which rely heavily on a careful dissection and improved understanding of the tangent cone of the feasible region of the program, bypass the combinatorial characterization that is intrinsic to Bstationarity.