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Exploiting sparsity in SDP relaxation for sensor network localization
 SIAM J. Optim
, 2009
"... Abstract. A sensor network localization problem can be formulated as a quadratic optimization problem (QOP). For quadratic optimization problems, semidefinite programming (SDP) relaxation by Lasserre with relaxation order 1 for general polynomial optimization problems (POPs) is known to be equivalen ..."
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Abstract. A sensor network localization problem can be formulated as a quadratic optimization problem (QOP). For quadratic optimization problems, semidefinite programming (SDP) relaxation by Lasserre with relaxation order 1 for general polynomial optimization problems (POPs) is known to be equivalent to the sparse SDP relaxation by Waki et al. with relaxation order 1, except the size and sparsity of the resulting SDP relaxation problems. We show that the sparse SDP relaxation applied to the QOP is at least as strong as the BiswasYe SDP relaxation for the sensor network localization problem. A sparse variant of the BiswasYe SDP relaxation, which is equivalent to the original BiswasYe SDP relaxation, is also derived. Numerical results are compared with the BiswasYe SDP relaxation and the edgebased SDP relaxation by Wang et al.. We show that the proposed sparse SDP relaxation is faster than the BiswasYe SDP relaxation. In fact, the computational efficiency in solving the resulting SDP problems increases as the number of anchors and/or the radio range grow. The proposed sparse SDP relaxation also provides more accurate solutions than the edgebased SDP relaxation when exact distances are given between sensors and anchors and there are only a small number of anchors. Key words. Sensor network localization problem, polynomial optimization problem, semidefinite relaxation, sparsity
An interiorpoint method for minimizing the sum of piecewiselinear convex functions
"... Abstract We consider the problem to minimize the sum of piecewiselinear convex functions under both linear and nonnegative constraints. We convert the piecewiselinear convex problem into a standard form linear programming problem (LP) and apply a primaldual interiorpoint method for the LP. From ..."
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Abstract We consider the problem to minimize the sum of piecewiselinear convex functions under both linear and nonnegative constraints. We convert the piecewiselinear convex problem into a standard form linear programming problem (LP) and apply a primaldual interiorpoint method for the LP. From the solution of the converted problem, we can obtain the solution of the original problem. We establish polynomial convergence of the interiorpoint method for the converted problem and devise the computaion of the Newton direction.
SERIES B: Operations ResearchB447 Exploiting Sparsity in SDP Relaxation for Sensor Network Localization
, 2008
"... Abstract. A sensor network localization problem can be formulated as a quadratic optimization problem (QOP). For quadratic optimization problems, semidefinite programming (SDP) relaxation by Lasserre with relaxation order 1 for general polynomial optimization problems (POPs) is known to be equivalen ..."
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Abstract. A sensor network localization problem can be formulated as a quadratic optimization problem (QOP). For quadratic optimization problems, semidefinite programming (SDP) relaxation by Lasserre with relaxation order 1 for general polynomial optimization problems (POPs) is known to be equivalent to the sparse SDP relaxation by Waki et al. with relaxation order 1, except the size and sparsity of the resulting SDP relaxation problems. We show that the sparse SDP relaxation applied to the QOP is at least as strong as the BiswasYe SDP relaxation for the sensor network localization problem. A sparse variant of the BiswasYe SDP relaxation, which is equivalent to the original BiswasYe SDP relaxation, is also derived. Numerical results are compared with the BiswasYe SDP relaxation and the edgebased SDP relaxation by Wang et al.. We show that the proposed sparse SDP relaxation is faster than the BiswasYe SDP relaxation. In fact, the computational efficiency in solving the resulting SDP problems increases as the number of anchors and/or the radio range grow. The proposed sparse SDP relaxation also provides more accurate solutions than the edgebased SDP relaxation when exact distances are given between sensors and anchors
On Handling Free Variables in InteriorPoint Methods for Conic Linear Optimization
, 2006
"... We revisit a regularization technique of Mészáros for handling free variables within interiorpoint methods for conic linear optimization. We propose a simple computational strategy, supported by a global convergence analysis, for handling the regularization. Using test problems from benchmark suites ..."
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We revisit a regularization technique of Mészáros for handling free variables within interiorpoint methods for conic linear optimization. We propose a simple computational strategy, supported by a global convergence analysis, for handling the regularization. Using test problems from benchmark suites and recent applications, we demonstrate that the modern code SDPT3 modified to incorporate the proposed regularization is able to achieve the same or significantly better accuracy over standard options of splitting variables, using a quadratic cone, and solving indefinite systems.
Right Type Departmental Bulletin Paper
"... It has been a longtime dream in electronic structure theory in physical chem$\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{y}/\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l} $ physics to compute ground state energies of atomic and molecular systems by employing a ..."
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It has been a longtime dream in electronic structure theory in physical chem$\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{y}/\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l} $ physics to compute ground state energies of atomic and molecular systems by employing a variational approach in which the twobody reduced density matrix (RDM) is the unknown variable. Realization of the RDM approach has benefited greatly from recent developments in semidefinite programming (SDP). We present the actual state of this new application of SDP as well as the formulations of these SDPs, which can be arbitrarily large. Numerical experiments using different SDP codes and formulations are given in order to seek for the best choices. The RDM method has several advantages including robustness and provision of high accuracy compared to traditional electronic structure methods, although its computational time and memory consum ption are still extremely large. 1