Yichuan Ding

Sensor Network Localization, Euclidean Distance Matrix completions, and graph realization

We study Semidefinite Programming, SDP, relaxations for Sensor Network Localization, SNL, with anchors and with noisy distance information. The main point of the paper is to view SNL as a (nearest) Euclidean Distance Matrix, EDM, completion problem that does not distinguish between the anchors and the sensors. We show that there are advantages for using the well studied EDM model. In fact, the set of anchors simply corresponds to a given fixed clique for the graph of the EDM problem.We next propose a method of projection when large cliques or dense subgraphs are identified. This projection reduces the size, and improves the stability, of the relaxation. In addition, by viewing the problem as an EDM completion problem, we are able to derive a new approximation scheme for the sensors from the SDP approximation. This yields, on average, better low rank approximations for the low dimensional realizations. This further emphasizes the theme that SNL is in fact just an EDM problem.We solve the SDP relaxations using a primal-dual interior/exterior-point algorithm based on the Gauss-Newton search direction. By not restricting iterations to the interior, we usually get lower rank optimal solutions and thus, better approximations for the SNL problem. We discuss the relative stability and strength of two formulations and the corresponding algorithms that are used. In particular, we show that the quadratic formulation arising from the SDP relaxation is better conditioned than the linearized form that is used in the literature.

Sensor network localization, euclidean distance matrix completions, and graph realization

We study Semidefinite Programming, SDP, relaxations for Sensor Network Localization, SNL, with anchors and with noisy distance information. The main point of the paper is to view SNL as a (nearest) Euclidean Distance Matrix, EDM, completion problem and to show the advantages for using this latter, well studied model. We first show that the current popular SDP relaxation is equivalent to known relaxations in the literature for EDM completions. The existence of anchors in the problem is not special. The set of anchors simply corresponds to a given fixed clique for the graph of the EDM problem. We next propose a method of projection when a large clique or a dense subgraph is identified in the underlying graph. This projection reduces the size, and improves the stability, of the relaxation. In addition, the projection/reduction procedure can be repeated for other given cliques of sensors or for sets of sensors, where many distances are known. Thus, further size reduction can be obtained.

A Low-Dimensional Semidefinite Relaxation for the Quadratic Assignment Problem

The quadratic assignment problem (QAP) is arguably one of the hardest NP-hard discrete optimization problems. Problems of dimension greater than 25 are still considered to be large scale. Current successful solution techniques use branch-and-bound methods, which rely on obtaining strong and inexpensive bounds. In this paper, we introduce a new semidefinite programming (SDP) relaxation for generating bounds for the QAP in the trace formulation. We apply majorization to obtain a relaxation of the orthogonal similarity set of the quadratic part of the objective function. This exploits the matrix structure of QAP and results in a relaxation with much smaller dimension than other current SDP relaxations. We compare the resulting bounds with several other computationally inexpensive bounds such as the convex quadratic programming relaxation (QPB). We find that our method provides stronger bounds on average and is adaptable for branch-and-bound methods.

On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming

We analyze two popular semidefinite programming relaxations for quadratically constrained quadratic programs with matrix variables. These relaxations are based on vector lifting and on matrix lifting; they are of different size and expense. We prove, under mild assumptions, that these two relaxations provide equivalent bounds. Thus, our results provide a theoretical guideline for how to choose a less expensive semidefinite programming relaxation and still obtain a strong bound. The main technique used to show the equivalence and that allows for the simplified constraints is the recognition of a class of nonchordal sparse patterns that admit a smaller representation of the positive semidefinite constraint.

Real-time tracking for sensor networks via sdp and gradient method

The sensor tracking problem is an important problem studied in many different fields. But many of those studies use analysis or machine learning method rather than optimization method. Recently, several approaches have been proposed to solve the static version of the tracking problem, the sensor network localization problem, via Semi-definite Programming(SDP). In this paper, we analyze a new real-time sensor tracking scheme by combining the SDP approach and the gradient method. We show that this approach provides fast and accurate tracking for network sensors. We also discuss the problem of extracting information from the moving sensors, which could be used to predict their movements.

A Nonasymptotic Approach to Analyzing Kidney Exchange Graphs

We propose a novel methodology to study kidney exchange. Using a random graph model of kidney exchange, we propose a nonasymptotic approach to quantifying the effectiveness of transplant chains in ...

A Fluid Model for an Overloaded Bipartite Queueing System with Heterogeneous Matching Utility

We consider a bipartite queueing system (BQS) with multiple types of servers and customers, where different customer-server combinations may generate different utilities. Whenever a server is available, it serves the customer with the highest index, which is the sum of a customers waiting index and the matching index. We call this an {\em M W} index. We assume that the waiting index is an increasing function of a customers waiting time and the matching index depends on both the customers and the servers types. We develop a fluid model to approximate the behavior of such a BQS system, and show that the fluid limit process can be computed over any finite horizon. We develop an efficient algorithm to check whether a steady state of the fluid process exists or not. When a steady state exists, the algorithm also computes one efficiently. We prove that there can be at most one steady state, and that the fluid limit process converges to the steady state under mild conditions. These results enable a policy designer to predict the behavior of a BQS when using an M W index, and to choose an indexing formula that optimizes a given set of performance metrics. We derive a closed-form M W index that optimizes the steady-state performance according to some well-known efficiency and fairness metrics.

A note on the trackability of dynamic sensor networks

Since the last decade, Semidefinite programming (SDP) has found its important application in locating the ad hoc wireless sensor networks. By choosing proper decomposition and computation schemes, SDP has been shown very efficient to handle the localization problem. Previous research also has shown that the SDP locates the sensor networks in Rd correctly provided the underlying framework is strong uniquely localizable. In this paper, we consider the localization problem in a more general and practical scenario, that is, the sensors are in movement following a certain trajectory. We show that given the initial position of each sensor and the instantaneous distance data, the dynamic sensor networks can be can be tracked correctly in the near future when the underlying framework is infinitesimal rigid and the trajectories of the sensors are subject to mild conditions. Our result also provides a way to approximate the sensor trajectories using Taylor series based on the distance data.