further relaxations of the semidefinite programming approach to sensor network localization 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 small size semidefinite submatrix cones which we call a sub sdp 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

small families of complex lines for testing holomorphic extendibility let b be the open unit ball in bbb c 2 this paper deals with the analog of hartogs separate analyticity theorem for cr functions on the sphere bb we prove such a theorem for functions in c infty bb if a b in overline b a not b and if f in c infty bb extends holomorphically into b along any complex line passing through either a or b then f extends holomorphically through b on the other hand for each k in bbb n there is a function f in c k bb which extends holomorphically into b along any complex line passing through either a or b yet f does not extend holomorphically through b more generally in the paper we obtain a fairly complete description of pairs of points a b in bbb c 2 a not b such that if f in c infty bb extends holomorphically into b along every complex line passing through either a or b that meets b then f extends holomorphically through b

exploiting sparsity in sdp relaxation for sensor network localization a sensor network localization problem can be formulated as a quadratic optimization problem qop for qops 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 for 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 biswas ye sdp relaxation for the sensor network localization problem a sparse variant of the biswas ye sdp relaxation which is equivalent to the original biswas ye sdp relaxation is also derived we compare numerically the sparse sdp relaxation applied to the qop the biswas ye sdp relaxation its sparse variant and the edge based sdp relaxation by wang et al to confirm the effectiveness of the proposed techniques for exploiting the sparsity in sdp relaxation for sensor network localization problems the sparse sdp relaxation applied to the qop is much faster than the biswas ye sdp relaxation and the sparse variant of the biswas ye sdp relaxation outperforms all other sdp relaxations in speed

testing families of complex lines for the unit ball abstract let p 1 and p 2 be two points in c 2 such that the complex line joining them is tangent to the unit sphere b 2 we prove that if a real analytic function f on b 2 extends holomorphically on every complex line through p 1 and p 2 then f extends holomorphically to the whole ball b 2

holomorphic extension from the sphere to the ball abstract real analytic functions on the boundary of the sphere which have separate holomorphic extension along the complex lines through a boundary point have holomorphic extension to the ball this was proved in baracco 2009 4 by an argument of cr geometry we give here an elementary proof based on the expansion in holomorphic and antiholomorphic powers

a distributed sdp approach for large scale noisy anchor free graph realization with applications to molecular conformation we propose a distributed algorithm for solving euclidean metric realization problems arising from large 3 d 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 are 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 3 d 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

the molecule problem exploiting structure in global optimization the molecule problem is that of determining the relative locations of a set of objects in euclidean space relying only upon a sparse set of pairwise distance measurements this np hard problem has applications in the determination of molecular conformation the molecule problem can be naturally expressed as a continuous global optimization problem but it also has a rich combinatorial structure this paper investigates how that structure can be exploited to simplify the optimization problem in particular we present a novel divide and conquer algorithm in which a large global optimization problem is replaced by a sequence of smaller ones since the cost of the optimization can grow exponentially with problem size this approach holds the promise of a substantial improvement in performance our algorithmic development relies upon some recently published results in graph theory we describe an implementation of this algorithm and report some results of its performance on a sample molecule

analyticity of functions analytic on circles abstract let be the open unit disc in c let p b and let f be a continuous function on which extends holomorphically from each circle in centered at the origin and from each circle in which passes through p then f is holomorphic on

on bar frameworks stress matrices and semidefinite programming a bar framework g p in r dimensional euclidean space is a graph g on the vertices 1 2 n where each vertex i is located at point p i in mathbb r r given a framework g p in mathbb r r a problem of great interest is that of determining whether or not there exists another framework g q not obtained from g p by a rigid motion such that q i q j 2 p i p j 2 for each edge i j of g this problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework g q is restricted to be in the same r dimensional space or not the stress matrix s of a bar framework g p plays a key role in these and other related problems in this paper semidefinite programming sdp theory is used to address in a unified manner several problems concerning universal rigidity new results are presented as well as new proofs of previously known theorems in particular we use the notion of sdp non degeneracy to obtain a sufficient condition for universal rigidity and we show that this condition yields the previously known sufficient condition for generic universal rigidity we present new results concerning positive semidefinite stress matrices and we use a semidefinite version of farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix s

testing analyticity on circles consider a continuous one parameter family of circles in a complex plane that contains two circles lying in the exterior of one another we prove that if a continuous function on the union of the circles extends holomorphically into each circle then the function is holomorphic in the interior of the union of the circles

orthogonal testing families and holomorphic extension from the sphere to the ball let mathbb b 2 denote the open unit ball in mathbb c 2 and let p in mathbb c 2 overline mathbb b 2 we prove that if f is an analytic function on the sphere partial mathbb b 2 that extends holomorphically in each variable separately and along each complex line through p then f is the trace of a holomorphic function in the ball

exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion a basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities the sparsity characterized with a chordal graph structure can be detected in the variable matrix or in a linear or nonlinear matrix inequality constraint of the problem we classify the sparsity in two types the domain space sparsity d space sparsity for the symmetric matrix variable in the objective and or constraint functions of the problem which is required to be positive semidefinite and the range space sparsity r space sparsity for a linear or nonlinear matrix inequality constraint of the problem four conversion methods are proposed in this framework two for exploiting the d space sparsity and the other two for exploiting the r space sparsity when applied to a polynomial semidefinite program sdp these conversion methods enhance the structured sparsity of the problem called the correlative sparsity as a result the resulting polynomial sdp can be solved more effectively by applying the sparse sdp relaxation preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

holomorphic extensions from open families of circles for a circle z c z c p write z w z a w a 2 0 z a p a continuous function f on extends holomorphically from into the disc bounded by if and only if the function f z z f z defined on z z z e r has a bounded holomorphic extension into a r in the paper we consider open connected families of circles c write u u e c and assume that a continuous function on u extends holomorphically from each e c we show that this happens if and only if the function f z z f z defined on z z z e u has a bounded holomorphic extension into the domain u r e q for each open family q compactly contained in c this allows us to use known facts from several complex variables in particular we use the edge of the wedge theorem to prove a theorem on real analyticity of such functions

the l p l p cr hartogs separate analyticity theorem for convex domains in this note a very general theorem of the cr hartogs type is proved for almost generic strictly convex domains in mathbf c mathbf n with real analytic boundary given such a domain d and given an l p function f on partial d which has holomorphic extensions on the slices of d by complex lines parallel to the coordinate axes f must be cr i e f has a holomorphic extension to d which is in the hardy space h p d this is the first general result of cr hartogs type which is not for the ball or some other domain with symmetries and holds for l 1 functions as a corollary the szeg kernel is shown to be the strong operator limit of pi 1 pi 2 n where pi i is the projection onto z i holomorphically extendible l 2 partial d functions in mathbf c mathbf 2 with a slightly more complicated formula in mathbf c mathbf n

meromorphic extensions from small families of circles and holomorphic extensions from spheres abstract let b be the open unit ball in c and let a b c be three points in c which do not lie in a complex line such that the complex line through a bmeetsb and such that if one of the points a b is inb and the other in c b then 6 1 and such that at least one of the numbers is different from 1 we prove that if a continuous function f on bb extends holomorphically into b along each complex line which meets a b c then f extends holomorphically through b this generalizes the recent result of l baracco who proved such a result in the case when the points a b c are contained in b

spaseloc an adaptive subproblem algorithm for scalable wireless sensor network localization an adaptive rule based algorithm spaseloc is described to solve localization problems for ad hoc wireless sensor networks a large problem is solved as a sequence of very small subproblems each of which is solved by semidefinite programming relaxation of a geometric optimization model the subproblems are generated according to a set of sensor anchor selection rules computational results compared with existing approaches show that the spaseloc algorithm scales well and provides excellent localization accuracy

universal rigidity and edge sparsification for sensor network localization owing to their high accuracy and ease of formulation there has been great interest in applying convex optimization techniques particularly that of semidefinite programming sdp relaxation to tackle the sensor network localization problem in recent years however a drawback of such techniques is that the resulting convex program is often expensive to solve in order to speed up computation various edge sparsification heuristics have been proposed whose aim is to reduce the number of edges in the input graph although these heuristics do reduce the size of the convex program and hence make it faster to solve they are often ad hoc in nature and do not preserve the localization properties of the input as such one often has to face a tradeoff between solution accuracy and computational effort in this paper we propose a novel edge sparsification heuristic that can provably preserve the localization properties of the original input at the heart of our heuristic is a graph decomposition procedure which allows us to identify certain sparse generically universally rigid subgraphs of the input graph our computational results show that the proposed approach can significantly reduce the computational and memory complexities of sdp based algorithms for solving the sensor network localization problem moreover it compares favorably with existing speedup approaches both in terms of accuracy and solution time

optimality of the paterson stockmeyer method for evaluating matrix polynomials and rational matrix functions abstract many state of the art algorithms reduce the computation of transcendental matrix functions to the evaluation of polynomial or rational approximants at a matrix argument this task can be accomplished efficiently by resorting to the paterson stockmeyer method an evaluation scheme originally developed for polynomials of matrices that extends quite naturally to rational functions an important feature of this technique is that the number of matrix multiplications required to evaluate an approximant of order n grows slower than n itself with the result that different approximants yield the same asymptotic computational cost we analyze the number of matrix multiplications required by the paterson stockmeyer method and by two widely used generalizations one for evaluating diagonal pade approximants of general functions and one specifically tailored to those of the exponential in all the three cases we identify the approximants of maximum order for any given computational cost

second order cone programming relaxation of sensor network localization the sensor network localization problem has been much studied recently biswas and ye proposed a semidefinite programming sdp relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed here we study a second order cone programming socp relaxation of this problem motivated by its simpler structure and its potential to be solved faster than sdp we show that the socp relaxation though weaker than the sdp relaxation has nice properties that make it useful as a problem preprocessor in particular sensors that are uniquely positioned among interior solutions of the socp relaxation are accurate up to the square root of the distance error thus these sensors which are easily identified are accurately positioned in our numerical simulation the interior solution found can accurately position up to 80 90p of the sensors we also propose a smoothing coordinate gradient descent method for finding an interior solution that is faster than an interior point method