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26
Robust Solutions To LeastSquares Problems With Uncertain Data
, 1997
"... . We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpret ..."
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Cited by 198 (14 self)
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. We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomialtime using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknownbutbounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worstcase residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Leastsquares, uncertainty, robustness, secondorder cone...
An efficient algorithm for minimizing a sum of Euclidean norms with applications
 SIAM Journal on Optimization
, 1997
"... Abstract. In recent years rich theories on polynomialtime interiorpoint algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum o ..."
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Cited by 33 (5 self)
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Abstract. In recent years rich theories on polynomialtime interiorpoint algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By transforming the problem into a standard convex programming problem in conic form, we show that an ɛoptimal solution can be computed efficiently using interiorpoint algorithms. As applications to this problem, polynomialtime algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree topology. In particular, by solving the Newton equation in linear time using Gaussian elimination on leaves of a tree, we present an algorithm which computes an ɛoptimal solution to the shortest network under a given full Steiner topology interconnecting N regular points, in O(N √ N(log(¯c/ɛ)+ log N)) arithmetic operations where ¯c is the largest pairwise distance among the given points. The previous bestknown result on this problem is a graphical algorithm which requires O(N 2) arithmetic operations under certain conditions. Key words. polynomial time, interiorpoint algorithm, minimizing a sum of Euclidean norms, Euclidean facilities location, shortest networks, Steiner minimum trees
An Efficient PrimalDual InteriorPoint Method for Minimizing a Sum of Euclidean Norms
 SIAM J. SCI. COMPUT
, 1998
"... The problem of minimizing a sum of Euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature. This nonsmooth optimization problem arises in many different kinds of modern scientific applications. We derive a primaldual inte ..."
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Cited by 32 (1 self)
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The problem of minimizing a sum of Euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature. This nonsmooth optimization problem arises in many different kinds of modern scientific applications. We derive a primaldual interiorpoint algorithm for the problem, by applying Newton's method directly to a system of nonlinear equations characterizing primal and dual feasibility and a perturbed complementarity condition. The main work at each step consists of solving a system of linear equations (the Schur complement equations). This Schur complement matrix is not symmetric, unlike in linear programming. We incorporate a Mehrotratype predictorcorrector scheme and present some experimental results comparing several variations of the algorithm, including, as one option, explicit symmetrization of the Schur complement with a skew corrector term. We also present results obtained from a code implemented to so...
Computing Limit Loads By Minimizing a Sum of Norms
 IFIP
, 1994
"... This paper treats the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition, such as, for example, the Mises condition. After discretization with the finite element method, using divergencefree elements for the plastic flow, the kinematic formulation ..."
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Cited by 20 (3 self)
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This paper treats the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition, such as, for example, the Mises condition. After discretization with the finite element method, using divergencefree elements for the plastic flow, the kinematic formulation turns into the problem of minimizing a sum of Euclidean vector norms, subject to a single linear constraint. This is a nonsmooth minimization problem, since many of the norms in the sum may vanish at the optimal point. However, efficient solution algorithms for this particular convex optimization problem have recently been developed. The method is applied to test problems in limit analysis in two different plane models: plane strain and plates. In the first case more than 80 percent of the terms in the sum are zero in the optimal solution, causing severe illconditioning. In the last case all terms are nonzero. In both cases the algorithm works very well, and problems are solved which are l...
An Efficient Algorithm for Minimizing a Sum of PNorms
 SIAM Journal on Optimization
, 1997
"... We study the problem of minimizing a sum of pnorms where p is a fixed real number in the interval [1; 1]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic ..."
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Cited by 15 (2 self)
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We study the problem of minimizing a sum of pnorms where p is a fixed real number in the interval [1; 1]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic form. Unlike those in most convex optimization problems, the cone for the pnorm problem is not selfdual unless p = 2. Nevertheless, we are able to construct two logarithmically homogeneous selfconcordant barrier functions for this problem. The barrier parameter of the first barrier function does not depend on p. The barrier parameter of the second barrier function increases with p. Using both barrier functions, we present a primaldual potential reduction algorithm to compute an ffloptimal solution in polynomial time that is independent of p. Computational experiences of a Matlab implementation are also reported. Key words. Shortest network, Steiner minimum trees, facilities location, po...
Computing the Minimum Cost Pipe Network Interconnecting One Sink and Many Sources
 SIAM Journal of Optimization
, 1999
"... Abstract. In this paper, we study the problem of computing the minimum cost pipe network interconnecting a given set of wells and a treatment site, where each well has a given capacity and the treatment site has a capacity that is no less than the sum of all the capacities of the wells. This is a ge ..."
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Cited by 10 (0 self)
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Abstract. In this paper, we study the problem of computing the minimum cost pipe network interconnecting a given set of wells and a treatment site, where each well has a given capacity and the treatment site has a capacity that is no less than the sum of all the capacities of the wells. This is a generalized Steiner minimum tree problem which has applications in communication networks and in groundwater treatment. We prove that there exists a minimum cost pipe network that is the minimum cost network under a full Steiner topology. For each given full Steiner topology, we can compute all the edge weights in linear time. A powerful interiorpoint algorithm is then used to find the minimum cost network under this given topology. We also prove a lower bound theorem which enables pruning in a backtrack method that partially enumerates the full Steiner topologies in search for a minimum cost pipe network. A heuristic ordering algorithm is proposed to enhance the performance of the backtrack algorithm. We then define the notion of koptimality and present an efficient (polynomial time) algorithm for checking 5optimality. We present a 5optimal heuristic algorithm for computing good solutions when the problem size is too large for the exact algorithm. Computational results are presented.
Enhancements in electrical impedance tomography (EIT) image reconstruction for 3D lung imaging
, 2007
"... Electrical Impedance Tomography (EIT) is an imaging technique which calculates the electrical conductivity distribution within a medium from electrical measurements made at a series of electrodes on the medium surface. Reconstruction of conductivity or conductivity change images requires the soluti ..."
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Cited by 5 (0 self)
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Electrical Impedance Tomography (EIT) is an imaging technique which calculates the electrical conductivity distribution within a medium from electrical measurements made at a series of electrodes on the medium surface. Reconstruction of conductivity or conductivity change images requires the solution of an illconditioned nonlinear inverse problem from noisy data. EIT is a hard problem as it is a particularly difficult example of attempting to recover a signal from noise. To date most EIT scanners and algorithms have been designed for 2D applications. This simplifying assumption was originally used due to the prohibitive computational complexity of solving the larger 3D problem. Contemporary PC’s can now calculate 3D solutions, however at the start of this thesis the prevailing algorithms in clinical use remain 2D models that rely on ad hoc tweaking to produce useful reconstructions. The aim of this thesis is to develop enhancements in EIT image reconstruction for 3D lung imaging; to remove some of the limitations that continue to impede its routine use in the clinic. The aim is attained through the systematic achievement of the following four
Two Heuristics for the Steiner Tree Problem
 JOURNAL OF GLOBAL OPTIMIZATION
, 1996
"... The Steiner tree problem is to find the tree with minimal Euclidean length spanning a set of fixed points in the plane, given the ability to add points (Steiner points). The problem is NPhard, so polynomialtime heuristics are desired. We present two such heuristics, both of which utilize an ef ..."
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Cited by 3 (0 self)
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The Steiner tree problem is to find the tree with minimal Euclidean length spanning a set of fixed points in the plane, given the ability to add points (Steiner points). The problem is NPhard, so polynomialtime heuristics are desired. We present two such heuristics, both of which utilize an efficient method for computing a locally optimal tree with a given topology. The first systematically inserts Steiner points between edges of the minimal spanning tree meeting at angles less than 120 degrees, performing a local optimization at the end. The second begins by finding the Steiner tree for three of the fixed points. Then, at each iteration, it introduces a new fixed point to the tree, connecting it to each possible edge by inserting a Steiner point, and minimizes over all connections, performing a local optimization for each. We present a variety of test cases that demonstrate the strengths and weaknesses of both algorithms.
`1based Construction of Polycube Maps from Complex Shapes
"... Polycube maps of triangle meshes have proved useful in a wide range of applications including texture mapping and hexahedral mesh generation. However, constructing either fully automatically or with limited user control a lowdistortion polycube from a detailed surface remains challenging in practi ..."
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Cited by 3 (0 self)
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Polycube maps of triangle meshes have proved useful in a wide range of applications including texture mapping and hexahedral mesh generation. However, constructing either fully automatically or with limited user control a lowdistortion polycube from a detailed surface remains challenging in practice. We propose a variational method for deforming an input triangle mesh into a polycube shape through minimization of the `1norm of the mesh normals, regularized via an asrigidaspossible volumetric distortion energy. Unlike previous work, our approach makes no assumption on the orientation, or on the presence of features in the input model. Userguided control over the resulting polycube map is also offered to increase design flexibility. We demonstrate the robustness, efficiency and controllability of our method on a variety of examples, and explore applications in hexahedral remeshing and quadrangulation.