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85
Penalized Weighted LeastSquares Image Reconstruction for Positron Emission Tomography
 IEEE TR. MED. IM
, 1994
"... This paper presents an image reconstruction method for positronemission tomography (PET) based on a penalized, weighted leastsquares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a leastsquares objective function is as approp ..."
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Cited by 86 (37 self)
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This paper presents an image reconstruction method for positronemission tomography (PET) based on a penalized, weighted leastsquares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a leastsquares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. We propose a simple databased method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive overrelaxation (+SOR) algorithm converges rapidly to the global minimum of the PWLS objective. Quantitative simulation results demonstrate that the bias/variance tradeoff of the PWLS+SOR method is comparable to the maximumlikelihood expectationmaximization (MLEM) method (but with fewer iterations), and is improved relative to the conventional filtered backprojection (FBP) method. Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminat...
Characterizations of strong regularity for variational inequalities over polyhedral convex sets
 SIAM J. OPTIMIZATION
, 1996
"... Linear and nonlinear variational inequality problems over a polyhedral convex set are analyzed parametrically. Robinson’s notion of strong regularity, as a criterion for the solution set to be a singleton depending Lipschitz continuously on the parameters, is characterized in terms of a new “critica ..."
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Cited by 46 (15 self)
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Linear and nonlinear variational inequality problems over a polyhedral convex set are analyzed parametrically. Robinson’s notion of strong regularity, as a criterion for the solution set to be a singleton depending Lipschitz continuously on the parameters, is characterized in terms of a new “critical face” condition and in other ways. The consequences for complementarity problems are worked out as a special case. Application is also made to standard nonlinear programming problems with parameters that include the canonical perturbations. In that framework a new characterization of strong regularity is obtained for the variational inequality associated with the KarushKuhnTucker conditions.
The Global Linear Convergence of a NonInterior PathFollowing Algorithm for Linear Complementarity Problems
 Mathematics of Operations Research
, 1997
"... A noninterior path following algorithm is proposed for the linear complementarity problem. The method employs smoothing techniques introduced by Kanzow. If the LCP is P 0 +R 0 and satisfies a nondegeneracy condition due to Fukushima, Luo, and Pang, then the algorithm is globally linearly converg ..."
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Cited by 31 (3 self)
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A noninterior path following algorithm is proposed for the linear complementarity problem. The method employs smoothing techniques introduced by Kanzow. If the LCP is P 0 +R 0 and satisfies a nondegeneracy condition due to Fukushima, Luo, and Pang, then the algorithm is globally linearly convergent. As with interior point path following methods, the convergence theory relies on the notion of a neighborhood for the central path. However, the choice of neighborhood differs significantly from that which appears in the interior point literature. Numerical experiments are presented that illustrate the significance of the neighborhood concept for this class of methods. 1 Introduction In this paper, we develop a noninterior path following method for the linear complementarity problem: LCP(q;M): Find (x ; y ) 2 IR n \Theta IR n satisfying Mx \Gamma y + q = 0; (1.1) x 0; y 0; (x ) T y = 0; (1.2) where M 2 IR n\Thetan and q 2 IR n . The global line...
Solution of General Linear Complementarity Problems via Nondifferentiable Concave Minimization
 Acta Mathematica Vietnamica
, 1997
"... Finite termination, at point satisfying the minimum principle necessary optimality condition, is established for a stepless (no line search) successive linearization algorithm (SLA) for minimizing a nondifferentiable concave function on a polyhedral set. The SLA is then applied to the general linear ..."
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Cited by 26 (11 self)
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Finite termination, at point satisfying the minimum principle necessary optimality condition, is established for a stepless (no line search) successive linearization algorithm (SLA) for minimizing a nondifferentiable concave function on a polyhedral set. The SLA is then applied to the general linear complementarity problem (LCP), formulated as minimizing a piecewiselinear concave error function on the usual polyhedral feasible region defining the LCP. When the feasible region is nonempty, the concave error function always has a global minimum at a vertex, and the minimum is zero if and only if the LCP is solvable. The SLA terminates at a solution or stationary point of the problem in a finite number of steps. A special case of the proposed algorithm [8] solved without failure 80 consecutive cases of the LCP formulation of the knapsack feasibilty problem, ranging in size between 10 and 3000. 1 Introduction We consider the classical linear complementarity problem (LCP) [4, 12, 5] 0 x ?...
Modified ProjectionType Methods For Monotone Variational Inequalities
 SIAM Journal on Control and Optimization
, 1996
"... . We propose new methods for solving the variational inequality problem where the underlying function F is monotone. These methods may be viewed as projectiontype methods in which the projection direction is modified by a strongly monotone mapping of the form I \Gamma ffF or, if F is affine with un ..."
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Cited by 26 (9 self)
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. We propose new methods for solving the variational inequality problem where the underlying function F is monotone. These methods may be viewed as projectiontype methods in which the projection direction is modified by a strongly monotone mapping of the form I \Gamma ffF or, if F is affine with underlying matrix M , of the form I + ffM T , with ff 2 (0; 1). We show that these methods are globally convergent and, if in addition a certain error bound based on the natural residual holds locally, the convergence is linear. Computational experience with the new methods is also reported. Key words. Monotone variational inequalities, projectiontype methods, error bound, linear convergence. AMS subject classifications. 49M45, 90C25, 90C33 1. Introduction. We consider the monotone variational inequality problem of finding an x 2 X satisfying F (x ) T (x \Gamma x ) 0 8x 2 X; (1) where X is a closed convex set in ! n and F is a monotone and continuous function from ! n to ...
On HomotopySmoothing Methods for Variational Inequalities
"... A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interiorpoint and smoothing methods, have been employed to solve the prob ..."
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Cited by 23 (5 self)
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A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interiorpoint and smoothing methods, have been employed to solve the problem. All of these methods use parametric functions and construct perturbed equations to approximate the problem. The solution to the perturbed system constitutes a smooth trajectory leading to the solution of the original variational inequality problem. The methods generate iterates to follow the trajectory. Among these methods ChenMangasarian and GabrielMor'e proposed a class of smooth functions to approximate F . In this paper, we study several properties of the trajectory defined by solutions of these smooth systems. We propose a homotopysmoothing method for solving the variational inequality problem, and show that the method converges globally and superlinearly under mild conditions. ...
Parametric Linear and Quadratic Optimization by Elimination
 UNIVERSITÄT PASSAU
, 1994
"... We propose a new elimination method for linear and quadratic optimization involving parametric coefficients. In comparison to the classical FourierMotzkin method that is of doubly exponential worstcase complexity our method is singly exponential in the worst case. Moreover it applies also to the ..."
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Cited by 22 (7 self)
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We propose a new elimination method for linear and quadratic optimization involving parametric coefficients. In comparison to the classical FourierMotzkin method that is of doubly exponential worstcase complexity our method is singly exponential in the worst case. Moreover it applies also to the minimization of a quadratic objective functions without convexity hypothesis under linear constraints, and to objective functions with arbitrary parametric coefficients. For problems with additive parameters the method is worstcase optimal. Examples computed in a REDUCEimplementation confirm the superiority of the method over FourierMotzkin and its applicability to problems of interesting size.
Approximation Algorithms for Quadratic Programming
, 1998
"... We consider the problem of approximating the global minimum of a general quadratic program (QP) with n variables subject to m ellipsoidal constraints. For m = 1, we rigorously show that an fflminimizer, where error ffl 2 (0; 1), can be obtained in polynomial time, meaning that the number of arithme ..."
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Cited by 22 (5 self)
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We consider the problem of approximating the global minimum of a general quadratic program (QP) with n variables subject to m ellipsoidal constraints. For m = 1, we rigorously show that an fflminimizer, where error ffl 2 (0; 1), can be obtained in polynomial time, meaning that the number of arithmetic operations is a polynomial in n, m, and log(1=ffl). For m 2, we present a polynomialtime (1 \Gamma 1 m 2 )approximation algorithm as well as a semidefinite programming relaxation for this problem. In addition, we present approximation algorithms for solving QP under the box constraints and the assignment polytope constraints. Key words. Quadratic programming, global minimizer, polynomialtime approximation algorithm The work of the first author was supported by the Australian Research Council; the second author was supported in part by the Department of Management Sciences of the University of Iowa where he performed this research during a research leave, and by the Natural Scien...
An Analytic Placer for MixedSize Placement and TimingDriven
 Placement”, Proc. Int. Conf. Computer Aided Design
"... We extend the APlace wirelengthdriven standardcell analytic placement framework of [21] to address timingdriven and mixedsize (“boulders and dust”) placement. Compared with timingdriven industry tools, evaluated by commercial detailed routing and STA, we achieve an average of 8.4 % reduction in c ..."
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Cited by 22 (3 self)
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We extend the APlace wirelengthdriven standardcell analytic placement framework of [21] to address timingdriven and mixedsize (“boulders and dust”) placement. Compared with timingdriven industry tools, evaluated by commercial detailed routing and STA, we achieve an average of 8.4 % reduction in cycle time and 7.5 % reduction in wirelength for a set of six industry testcases. For mixedsize placement, we achieve an average of 4 % wirelength reduction on ISPD02 mixedsize placement benchmarks [18] compared to results of the leadingedge solver, Feng Shui (v2.4) [25]. We are currently evaluating our placer on industry testcases that combine the challenges of timing constraints, large instance sizes, and embedded blocks (both fixed and unfixed). 1