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This paper presents an image reconstruction method for positron-emission tomography (PET) based on a penalized, weighted least-squares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a least-squares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. We propose a simple data-based method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive over-relaxation (+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 maximum-likelihood expectation-maximization (ML-EM) 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...

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 Karush-Kuhn-Tucker conditions.

A non--interior 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 non--degeneracy 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 non--interior 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...

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 ?...

We propose a new elimination method for linear and quadratic optimization involving parametric coefficients. In comparison to the classical Fourier-Motzkin method that is of doubly exponential worst-case 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 worst-case optimal. Examples computed in a REDUCE-implementation confirm the superiority of the method over Fourier-Motzkin and its applicability to problems of interesting size.

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 ffl-minimizer, 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 polynomial-time (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, polynomial-time 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...

. We propose new methods for solving the variational inequality problem where the underlying function F is monotone. These methods may be viewed as projection-type 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, projection-type 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 ...

We extend the APlace wirelength-driven standard-cell analytic placement framework of [21] to address timing-driven 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 mixed-size placement, we achieve an average of 4 % wirelength reduction on ISPD02 mixed-size placement benchmarks [18] compared to results of the leading-edge 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

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 interior-point 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 Chen-Mangasarian and Gabriel-Mor'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 homotopy-smoothing method for solving the variational inequality problem, and show that the method converges globally and superlinearly under mild conditions. ...