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Linear complementarity, linear and nonlinear programming
01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 993 citation(s) till now. The article focuses on the topic(s): Mixed complementarity problem & Complementarity theory.
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TL;DR: The field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory as mentioned in this paper.
Abstract: Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.
1,571 citations
TL;DR: A special class of indefinite quadratic programs is constructed, with simple constraints and integer data, and it is shown that checking (a) or (b) on this class is NP-complete.
Abstract: In continuous variable, smooth, nonconvex nonlinear programming, we analyze the complexity of checking whether(a)a given feasible solution is not a local minimum, and(b)the objective function is not bounded below on the set of feasible solutions.
We construct a special class of indefinite quadratic programs, with simple constraints and integer data, and show that checking (a) or (b) on this class is NP-complete. As a corollary, we show that checking whether a given integer square matrix is not copositive, is NP-complete.
1,002 citations
17 Jul 2013
TL;DR: The Multi-Parametric Toolbox is a collection of algorithms for modeling, control, analysis, and deployment of constrained optimal controllers developed under Matlab that features a powerful geometric library that extends the application of the toolbox beyond optimal control to various problems arising in computational geometry.
Abstract: The Multi-Parametric Toolbox is a collection of algorithms for modeling, control, analysis, and deployment of constrained optimal controllers developed under Matlab. It features a powerful geometric library that extends the application of the toolbox beyond optimal control to various problems arising in computational geometry. The new version 3.0 is a complete rewrite of the original toolbox with a more flexible structure that offers faster integration of new algorithms. The numerical side of the toolbox has been improved by adding interfaces to state of the art solvers and by incorporation of a new parametric solver that relies on solving linear-complementarity problems. The toolbox provides algorithms for design and implementation of real-time model predictive controllers that have been extensively tested.
850 citations
Cites methods from "Linear complementarity, linear and ..."
...Hence a quick way to specify an LTI system described by the state-update equation x(t+1) = Ax(t)+Bu(t) is to use sys = LTISystem(’A’, A, ’B’, B) Such a syntax conveniently allows to specify autonomous systems as well....
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TL;DR: A new solution for the label fusion problem in which weighted voting is formulated in terms of minimizing the total expectation of labeling error and in which pairwise dependency between atlases is explicitly modeled as the joint probability of two atlas making a segmentation error at a voxel is proposed.
Abstract: Multi-atlas segmentation is an effective approach for automatically labeling objects of interest in biomedical images. In this approach, multiple expert-segmented example images, called atlases, are registered to a target image, and deformed atlas segmentations are combined using label fusion. Among the proposed label fusion strategies, weighted voting with spatially varying weight distributions derived from atlas-target intensity similarity have been particularly successful. However, one limitation of these strategies is that the weights are computed independently for each atlas, without taking into account the fact that different atlases may produce similar label errors. To address this limitation, we propose a new solution for the label fusion problem in which weighted voting is formulated in terms of minimizing the total expectation of labeling error and in which pairwise dependency between atlases is explicitly modeled as the joint probability of two atlases making a segmentation error at a voxel. This probability is approximated using intensity similarity between a pair of atlases and the target image in the neighborhood of each voxel. We validate our method in two medical image segmentation problems: hippocampus segmentation and hippocampus subfield segmentation in magnetic resonance (MR) images. For both problems, we show consistent and significant improvement over label fusion strategies that assign atlas weights independently.
722 citations
Cites methods from "Linear complementarity, linear and ..."
...When Mx is not full rank, the weights can be estimated using quadratic programming optimization [27]....
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TL;DR: Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminated by the PWLS+SOR method, and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over FBP.
Abstract: 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, the author argues statistically that a least-squares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. The author proposes 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 eliminated by the PWLS+SOR method, and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over FBP. >
657 citations
Cites methods from "Linear complementarity, linear and ..."
...Then without the nonnegativity constraint, the SOR method can be compactly written [47]: i+1 = (1 !) i + !D 1(A0 1y L0 i L i+1);...
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...A detailed discussion of +SOR is given in [47,40], so we only summarize the algorithm here....
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