Michael Saunders
Professor (Research) of Management Science and Engineering, Emeritus
Bio
Saunders develops mathematical methods for solving large-scale constrained optimization problems and large systems of equations. He also implements such methods as general-purpose software to allow their use in many areas of engineering, science, and business. He is co-developer of the large-scale optimizers MINOS, SNOPT, SQOPT, PDCO, the dense QP and NLP solvers LSSOL, QPOPT, NPSOL, and the linear equation solvers SYMMLQ, MINRES, MINRES-QLP, LSQR, LSMR, LSLQ, LNLQ, LSRN, LUSOL.
Honors & Awards
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Orchard-Hays Prize, MPS (1985)
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Highly Cited Researcher, Computer Science, ISI (2004)
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Highly Cited Researcher, Mathematics, ISI (2007)
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Honorary Fellow, RSNZ (2007)
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Linear Algebra Prize, SIAM (2012)
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Invention Hall of Fame, OTL, Stanford University (2012)
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Fellow, SIAM (2013)
Boards, Advisory Committees, Professional Organizations
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Member, ACM (1982 - Present)
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Associate Editor, NACO (2010 - 2016)
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Member, INFORMS (2010 - Present)
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Member, ORSNZ (1990 - Present)
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Member, SIAM (1980 - Present)
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Associate Editor, ACM TOMS (1982 - 2004)
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Associate Editor, SIAM Journal on Optimization (1989 - 2002)
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Associate Editor, OPTE (1999 - Present)
Professional Education
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B.Sc. (Hons), Canterbury, Mathematics (1965)
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MS, Stanford University, Computer Science (1970)
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PhD, Stanford University, Computer Science (1972)
All Publications
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IMPLEMENTING A SMOOTH EXACT PENALTY FUNCTION FOR EQUALITY-CONSTRAINED NONLINEAR OPTIMIZATION
SIAM JOURNAL ON SCIENTIFIC COMPUTING
2020; 42 (3): A1809–A1835
View details for DOI 10.1137/19M1238265
View details for Web of Science ID 000551255700006
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IMPLEMENTING A SMOOTH EXACT PENALTY FUNCTION FOR GENERAL CONSTRAINED NONLINEAR OPTIMIZATION
SIAM JOURNAL ON SCIENTIFIC COMPUTING
2020; 42 (3): A1836–A1859
View details for DOI 10.1137/19M1255069
View details for Web of Science ID 000551255700011
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Analysis of the Regularization Parameters of Primal-Dual Interior Method for Convex Objectives Applied to H-1 Low Field Nuclear Magnetic Resonance Data Processing (vol 49, pg 1129, 2018)
APPLIED MAGNETIC RESONANCE
2019; 50 (1-3): 521
View details for DOI 10.1007/s00723-018-1072-4
View details for Web of Science ID 000458124800037
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EUCLIDEAN-NORM ERROR BOUNDS FOR SYMMLQ AND CG
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
2019; 40 (1): 235–53
View details for DOI 10.1137/16M1094816
View details for Web of Science ID 000462583900011
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LSLQ: AN ITERATIVE METHOD FOR LINEAR LEAST-SQUARES WITH AN ERROR MINIMIZATION PROPERTY
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
2019; 40 (1): 254–75
View details for DOI 10.1137/17M1113552
View details for Web of Science ID 000462583900012
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Stabilized Optimization Via an NCL Algorithm
SPRINGER. 2018: 173–91
View details for DOI 10.1007/978-3-319-90026-1_8
View details for Web of Science ID 000448498900008
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Reliable and efficient solution of genome-scale models of Metabolism and macromolecular Expression
SCIENTIFIC REPORTS
2017; 7
Abstract
Constraint-Based Reconstruction and Analysis (COBRA) is currently the only methodology that permits integrated modeling of Metabolism and macromolecular Expression (ME) at genome-scale. Linear optimization computes steady-state flux solutions to ME models, but flux values are spread over many orders of magnitude. Data values also have greatly varying magnitudes. Standard double-precision solvers may return inaccurate solutions or report that no solution exists. Exact simplex solvers based on rational arithmetic require a near-optimal warm start to be practical on large problems (current ME models have 70,000 constraints and variables and will grow larger). We have developed a quadruple-precision version of our linear and nonlinear optimizer MINOS, and a solution procedure (DQQ) involving Double and Quad MINOS that achieves reliability and efficiency for ME models and other challenging problems tested here. DQQ will enable extensive use of large linear and nonlinear models in systems biology and other applications involving multiscale data.
View details for DOI 10.1038/srep40863
View details for Web of Science ID 000392188100001
View details for PubMedID 28098205
View details for PubMedCentralID PMC5241643
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Conditions for duality between fluxes and concentrations in biochemical networks
JOURNAL OF THEORETICAL BIOLOGY
2016; 409: 1-10
Abstract
Mathematical and computational modelling of biochemical networks is often done in terms of either the concentrations of molecular species or the fluxes of biochemical reactions. When is mathematical modelling from either perspective equivalent to the other? Mathematical duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one manner. We present a novel stoichiometric condition that is necessary and sufficient for duality between unidirectional fluxes and concentrations. Our numerical experiments, with computational models derived from a range of genome-scale biochemical networks, suggest that this flux-concentration duality is a pervasive property of biochemical networks. We also provide a combinatorial characterisation that is sufficient to ensure flux-concentration duality.The condition prescribes that, for every two disjoint sets of molecular species, there is at least one reaction complex that involves species from only one of the two sets. When unidirectional fluxes and molecular species concentrations are dual vectors, this implies that the behaviour of the corresponding biochemical network can be described entirely in terms of either concentrations or unidirectional fluxes.
View details for DOI 10.1016/j.jtbi.2016.06.033
View details for Web of Science ID 000385471800001
View details for PubMedID 27345817
View details for PubMedCentralID PMC5048525
- Laplace inversion of low-resolution NMR relaxometry data using sparse representation methods Concepts in Magnetic Resonance Part A 2013; 42A:3: 72-88
- Novel 1H low field nuclear magnetic resonance applications for the field of biodiesel Biotechnologyfor Biofuels 2013; 6:55: 20
- LSRN: a parallel iterative solver for strongly over- or under-determined systems SIAM J. Sci. Comp. 2013; 36 (2): C95-C118
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A variational principle for computing nonequilibrium fluxes and potentials in genome-scale biochemical networks
JOURNAL OF THEORETICAL BIOLOGY
2012; 292: 71-77
Abstract
We derive a convex optimization problem on a steady-state nonequilibrium network of biochemical reactions, with the property that energy conservation and the second law of thermodynamics both hold at the problem solution. This suggests a new variational principle for biochemical networks that can be implemented in a computationally tractable manner. We derive the Lagrange dual of the optimization problem and use strong duality to demonstrate that a biochemical analogue of Tellegen's theorem holds at optimality. Each optimal flux is dependent on a free parameter that we relate to an elementary kinetic parameter when mass action kinetics is assumed.
View details for DOI 10.1016/j.jtbi.2011.09.029
View details for Web of Science ID 000297450100008
View details for PubMedID 21983269
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LSMR: AN ITERATIVE ALGORITHM FOR SPARSE LEAST-SQUARES PROBLEMS
SIAM JOURNAL ON SCIENTIFIC COMPUTING
2011; 33 (5): 2950-2971
View details for Web of Science ID 000296591200039
- SNOPT: An SQP algorithm for large-scaleconstrained optimization, SIGEST article SIAM Rev. 2005; 1 (47): 99-131
- Atomic decomposition by basis pursuit, SIGEST article SIAM Rev. 2001; 1 (43): 129-159
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Properties of semi-conjugate gradient methods for solving unsymmetric positive definite linear systems
OPTIMIZATION METHODS & SOFTWARE
2023
View details for DOI 10.1080/10556788.2023.2189716
View details for Web of Science ID 000975713500001
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HyKKT: a hybrid direct-iterative method for solving KKT linear systems
OPTIMIZATION METHODS & SOFTWARE
2022
View details for DOI 10.1080/10556788.2022.2124990
View details for Web of Science ID 000860640100001
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Linear solvers for power grid optimization problems: A review of GPU-accelerated linear solvers
PARALLEL COMPUTING
2022; 111
View details for DOI 10.1016/j.parco.2021.102870
View details for Web of Science ID 000800000400004
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LARGE-SCALE OPTIMIZATION WITH LINEAR EQUALITY CONSTRAINTS USING REDUCED COMPACT REPRESENTATION\ast
SIAM JOURNAL ON SCIENTIFIC COMPUTING
2022; 44 (1): A103-A127
View details for DOI 10.1137/21M1393819
View details for Web of Science ID 000773632300005
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Linear systems arising in interior methods for convex optimization: a symmetric formulation with bounded condition number
OPTIMIZATION METHODS & SOFTWARE
2021
View details for DOI 10.1080/10556788.2021.1965599
View details for Web of Science ID 000705433600001
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Simulation-Based Sensitivity Analysis of Regularization Parameters for Robust Reconstruction of Complex Material's T-1 - (T2H)-H-1 LF-NMR Energy Relaxation Signals
APPLIED MAGNETIC RESONANCE
2019
View details for DOI 10.1007/s00723-019-01173-1
View details for Web of Science ID 000497186200001
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Creation and analysis of biochemical constraint-based models using the COBRA Toolbox v.3.0.
Nature protocols
2019
Abstract
Constraint-based reconstruction and analysis (COBRA) provides a molecular mechanistic framework for integrative analysis of experimental molecular systems biology data and quantitative prediction of physicochemically and biochemically feasible phenotypic states. The COBRA Toolbox is a comprehensive desktop software suite of interoperable COBRA methods. It has found widespread application in biology, biomedicine, and biotechnology because its functions can be flexibly combined to implement tailored COBRA protocols for any biochemical network. This protocol is an update to the COBRA Toolbox v.1.0 and v.2.0. Version 3.0 includes new methods for quality-controlled reconstruction, modeling, topological analysis, strain and experimental design, and network visualization, as well as network integration of chemoinformatic, metabolomic, transcriptomic, proteomic, and thermochemical data. New multi-lingual code integration also enables an expansion in COBRA application scope via high-precision, high-performance, and nonlinear numerical optimization solvers for multi-scale, multi-cellular, and reaction kinetic modeling, respectively. This protocol provides an overview of all these new features and can be adapted to generate and analyze constraint-based models in a wide variety of scenarios. The COBRA Toolbox v.3.0 provides an unparalleled depth of COBRA methods.
View details for PubMedID 30787451
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DynamicME: dynamic simulation and refinement of integrated models of metabolism and protein expression.
BMC systems biology
2019; 13 (1): 2
Abstract
BACKGROUND: Genome-scale models of metabolism and macromolecular expression (ME models) enable systems-level computation of proteome allocation coupled to metabolic phenotype.RESULTS: We develop DynamicME, an algorithm enabling time-course simulation of cell metabolism and protein expression. DynamicME correctly predicted the substrate utilization hierarchy on a mixed carbon substrate medium. We also found good agreement between predicted and measured time-course expression profiles. ME models involve considerably more parameters than metabolic models (M models). We thus generate an ensemble of models (each model having its rate constants perturbed), and then analyze the models by identifying archetypal time-course metabolite concentration profiles. Furthermore, we use a metaheuristic optimization method to calibrate ME model parameters using time-course measurements such as from a (fed-) batch culture. Finally, we show that constraints on protein concentration dynamics ("inertia") alter the metabolic response to environmental fluctuations, including increased substrate-level phosphorylation and lowered oxidative phosphorylation.CONCLUSIONS: Overall, DynamicME provides a novel method for understanding proteome allocation and metabolism under complex and transient environments, and to utilize time-course cell culture data for model-based interpretation or model refinement.
View details for PubMedID 30626386
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LNLQ: AN ITERATIVE METHOD FOR LEAST-NORM PROBLEMS WITH AN ERROR MINIMIZATION PROPERTY
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
2019; 40 (3): 1102–24
View details for DOI 10.1137/18M1194948
View details for Web of Science ID 000487856100012
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Estimating Cellular Goals from High-Dimensional Biological Data
ASSOC COMPUTING MACHINERY. 2019: 2202–11
View details for DOI 10.1145/3292500.3330775
View details for Web of Science ID 000485562502026
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Analysis of the Regularization Parameters of Primal-Dual Interior Method for Convex Objectives Applied to H-1 Low Field Nuclear Magnetic Resonance Data Processing
APPLIED MAGNETIC RESONANCE
2018; 49 (10): 1129–50
View details for DOI 10.1007/s00723-018-1048-4
View details for Web of Science ID 000444236900007
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Principles of proteome allocation are revealed using proteomic data and genome-scale models
SCIENTIFIC REPORTS
2016; 6
Abstract
Integrating omics data to refine or make context-specific models is an active field of constraint-based modeling. Proteomics now cover over 95% of the Escherichia coli proteome by mass. Genome-scale models of Metabolism and macromolecular Expression (ME) compute proteome allocation linked to metabolism and fitness. Using proteomics data, we formulated allocation constraints for key proteome sectors in the ME model. The resulting calibrated model effectively computed the "generalist" (wild-type) E. coli proteome and phenotype across diverse growth environments. Across 15 growth conditions, prediction errors for growth rate and metabolic fluxes were 69% and 14% lower, respectively. The sector-constrained ME model thus represents a generalist ME model reflecting both growth rate maximization and "hedging" against uncertain environments and stresses, as indicated by significant enrichment of these sectors for the general stress response sigma factor σ(S). Finally, the sector constraints represent a general formalism for integrating omics data from any experimental condition into constraint-based ME models. The constraints can be fine-grained (individual proteins) or coarse-grained (functionally-related protein groups) as demonstrated here. This flexible formalism provides an accessible approach for narrowing the gap between the complexity captured by omics data and governing principles of proteome allocation described by systems-level models.
View details for DOI 10.1038/srep36734
View details for Web of Science ID 000388074400001
View details for PubMedID 27857205
View details for PubMedCentralID PMC5114563
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solveME: fast and reliable solution of nonlinear ME models.
BMC bioinformatics
2016; 17 (1): 391
Abstract
Genome-scale models of metabolism and macromolecular expression (ME) significantly expand the scope and predictive capabilities of constraint-based modeling. ME models present considerable computational challenges: they are much (>30 times) larger than corresponding metabolic reconstructions (M models), are multiscale, and growth maximization is a nonlinear programming (NLP) problem, mainly due to macromolecule dilution constraints.Here, we address these computational challenges. We develop a fast and numerically reliable solution method for growth maximization in ME models using a quad-precision NLP solver (Quad MINOS). Our method was up to 45 % faster than binary search for six significant digits in growth rate. We also develop a fast, quad-precision flux variability analysis that is accelerated (up to 60× speedup) via solver warm-starts. Finally, we employ the tools developed to investigate growth-coupled succinate overproduction, accounting for proteome constraints.Just as genome-scale metabolic reconstructions have become an invaluable tool for computational and systems biologists, we anticipate that these fast and numerically reliable ME solution methods will accelerate the wide-spread adoption of ME models for researchers in these fields.
View details for DOI 10.1186/s12859-016-1240-1
View details for PubMedID 27659412
View details for PubMedCentralID PMC5034503
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Heart rate analysis by sparse representation for acute pain detection
MEDICAL & BIOLOGICAL ENGINEERING & COMPUTING
2016; 54 (4): 595-606
Abstract
Objective pain assessment methods pose an advantage over the currently used subjective pain rating tools. Advanced signal processing methodologies, including the wavelet transform (WT) and the orthogonal matching pursuit algorithm (OMP), were developed in the past two decades. The aim of this study was to apply and compare these time-specific methods to heart rate samples of healthy subjects for acute pain detection. Fifteen adult volunteers participated in a study conducted in the pain clinic at a single center. Each subject's heart rate was sampled for 5-min baseline, followed by a cold pressor test (CPT). Analysis was done by the WT and the OMP algorithm with a Fourier/Wavelet dictionary separately. Data from 11 subjects were analyzed. Compared to baseline, The WT analysis showed a significant coefficients' density increase during the pain incline period (p < 0.01) and the entire CPT (p < 0.01), with significantly higher coefficient amplitudes. The OMP analysis showed a significant wavelet coefficients' density increase during pain incline and decline periods (p < 0.01, p < 0.05) and the entire CPT (p < 0.001), with suggestive higher amplitudes. Comparison of both methods showed that during the baseline there was a significant reduction in wavelet coefficient density using the OMP algorithm (p < 0.001). Analysis by the two-way ANOVA with repeated measures showed a significant proportional increase in wavelet coefficients during the incline period and the entire CPT using the OMP algorithm (p < 0.01). Both methods provided accurate and non-delayed detection of pain events. Statistical analysis proved the OMP to be by far more specific allowing the Fourier coefficients to represent the signal's basic harmonics and the wavelet coefficients to focus on the time-specific painful event. This is an initial study using OMP for pain detection; further studies need to prove the efficiency of this system in different settings.
View details for DOI 10.1007/s11517-015-1350-3
View details for Web of Science ID 000373021100004
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Heart rate analysis by sparse representation for acute pain detection.
Medical & biological engineering & computing
2016; 54 (4): 595-606
Abstract
Objective pain assessment methods pose an advantage over the currently used subjective pain rating tools. Advanced signal processing methodologies, including the wavelet transform (WT) and the orthogonal matching pursuit algorithm (OMP), were developed in the past two decades. The aim of this study was to apply and compare these time-specific methods to heart rate samples of healthy subjects for acute pain detection. Fifteen adult volunteers participated in a study conducted in the pain clinic at a single center. Each subject's heart rate was sampled for 5-min baseline, followed by a cold pressor test (CPT). Analysis was done by the WT and the OMP algorithm with a Fourier/Wavelet dictionary separately. Data from 11 subjects were analyzed. Compared to baseline, The WT analysis showed a significant coefficients' density increase during the pain incline period (p < 0.01) and the entire CPT (p < 0.01), with significantly higher coefficient amplitudes. The OMP analysis showed a significant wavelet coefficients' density increase during pain incline and decline periods (p < 0.01, p < 0.05) and the entire CPT (p < 0.001), with suggestive higher amplitudes. Comparison of both methods showed that during the baseline there was a significant reduction in wavelet coefficient density using the OMP algorithm (p < 0.001). Analysis by the two-way ANOVA with repeated measures showed a significant proportional increase in wavelet coefficients during the incline period and the entire CPT using the OMP algorithm (p < 0.01). Both methods provided accurate and non-delayed detection of pain events. Statistical analysis proved the OMP to be by far more specific allowing the Fourier coefficients to represent the signal's basic harmonics and the wavelet coefficients to focus on the time-specific painful event. This is an initial study using OMP for pain detection; further studies need to prove the efficiency of this system in different settings.
View details for DOI 10.1007/s11517-015-1350-3
View details for PubMedID 26264057
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A Practical Factorization of a Schur Complement for PDE-Constrained Distributed Optimal Control
JOURNAL OF SCIENTIFIC COMPUTING
2015; 65 (2): 576-597
View details for DOI 10.1007/s10915-014-9976-0
View details for Web of Science ID 000362911900007
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Do genome-scale models need exact solvers or clearer standards?
MOLECULAR SYSTEMS BIOLOGY
2015; 11 (10): 831
View details for PubMedID 26467284
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Systems biology definition of the core proteome of metabolism and expression is consistent with high-throughput data.
Proceedings of the National Academy of Sciences of the United States of America
2015; 112 (34): 10810-10815
Abstract
Finding the minimal set of gene functions needed to sustain life is of both fundamental and practical importance. Minimal gene lists have been proposed by using comparative genomics-based core proteome definitions. A definition of a core proteome that is supported by empirical data, is understood at the systems-level, and provides a basis for computing essential cell functions is lacking. Here, we use a systems biology-based genome-scale model of metabolism and expression to define a functional core proteome consisting of 356 gene products, accounting for 44% of the Escherichia coli proteome by mass based on proteomics data. This systems biology core proteome includes 212 genes not found in previous comparative genomics-based core proteome definitions, accounts for 65% of known essential genes in E. coli, and has 78% gene function overlap with minimal genomes (Buchnera aphidicola and Mycoplasma genitalium). Based on transcriptomics data across environmental and genetic backgrounds, the systems biology core proteome is significantly enriched in nondifferentially expressed genes and depleted in differentially expressed genes. Compared with the noncore, core gene expression levels are also similar across genetic backgrounds (two times higher Spearman rank correlation) and exhibit significantly more complex transcriptional and posttranscriptional regulatory features (40% more transcription start sites per gene, 22% longer 5'UTR). Thus, genome-scale systems biology approaches rigorously identify a functional core proteome needed to support growth. This framework, validated by using high-throughput datasets, facilitates a mechanistic understanding of systems-level core proteome function through in silico models; it de facto defines a paleome.
View details for DOI 10.1073/pnas.1501384112
View details for PubMedID 26261351
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Study of liquid-phase molecular packing interactions and morphology of fatty acid methyl esters (biodiesel).
Biotechnology for biofuels
2015; 8: 12-?
Abstract
(1)H low field nuclear magnetic resonance (LF-NMR) relaxometry has been suggested as a tool to distinguish between different molecular ensembles in complex systems with differential segmental or whole molecular motion and/or different morphologies. In biodiesel applications the molecular structure versus liquid-phase packing morphologies of fatty acid methyl esters (FAMEs) influences physico-chemical characteristics of the fuel, including flow properties, operability during cold weather, blending, and more. Still, their liquid morphological structures have scarcely been studied. It was therefore the objective of this work to explore the potential of this technology for characterizing the molecular organization of FAMEs in the liquid phase. This was accomplished by using a combination of supporting advanced technologies.We show that pure oleic acid (OA) and methyl oleate (MO) standards exhibited both similarities and differences in the (1)H LF-NMR relaxation times (T2s) and peak areas, for a range of temperatures. Based on X-ray measurements, both molecules were found to possess a liquid crystal-like order, although a larger fluidity was found for MO, because as the temperature is increased, MO molecules separate both longitudinally and transversely from one another. In addition, both molecules exhibited a preferred direction of diffusion based on the apparent hydrodynamic radius. The close molecular packing arrangement and interactions were found to affect the translational and segmental motions of the molecules, as a result of dimerization of the head group in OA as opposed to weaker polar interactions in MO.A comprehensive model for the liquid crystal-like arrangement of FAMEs in the liquid phase is suggested. The differences in translational and segmental motions of the molecules were rationalized by the differences in the (1)H LF-NMR T2 distributions of OA and MO, which was further supported by (13)C high field (HF)-NMR spectra and (1)H HF-NMR relaxation. The proposed assignment allows for material characterization based on parameters that contribute to properties in applications such as biodiesel fuels.
View details for DOI 10.1186/s13068-014-0194-7
View details for PubMedID 25688289
View details for PubMedCentralID PMC4329664
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LSRN: A PARALLEL ITERATIVE SOLVER FOR STRONGLY OVER- OR UNDERDETERMINED SYSTEMS.
SIAM journal on scientific computing : a publication of the Society for Industrial and Applied Mathematics
2014; 36 (2): C95-C118
Abstract
We describe a parallel iterative least squares solver named LSRN that is based on random normal projection. LSRN computes the min-length solution to min x∈ℝ n ‖Ax - b‖2, where A ∈ ℝ m × n with m ≫ n or m ≪ n, and where A may be rank-deficient. Tikhonov regularization may also be included. Since A is involved only in matrix-matrix and matrix-vector multiplications, it can be a dense or sparse matrix or a linear operator, and LSRN automatically speeds up when A is sparse or a fast linear operator. The preconditioning phase consists of a random normal projection, which is embarrassingly parallel, and a singular value decomposition of size ⌈γ min(m, n)⌉ × min(m, n), where γ is moderately larger than 1, e.g., γ = 2. We prove that the preconditioned system is well-conditioned, with a strong concentration result on the extreme singular values, and hence that the number of iterations is fully predictable when we apply LSQR or the Chebyshev semi-iterative method. As we demonstrate, the Chebyshev method is particularly efficient for solving large problems on clusters with high communication cost. Numerical results show that on a shared-memory machine, LSRN is very competitive with LAPACK's DGELSD and a fast randomized least squares solver called Blendenpik on large dense problems, and it outperforms the least squares solver from SuiteSparseQR on sparse problems without sparsity patterns that can be exploited to reduce fill-in. Further experiments show that LSRN scales well on an Amazon Elastic Compute Cloud cluster.
View details for DOI 10.1137/120866580
View details for PubMedID 25419094
View details for PubMedCentralID PMC4238893
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Algorithm 937: MINRES-QLP for Symmetric and Hermitian Linear Equations and Least-Squares Problems
ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
2014; 40 (2)
Abstract
We describe algorithm MINRES-QLP and its FORTRAN 90 implementation for solving symmetric or Hermitian linear systems or least-squares problems. If the system is singular, MINRES-QLP computes the unique minimum-length solution (also known as the pseudoinverse solution), which generally eludes MINRES. In all cases, it overcomes a potential instability in the original MINRES algorithm. A positive-definite pre-conditioner may be supplied. Our FORTRAN 90 implementation illustrates a design pattern that allows users to make problem data known to the solver but hidden and secure from other program units. In particular, we circumvent the need for reverse communication. Example test programs input and solve real or complex problems specified in Matrix Market format. While we focus here on a FORTRAN 90 implementation, we also provide and maintain MATLAB versions of MINRES and MINRES-QLP.
View details for DOI 10.1145/2527267
View details for Web of Science ID 000333653400008
View details for PubMedCentralID PMC4199394
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Algorithm 937: MINRES-QLP for Symmetric and Hermitian Linear Equations and Least-Squares Problems.
ACM transactions on mathematical software. Association for Computing Machinery
2014; 40 (2)
Abstract
We describe algorithm MINRES-QLP and its FORTRAN 90 implementation for solving symmetric or Hermitian linear systems or least-squares problems. If the system is singular, MINRES-QLP computes the unique minimum-length solution (also known as the pseudoinverse solution), which generally eludes MINRES. In all cases, it overcomes a potential instability in the original MINRES algorithm. A positive-definite pre-conditioner may be supplied. Our FORTRAN 90 implementation illustrates a design pattern that allows users to make problem data known to the solver but hidden and secure from other program units. In particular, we circumvent the need for reverse communication. Example test programs input and solve real or complex problems specified in Matrix Market format. While we focus here on a FORTRAN 90 implementation, we also provide and maintain MATLAB versions of MINRES and MINRES-QLP.
View details for DOI 10.1145/2527267
View details for PubMedID 25328255
View details for PubMedCentralID PMC4199394
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LSRN: A PARALLEL ITERATIVE SOLVER FOR STRONGLY OVER- OR UNDERDETERMINED SYSTEMS
SIAM JOURNAL ON SCIENTIFIC COMPUTING
2014; 36 (2): C95-C118
Abstract
We describe a parallel iterative least squares solver named LSRN that is based on random normal projection. LSRN computes the min-length solution to min x∈ℝ n ‖Ax - b‖2, where A ∈ ℝ m × n with m ≫ n or m ≪ n, and where A may be rank-deficient. Tikhonov regularization may also be included. Since A is involved only in matrix-matrix and matrix-vector multiplications, it can be a dense or sparse matrix or a linear operator, and LSRN automatically speeds up when A is sparse or a fast linear operator. The preconditioning phase consists of a random normal projection, which is embarrassingly parallel, and a singular value decomposition of size ⌈γ min(m, n)⌉ × min(m, n), where γ is moderately larger than 1, e.g., γ = 2. We prove that the preconditioned system is well-conditioned, with a strong concentration result on the extreme singular values, and hence that the number of iterations is fully predictable when we apply LSQR or the Chebyshev semi-iterative method. As we demonstrate, the Chebyshev method is particularly efficient for solving large problems on clusters with high communication cost. Numerical results show that on a shared-memory machine, LSRN is very competitive with LAPACK's DGELSD and a fast randomized least squares solver called Blendenpik on large dense problems, and it outperforms the least squares solver from SuiteSparseQR on sparse problems without sparsity patterns that can be exploited to reduce fill-in. Further experiments show that LSRN scales well on an Amazon Elastic Compute Cloud cluster.
View details for DOI 10.1137/120866580
View details for Web of Science ID 000335817600030
View details for PubMedCentralID PMC4238893
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PROXIMAL NEWTON-TYPE METHODS FOR MINIMIZING COMPOSITE FUNCTIONS
SIAM JOURNAL ON OPTIMIZATION
2014; 24 (3): 1420-1443
View details for DOI 10.1137/130921428
View details for Web of Science ID 000343229000019
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Robust flux balance analysis of multiscale biochemical reaction networks
BMC BIOINFORMATICS
2013; 14
Abstract
Biological processes such as metabolism, signaling, and macromolecular synthesis can be modeled as large networks of biochemical reactions. Large and comprehensive networks, like integrated networks that represent metabolism and macromolecular synthesis, are inherently multiscale because reaction rates can vary over many orders of magnitude. They require special methods for accurate analysis because naive use of standard optimization systems can produce inaccurate or erroneously infeasible results.We describe techniques enabling off-the-shelf optimization software to compute accurate solutions to the poorly scaled optimization problems arising from flux balance analysis of multiscale biochemical reaction networks. We implement lifting techniques for flux balance analysis within the openCOBRA toolbox and demonstrate our techniques using the first integrated reconstruction of metabolism and macromolecular synthesis for E. coli.Our techniques enable accurate flux balance analysis of multiscale networks using off-the-shelf optimization software. Although we describe lifting techniques in the context of flux balance analysis, our methods can be used to handle a variety of optimization problems arising from analysis of multiscale network reconstructions.
View details for DOI 10.1186/1471-2105-14-240
View details for Web of Science ID 000322915900001
View details for PubMedID 23899245
View details for PubMedCentralID PMC3750362
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Equispaced Pareto front construction for constrained bi-objective optimization
MATHEMATICAL AND COMPUTER MODELLING
2013; 57 (9-10): 2122-2131
View details for DOI 10.1016/j.mcm.2010.12.044
View details for Web of Science ID 000317262100010
- Robust flux balance analysis of multiscale biochemical reaction networks BMC Bioinformatics 2013; 14:240: 6
- CG versus MINRES: An empirical comparison SQUJournal for Science 2012; 17:1: 44-62
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A Higher-Order Generalized Singular Value Decomposition for Comparison of Global mRNA Expression from Multiple Organisms
PLOS ONE
2011; 6 (12)
Abstract
The number of high-dimensional datasets recording multiple aspects of a single phenomenon is increasing in many areas of science, accompanied by a need for mathematical frameworks that can compare multiple large-scale matrices with different row dimensions. The only such framework to date, the generalized singular value decomposition (GSVD), is limited to two matrices. We mathematically define a higher-order GSVD (HO GSVD) for N≥2 matrices D(i)∈R(m(i) × n), each with full column rank. Each matrix is exactly factored as D(i)=U(i)Σ(i)V(T), where V, identical in all factorizations, is obtained from the eigensystem SV=VΛ of the arithmetic mean S of all pairwise quotients A(i)A(j)(-1) of the matrices A(i)=D(i)(T)D(i), i≠j. We prove that this decomposition extends to higher orders almost all of the mathematical properties of the GSVD. The matrix S is nondefective with V and Λ real. Its eigenvalues satisfy λ(k)≥1. Equality holds if and only if the corresponding eigenvector v(k) is a right basis vector of equal significance in all matrices D(i) and D(j), that is σ(i,k)/σ(j,k)=1 for all i and j, and the corresponding left basis vector u(i,k) is orthogonal to all other vectors in U(i) for all i. The eigenvalues λ(k)=1, therefore, define the "common HO GSVD subspace." We illustrate the HO GSVD with a comparison of genome-scale cell-cycle mRNA expression from S. pombe, S. cerevisiae and human. Unlike existing algorithms, a mapping among the genes of these disparate organisms is not required. We find that the approximately common HO GSVD subspace represents the cell-cycle mRNA expression oscillations, which are similar among the datasets. Simultaneous reconstruction in the common subspace, therefore, removes the experimental artifacts, which are dissimilar, from the datasets. In the simultaneous sequence-independent classification of the genes of the three organisms in this common subspace, genes of highly conserved sequences but significantly different cell-cycle peak times are correctly classified.
View details for DOI 10.1371/journal.pone.0028072
View details for Web of Science ID 000299684700003
View details for PubMedID 22216090
View details for PubMedCentralID PMC3245232
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MINRES-QLP: A KRYLOV SUBSPACE METHOD FOR INDEFINITE OR SINGULAR SYMMETRIC SYSTEMS
SIAM JOURNAL ON SCIENTIFIC COMPUTING
2011; 33 (4): 1810-1836
View details for DOI 10.1137/100787921
View details for Web of Science ID 000294293200016
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Nonconservative Robust Control: Optimized and Constrained Sensitivity Functions
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY
2009; 17 (2): 298-308
View details for DOI 10.1109/TCST.2008.924564
View details for Web of Science ID 000263832000004
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STABILIZING POLICY IMPROVEMENT FOR LARGE-SCALE INFINITE-HORIZON DYNAMIC PROGRAMMING
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
2009; 31 (2): 434-459
View details for DOI 10.1137/060653305
View details for Web of Science ID 000267745500012
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Variational Bayesian image restoration based on a product of t-distributions image prior
IEEE TRANSACTIONS ON IMAGE PROCESSING
2008; 17 (10): 1795-1805
Abstract
Image priors based on products have been recognized to offer many advantages because they allow simultaneous enforcement of multiple constraints. However, they are inconvenient for Bayesian inference because it is hard to find their normalization constant in closed form. In this paper, a new Bayesian algorithm is proposed for the image restoration problem that bypasses this difficulty. An image prior is defined by imposing Student-t densities on the outputs of local convolutional filters. A variational methodology, with a constrained expectation step, is used to infer the restored image. Numerical experiments are shown that compare this methodology to previous ones and demonstrate its advantages.
View details for DOI 10.1109/TIP.2008.2002828
View details for Web of Science ID 000259372100005
View details for PubMedID 18784028
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George B. Dantzig and systems optimization
DISCRETE OPTIMIZATION
2008; 5 (2): 151-158
View details for DOI 10.1016/j.disopt.2007.01.002
View details for Web of Science ID 000255475400002
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Discussion: The Dantzig selector: Statistical estimation when p is much larger than n
ANNALS OF STATISTICS
2007; 35 (6): 2385-2391
View details for DOI 10.1214/00905360700000479
View details for Web of Science ID 000253077800007
- Commentary on Methods for modifying matrix factorizations Milestones in Matrix Computation: Selected Works of Gene H. Golub With Commentaries edited by Chan, R., H., Greif, C., O'Leary, D., P. Oxford University Press. 2007: 310–310
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SpaseLoc: An adaptive subproblem algorithm for scalable wireless sensor network localization
SIAM JOURNAL ON OPTIMIZATION
2006; 17 (4): 1102-1128
View details for DOI 10.1137/040621600
View details for Web of Science ID 000244631800007
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SNOPT: An SQP algorithm for large-scale constrained optimization (Reprinted from SIAM Journal Optimization, vol 12, pg 979-1006, 2002)
SIAM REVIEW
2005; 47 (1): 99-131
View details for DOI 10.1137/S0036144504446096
View details for Web of Science ID 000227119200005
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Sparsity and smoothness via the fused lasso
JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
2005; 67: 91-108
View details for Web of Science ID 000225686900006
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A globally convergent linearly constrained Lagrangian method for nonlinear optimization
SIAM JOURNAL ON OPTIMIZATION
2005; 15 (3): 863-897
View details for DOI 10.1137/S1052623402419789
View details for Web of Science ID 000229826800011
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A bisection algorithm for the mixed mu upper bound and its supremum
American Control Conference
IEEE. 2004: 2665–2670
View details for Web of Science ID 000224688300453
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Subspace preconditioned LSQR for discrete ill-posed problems
Conference on Computational Linear Algebra with Applications
SPRINGER. 2003: 975–89
View details for Web of Science ID 000188719300011
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SNOPT: An SQP algorithm for large-scale constrained optimization
SIAM JOURNAL ON OPTIMIZATION
2002; 12 (4): 979-1006
View details for Web of Science ID 000175810600007
- Global controller optimization using Horowitz bounds 2002
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Atomic decomposition by basis pursuit
SIAM REVIEW
2001; 43 (1): 129-159
View details for Web of Science ID 000167366100008
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Atomic decomposition by basis pursuit
SIAM JOURNAL ON SCIENTIFIC COMPUTING
1998; 20 (1): 33-61
View details for Web of Science ID 000075434800003
- SNOPT: A Fortran software package to solve large-scale optimization problems 1998
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OSSE mapping of galactic 511 keV positron annihilation line emission
ASTROPHYSICAL JOURNAL
1997; 491 (2): 725-748
View details for Web of Science ID 000071152600025
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Computing projections with LSQR
BIT NUMERICAL MATHEMATICS
1997; 37 (1): 96-104
View details for Web of Science ID A1997WK05500008
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Non-parametric estimates of high energy gamma-ray source distributions
4th Compton Symposium
AIP PRESS. 1997: 1601–5
View details for Web of Science ID 000071400800240
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On the stability of Cholesky factorization for symmetric quasidefinite systems
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
1996; 17 (1): 35-46
View details for Web of Science ID A1996TV10700002
- SQP methods for large-scale optimization 1996
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Cholesky-based methods for sparse least squares: The benefits of regularization
AMS/IMS/SIAM Summer Research Conference on Linear and Nonlinear Conjugate Gradient-Related Methods
SIAM. 1996: 92–100
View details for Web of Science ID A1996BF52D00008
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Solution of sparse rectangular systems using LSQR and Craig
BIT NUMERICAL MATHEMATICS
1995; 35 (4): 588-604
View details for Web of Science ID A1995TL73600010
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Primal-dual methods for linear programming
MATHEMATICAL PROGRAMMING
1995; 70 (3): 251-277
View details for Web of Science ID A1995TK93600002
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A PRACTICAL INTERIOR-POINT METHOD FOR CONVEX-PROGRAMMING
SIAM JOURNAL ON OPTIMIZATION
1995; 5 (1): 149-171
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- MINOS(IIS) version 4.2: Analyzing infeasibilities inlinear programming Eur. J. Oper. Res. 1995; 81: 217-218
- Fortran software for optimization 1995
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THE SIMPLEX ALGORITHM WITH A NEW PRIMAL AND DUAL PIVOT RULE
OPERATIONS RESEARCH LETTERS
1994; 16 (3): 121-127
View details for Web of Science ID A1994PV30700001
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SOLVING REDUCED KKT SYSTEMS IN BARRIER METHODS FOR LINEAR-PROGRAMMING
15th Dundee Conference on Numerical Analysis
LONGMAN SCIENTIFIC & TECHNICAL. 1994: 89–104
View details for Web of Science ID A1994BA91K00006
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Fortran software for optimization
1995 NSF Design and Manufacturing Grantees Conference
SOC MANUFACTURING ENGINEERS. 1994: 31–32
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- Major Cholesky would feel proud ORSA J. Comput. 1994; 6: 23-27
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PRECONDITIONERS FOR INDEFINITE SYSTEMS ARISING IN OPTIMIZATION
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
1992; 13 (1): 292-311
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- Some theoretical properties of an augmented Lagrangian merit function Advances in Optimization and Parallel Computing edited by Pardalos, P., M. North-Holland, Amsterdam. 1992: 101–128
- The applicationof nonlinear programming and collocation to optimal aeroassisted orbital transfers 1992
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A BLOCK-LU UPDATE FOR LARGE-SCALE LINEAR-PROGRAMMING
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
1992; 13 (1): 191-201
View details for Web of Science ID A1992HC84600016
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INERTIA-CONTROLLING METHODS FOR GENERAL QUADRATIC-PROGRAMMING
SIAM REVIEW
1991; 33 (1): 1-36
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- An adaptive primal-dual method for linear programming Math.Prog. Soc., Committee on Algorithms Newsletter 1991; 19: 7-16
- A Schur-complement method forsparse quadratic programming Reliable Numerical Computation edited by Cox, M., G., Hammarling, S. Oxford University Press, Oxford and New York. 1990: 113–138
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A PRACTICAL ANTICYCLING PROCEDURE FOR LINEARLY CONSTRAINED OPTIMIZATION
MATHEMATICAL PROGRAMMING
1989; 45 (3): 437-474
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- Constrained nonlinear programming Optimization Handbooks in Operations Research and Management Science edited by Nemhauser, G., L., G., A., H., Kan, R. North-Holland, Amsterdam. 1989: 171–210
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2 CONJUGATE-GRADIENT-TYPE METHODS FOR UNSYMMETRIC LINEAR-EQUATIONS
SIAM JOURNAL ON NUMERICAL ANALYSIS
1988; 25 (4): 927-940
View details for Web of Science ID A1988P634900009
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RECENT DEVELOPMENTS IN CONSTRAINED OPTIMIZATION
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
1988; 22 (2-3): 257-270
View details for Web of Science ID A1988P398600009
- Two conjugate-gradient-type methods forunsymmetric linear equations SIAM J. Numer. Anal. 1988; 25: 927-940
- GAMS/MINOS GAMS: A User's Guide edited by Brooke, A., Kendrick, D., Meeraus, A. The Scientic Press. 1988: 201–224
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MAINTAINING LU FACTORS OF A GENERAL SPARSE-MATRIX
LINEAR ALGEBRA AND ITS APPLICATIONS
1987; 88-9: 239-270
View details for Web of Science ID A1987G801700014
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ON PROJECTED NEWTON BARRIER METHODS FOR LINEAR-PROGRAMMING AND AN EQUIVALENCE TO KARMARKAR PROJECTIVE METHOD
MATHEMATICAL PROGRAMMING
1986; 36 (2): 183-209
View details for Web of Science ID A1986F105800006
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CONSIDERATIONS OF NUMERICAL-ANALYSIS IN A SEQUENTIAL QUADRATIC-PROGRAMMING METHOD
LECTURE NOTES IN MATHEMATICS
1986; 1230: 46-62
View details for Web of Science ID A1986G659700004
- Considerations of numerical analysis in sequential quadratic programming methods Numerical Analysis edited by Hennart, J., P. Springer-Verlag, New York and London. 1986: 46–62
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PROPERTIES OF A REPRESENTATION OF A BASIS FOR THE NULL SPACE
MATHEMATICAL PROGRAMMING
1985; 33 (2): 172-186
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- Software and its relationship tomethods Numerical Optimization 1984 edited by Boggs, P., T., Byrd, R., H., B., R. SIAM, Philadelphia. 1985: 139–159
- Model building and practical aspects of nonlinear programming Computational Mathematical Programming edited by Schittkowski, K. NATO ASI, Springer-Verlag,Berlin and New York. 1985: 209–247
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A WEIGHTED GRAM-SCHMIDT METHOD FOR CONVEX QUADRATIC-PROGRAMMING
MATHEMATICAL PROGRAMMING
1984; 30 (2): 176-195
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- Sequential quadratic programming methods for nonlinear programming Computer Aided Analysis and Optimization of Mechanical System Dynamics edited by Haug, E., J. NATO ASI. 1984: 679–697
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TRENDS IN NONLINEAR-PROGRAMMING SOFTWARE
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
1984; 17 (2): 141-149
View details for Web of Science ID A1984TF70000001
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PROCEDURES FOR OPTIMIZATION PROBLEMS WITH A MIXTURE OF BOUNDS AND GENERAL LINEAR CONSTRAINTS
ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
1984; 10 (3): 282-298
View details for Web of Science ID A1984TU57100006
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SPARSE-MATRIX METHODS IN OPTIMIZATION
SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING
1984; 5 (3): 562-589
View details for Web of Science ID A1984TG34100006
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AQUIFER RECLAMATION DESIGN - THE USE OF CONTAMINANT TRANSPORT SIMULATION COMBINED WITH NONLINEAR-PROGRAMMING
WATER RESOURCES RESEARCH
1984; 20 (4): 415-427
View details for Web of Science ID A1984SM88600001
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COMPUTING FORWARD-DIFFERENCE INTERVALS FOR NUMERICAL OPTIMIZATION
SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING
1983; 4 (2): 310-321
View details for Web of Science ID A1983QQ77800015
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ALGORITHM-583 - LSQR - SPARSE LINEAR-EQUATIONS AND LEAST-SQUARES PROBLEMS
ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
1982; 8 (2): 195-209
View details for Web of Science ID A1982PE14000007
- Software for constrained optimization Nonlinear Optimization 1981 edited by Powell, M. J., D. Academic Press, London and New York. 1982: 381–393
- Linearly constrained optimization Nonlinear Optimization 1981 edited by Powell, M. J., D. Academic Press, London and NewYork. 1982: 123–139
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LSQR - AN ALGORITHM FOR SPARSE LINEAR-EQUATIONS AND SPARSE LEAST-SQUARES
ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
1982; 8 (1): 43-71
View details for Web of Science ID A1982NH42200005
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A NOTE ON A SUFFICIENT-DECREASE CRITERION FOR A NON-DERIVATIVE STEP-LENGTH PROCEDURE
MATHEMATICAL PROGRAMMING
1982; 23 (3): 349-352
View details for Web of Science ID A1982NX42400006
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A PROJECTED LAGRANGIAN ALGORITHM AND ITS IMPLEMENTATION FOR SPARSE NON-LINEAR CONSTRAINTS
MATHEMATICAL PROGRAMMING STUDY
1982; 16 (MAR): 84-117
View details for Web of Science ID A1982NR90100006
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ASPECTS OF MATHEMATICAL-MODELING RELATED TO OPTIMIZATION
APPLIED MATHEMATICAL MODELLING
1981; 5 (2): 71-83
View details for Web of Science ID A1981LJ58400002
- QP-based methods for large-scale nonlinearly constrained optimization Nonlinear Programming 4 edited by Mangasarian, O., L., Meyer, R., R., M., S. Academic Press London and New York. 1981: 57–98
- A numerical investigation of ellipsoid algorithms for large-scale linear programming Large-scale Linear Programming edited by Dantzig, G., B., Dempster, M., A.H., Kallio, M. axenburg, Austria. 1981: 487–509
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TOWARDS A GENERALIZED SINGULAR VALUE DECOMPOSITION
SIAM JOURNAL ON NUMERICAL ANALYSIS
1981; 18 (3): 398-405
View details for Web of Science ID A1981LT69800003
- Methods for large-scale nonlinear optimization Electric PowerProblems: The Mathematical Challenge edited by Erisman, A., M., Neves, K., W., Dwarakanath, M., H. SIAM, Philadelphia. 1980: 352–377
- Sparse least squares by conjugate gradients: a comparison of preconditioning methods 1979
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LARGE-SCALE LINEARLY CONSTRAINED OPTIMIZATION
MATHEMATICAL PROGRAMMING
1978; 14 (1): 41-72
View details for Web of Science ID A1978EM23400004
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LEAST-SQUARES ESTIMATION OF DISCRETE LINEAR DYNAMIC-SYSTEMS USING ORTHOGONAL TRANSFORMATIONS
SIAM JOURNAL ON NUMERICAL ANALYSIS
1977; 14 (2): 180-193
View details for Web of Science ID A1977DC77900002
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NONLINEAR OPTIMIZATION SUBJECT TO LINEAR-PROGRAMMING CONSTRAINTS
SIAM PUBLICATIONS. 1976: 825–26
View details for Web of Science ID A1976CH15400157
- A fast, stable implementation of the simplex method using Bartels-Golub updating Sparse Matrix Computations edited by Bunch, J., R., Rose, D., J. Academic Press. 1976: 213–226
- The complexity of LU updating in the simplex method The Complexity of Computational Problem Solving edited by Brent, R., P. University of Queensland Press. 1976: 214–230
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SOLUTION OF SPARSE INDEFINITE SYSTEMS OF LINEAR EQUATIONS
SIAM JOURNAL ON NUMERICAL ANALYSIS
1975; 12 (4): 617-629
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- Methods for computing and modifying the LDV factors of a matrix Math. Comput. 1975; 29: 1051-1077
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METHODS FOR COMPUTING AND MODIFYING LDV FACTORS OF A MATRIX
MATHEMATICS OF COMPUTATION
1975; 29 (132): 1051-1077
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METHODS FOR MODIFYING MATRIX FACTORIZATIONS
MATHEMATICS OF COMPUTATION
1974; 28 (126): 505-535
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- Numerical stability in large-scale linear programming Approximation and Accuracy edited by deHoog, F., R., Jarvis, C., L. University of Queensland Press. 1973: 144–158
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