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.

Academic Appointments


  • Professor Emeritus, Management Science and Engineering

Honors & Awards


  • Orchard-Hays Prize, MPS (1985)
  • Highly Cited Researcher, Computer Science, ISI (2004)
  • Highly Cited Researcher, Mathematics, ISI (2007)
  • Honorary Fellow, RSNZ (2007)
  • Linear Algebra Prize, SIAM (2012)
  • Invention Hall of Fame, OTL, Stanford University (2012)
  • Fellow, SIAM (2013)

Boards, Advisory Committees, Professional Organizations


  • Member, ACM (1982 - Present)
  • Associate Editor, NACO (2010 - 2016)
  • Member, INFORMS (2010 - Present)
  • Member, ORSNZ (1990 - Present)
  • Member, SIAM (1980 - Present)
  • Associate Editor, ACM TOMS (1982 - 2004)
  • Associate Editor, SIAM Journal on Optimization (1989 - 2002)
  • Associate Editor, OPTE (1999 - Present)

Program Affiliations


  • Institute for Computational and Mathematical Engineering (ICME)

Professional Education


  • B.Sc. (Hons), Canterbury, Mathematics (1965)
  • MS, Stanford University, Computer Science (1970)
  • PhD, Stanford University, Computer Science (1972)

2018-19 Courses


Stanford Advisees


All Publications


  • 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 Campisi-Pinto, S., Levi, O., Benson, D., Cohen, M., Resende, M., Saunders, M., Linder, C., Wiesman, Z. 2019; 50 (1-3): 521
  • EUCLIDEAN-NORM ERROR BOUNDS FOR SYMMLQ AND CG SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS Estrin, R., Orban, D., Saunders, M. 2019; 40 (1): 235–53

    View details for DOI 10.1137/16M1094816

    View details for Web of Science ID 000462583900011

  • LSLQ: AN ITERATIVE METHOD FOR LINEAR LEAST-SQUARES WITH AN ERROR MINIMIZATION PROPERTY SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS Estrin, R., Orban, D., Saunders, M. A. 2019; 40 (1): 254–75

    View details for DOI 10.1137/17M1113552

    View details for Web of Science ID 000462583900012

  • Reliable and efficient solution of genome-scale models of Metabolism and macromolecular Expression SCIENTIFIC REPORTS Ma, D., Yang, L., Fleming, R. M., Thiele, I., Palsson, B. O., Saunders, M. A. 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

  • Conditions for duality between fluxes and concentrations in biochemical networks JOURNAL OF THEORETICAL BIOLOGY Fleming, R. M., Vlassis, N., Thiele, I., Saunders, M. A. 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 Berman, P., Levi, O., Parmet, Y., Saunders, M., Wiesman, Z. 2013; 42A:3: 72-88
  • Novel 1H low field nuclear magnetic resonance applications for the field of biodiesel Biotechnologyfor Biofuels Berman, P., Leshem, A., Etziony, O., Levi, O., Parmet, Y., Saunders, M., Wiesman, Z. 2013; 6:55: 20
  • LSRN: a parallel iterative solver for strongly over- or under-determined systems SIAM J. Sci. Comp. Meng, X., Saunders, M. A., Mahoney, M. W. 2013; 36 (2): C95-C118
  • A variational principle for computing nonequilibrium fluxes and potentials in genome-scale biochemical networks JOURNAL OF THEORETICAL BIOLOGY Fleming, R. M., MAES, C. M., Saunders, M. A., Ye, Y., Palsson, B. O. 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

  • LSMR: AN ITERATIVE ALGORITHM FOR SPARSE LEAST-SQUARES PROBLEMS SIAM JOURNAL ON SCIENTIFIC COMPUTING Fong, D. C., Saunders, M. 2011; 33 (5): 2950-2971
  • SNOPT: An SQP algorithm for large-scaleconstrained optimization, SIGEST article SIAM Rev. Gill, P., E., Murray, W., Saunders, M., A. 2005; 1 (47): 99-131
  • Atomic decomposition by basis pursuit, SIGEST article SIAM Rev. Chen, S., S., Donoho, D., L., Saunders, M., A. 2001; 1 (43): 129-159
  • Creation and analysis of biochemical constraint-based models using the COBRA Toolbox v.3.0. Nature protocols Heirendt, L., Arreckx, S., Pfau, T., Mendoza, S. N., Richelle, A., Heinken, A., Haraldsdottir, H. S., Wachowiak, J., Keating, S. M., Vlasov, V., Magnusdottir, S., Ng, C. Y., Preciat, G., Zagare, A., Chan, S. H., Aurich, M. K., Clancy, C. M., Modamio, J., Sauls, J. T., Noronha, A., Bordbar, A., Cousins, B., El Assal, D. C., Valcarcel, L. V., Apaolaza, I., Ghaderi, S., Ahookhosh, M., Ben Guebila, M., Kostromins, A., Sompairac, N., Le, H. M., Ma, D., Sun, Y., Wang, L., Yurkovich, J. T., Oliveira, M. A., Vuong, P. T., El Assal, L. P., Kuperstein, I., Zinovyev, A., Hinton, H. S., Bryant, W. A., Aragon Artacho, F. J., Planes, F. J., Stalidzans, E., Maass, A., Vempala, S., Hucka, M., Saunders, M. A., Maranas, C. D., Lewis, N. E., Sauter, T., Palsson, B. O., Thiele, I., Fleming, R. M. 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

  • DynamicME: dynamic simulation and refinement of integrated models of metabolism and protein expression. BMC systems biology Yang, L., Ebrahim, A., Lloyd, C. J., Saunders, M. A., Palsson, B. O. 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

  • 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 Campisi-Pinto, S., Levi, O., Benson, D., Cohen, M., Resende, M., Saunders, M., Linder, C., Wiesman, Z. 2018; 49 (10): 1129–50
  • Sparse least squares by conjugate gradients: a comparison of preconditioning methods M. edited by J. University of Waterloo, Waterloo, Ontario, Canada
  • Principles of proteome allocation are revealed using proteomic data and genome-scale models SCIENTIFIC REPORTS Yang, L., Yurkovich, J. T., Lloyd, C. J., Ebrahim, A., Saunders, M. A., Palsson, B. O. 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

  • Heart rate analysis by sparse representation for acute pain detection MEDICAL & BIOLOGICAL ENGINEERING & COMPUTING Tejman-Yarden, S., Levi, O., Beizerov, A., Parmet, Y., Tu Nguyen, T., Saunders, M., Rudich, Z., Perry, J. C., Baker, D. G., Moeller-Bertram, T. 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

  • Heart rate analysis by sparse representation for acute pain detection. Medical & biological engineering & computing Tejman-Yarden, S., Levi, O., Beizerov, A., Parmet, Y., Nguyen, T., Saunders, M., Rudich, Z., Perry, J. C., Baker, D. G., Moeller-Bertram, T. 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

  • solveME: fast and reliable solution of nonlinear ME models. BMC bioinformatics Yang, L., Ma, D., Ebrahim, A., Lloyd, C. J., Saunders, M. A., Palsson, B. O. 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 PubMedID 27659412

  • A Practical Factorization of a Schur Complement for PDE-Constrained Distributed Optimal Control JOURNAL OF SCIENTIFIC COMPUTING Choi, Y., Farhat, C., Murray, W., Saunders, M. 2015; 65 (2): 576-597
  • Do genome-scale models need exact solvers or clearer standards? MOLECULAR SYSTEMS BIOLOGY Ebrahim, A., Almaas, E., Bauer, E., Bordbar, A., Burgard, A. P., Chang, R. L., Draeger, A., Famili, I., Feist, A. M., Fleming, R. T., Fong, S. S., Hatzimanikatis, V., Herrgard, M. J., Holder, A., Hucka, M., Hyduke, D., Jamshidi, N., Lee, S., Le Novere, N., Lerman, J. A., Lewis, N. E., Ma, D., Mahadevan, R., Maranas, C., Nagarajan, H., Navid, A., Nielsen, J., Nielsen, L. K., Nogales, J., Noronha, A., Pal, C., Palsson, B. O., Papin, J. A., Patil, K. R., Price, N. D., Reed, J. L., Saunders, M., Senger, R. S., Sonnenschein, N., Sun, Y., Thiele, I. 2015; 11 (10): 831

    View details for PubMedID 26467284

  • 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 Yang, L., Tan, J., O'Brien, E. J., Monk, J. M., Kim, D., Li, H. J., Charusanti, P., Ebrahim, A., Lloyd, C. J., Yurkovich, J. T., Du, B., Dräger, A., Thomas, A., Sun, Y., Saunders, M. A., Palsson, B. O. 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

  • 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 Yang, L., Tan, J., O'Brien, E. J., Monk, J. M., Kim, D., Li, H. J., Charusanti, P., Ebrahim, A., Lloyd, C. J., Yurkovich, J. T., Du, B., Draeger, A., Thomas, A., Sun, Y., Saunders, M. A., Palsson, B. O. 2015; 112 (34): 10810-10815
  • Study of liquid-phase molecular packing interactions and morphology of fatty acid methyl esters (biodiesel). Biotechnology for biofuels Berman, P., Meiri, N., Colnago, L. A., Moraes, T. B., Linder, C., Levi, O., Parmet, Y., Saunders, M., Wiesman, Z. 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

  • Algorithm 937: MINRES-QLP for Symmetric and Hermitian Linear Equations and Least-Squares Problems ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE Choi, S. T., Saunders, M. A. 2014; 40 (2)

    View details for DOI 10.1145/2527267

    View details for Web of Science ID 000333653400008

  • LSRN: A PARALLEL ITERATIVE SOLVER FOR STRONGLY OVER- OR UNDERDETERMINED SYSTEMS SIAM JOURNAL ON SCIENTIFIC COMPUTING Meng, X., Saunders, M. A., Mahoney, M. W. 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

  • Algorithm 937: MINRES-QLP for Symmetric and Hermitian Linear Equations and Least-Squares Problems. ACM transactions on mathematical software. Association for Computing Machinery Choi, S. T., Saunders, M. A. 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 PubMedID 25328255

    View details for PubMedCentralID PMC4199394

  • 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 Meng, X., Saunders, M. A., Mahoney, M. W. 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 PubMedID 25419094

    View details for PubMedCentralID PMC4238893

  • PROXIMAL NEWTON-TYPE METHODS FOR MINIMIZING COMPOSITE FUNCTIONS SIAM JOURNAL ON OPTIMIZATION Lee, J. D., Sun, Y., Saunders, M. A. 2014; 24 (3): 1420-1443

    View details for DOI 10.1137/130921428

    View details for Web of Science ID 000343229000019

  • Robust flux balance analysis of multiscale biochemical reaction networks BMC BIOINFORMATICS Sun, Y., Fleming, R. M., Thiele, I., Saunders, M. A. 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

  • Equispaced Pareto front construction for constrained bi-objective optimization MATHEMATICAL AND COMPUTER MODELLING Pereyra, V., Saunders, M., Castillo, J. 2013; 57 (9-10): 2122-2131
  • Robust flux balance analysis of multiscale biochemical reaction networks BMC Bioinformatics Fleming, R. M., Saunders, M. A. 2013; 14:240: 6
  • CG versus MINRES: An empirical comparison SQUJournal for Science Fong, D., C.-L., Saunders, M., A. 2012; 17:1: 44-62
  • A Higher-Order Generalized Singular Value Decomposition for Comparison of Global mRNA Expression from Multiple Organisms PLOS ONE Ponnapalli, S. P., Saunders, M. A., Van Loan, C. F., Alter, O. 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

  • MINRES-QLP: A KRYLOV SUBSPACE METHOD FOR INDEFINITE OR SINGULAR SYMMETRIC SYSTEMS SIAM JOURNAL ON SCIENTIFIC COMPUTING Choi, S. T., Paige, C. C., Saunders, M. A. 2011; 33 (4): 1810-1836

    View details for DOI 10.1137/100787921

    View details for Web of Science ID 000294293200016

  • Nonconservative Robust Control: Optimized and Constrained Sensitivity Functions IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY Fransson, C., Wik, T., Lennartson, B., Saunders, M., Gutman, P. 2009; 17 (2): 298-308
  • STABILIZING POLICY IMPROVEMENT FOR LARGE-SCALE INFINITE-HORIZON DYNAMIC PROGRAMMING SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS O'Sullivan, M. J., Saunders, M. A. 2009; 31 (2): 434-459

    View details for DOI 10.1137/060653305

    View details for Web of Science ID 000267745500012

  • Variational Bayesian image restoration based on a product of t-distributions image prior IEEE TRANSACTIONS ON IMAGE PROCESSING Chantas, G., Galatsanos, N., Likas, A., Saunders, M. 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

  • George B. Dantzig and systems optimization DISCRETE OPTIMIZATION Gill, P. E., Murray, W., Saunders, M. A., Tomlin, J. A., Wright, M. H. 2008; 5 (2): 151-158
  • Discussion: The Dantzig selector: Statistical estimation when p is much larger than n ANNALS OF STATISTICS Friedlander, M. P., Saunders, M. A. 2007; 35 (6): 2385-2391
  • Commentary on Methods for modifying matrix factorizations Milestones in Matrix Computation: Selected Works of Gene H. Golub With Commentaries Saunders, M., A. edited by Chan, R., H., Greif, C., O'Leary, D., P. Oxford University Press. 2007: 310–310
  • SpaseLoc: An adaptive subproblem algorithm for scalable wireless sensor network localization SIAM JOURNAL ON OPTIMIZATION Carter, M. W., Jin, H. H., Saunders, M. A., Ye, Y. 2006; 17 (4): 1102-1128

    View details for DOI 10.1137/040621600

    View details for Web of Science ID 000244631800007

  • SNOPT: An SQP algorithm for large-scale constrained optimization (Reprinted from SIAM Journal Optimization, vol 12, pg 979-1006, 2002) SIAM REVIEW Gill, P. E., Murray, W., Saunders, M. A. 2005; 47 (1): 99-131
  • A globally convergent linearly constrained Lagrangian method for nonlinear optimization SIAM JOURNAL ON OPTIMIZATION Friedlander, M. P., Saunders, M. A. 2005; 15 (3): 863-897
  • Sparsity and smoothness via the fused lasso JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., Knight, K. 2005; 67: 91-108
  • A bisection algorithm for the mixed mu upper bound and its supremum American Control Conference Fransson, C. M., Saunders, M. A. IEEE. 2004: 2665–2670
  • Subspace preconditioned LSQR for discrete ill-posed problems Conference on Computational Linear Algebra with Applications Jacobsen, M., Hansen, P. C., Saunders, M. A. SPRINGER. 2003: 975–89
  • SNOPT: An SQP algorithm for large-scale constrained optimization SIAM JOURNAL ON OPTIMIZATION Gill, P. E., Murray, W., Saunders, M. A. 2002; 12 (4): 979-1006
  • Global controller optimization using Horowitz bounds Fransson, C., M., Lennartson, B., Wik, T., Holmstrom, K., Saunders, M., Gutman, P., O. 2002
  • Atomic decomposition by basis pursuit SIAM REVIEW Chen, S. S., Donoho, D. L., Saunders, M. A. 2001; 43 (1): 129-159
  • Atomic decomposition by basis pursuit SIAM JOURNAL ON SCIENTIFIC COMPUTING Chen, S. S., Donoho, D. L., Saunders, M. A. 1998; 20 (1): 33-61
  • SNOPT: A Fortran software package to solve large-scale optimization problems Gill, P., E., Murray, W., Saunders, M., A. 1998
  • OSSE mapping of galactic 511 keV positron annihilation line emission ASTROPHYSICAL JOURNAL Purcell, W. R., Cheng, L. X., Dixon, D. D., Kinzer, R. L., Kurfess, J. D., Leventhal, M., Saunders, M. A., Skibo, J. G., Smith, D. M., Tueller, J. 1997; 491 (2): 725-748
  • Computing projections with LSQR BIT NUMERICAL MATHEMATICS Saunders, M. A. 1997; 37 (1): 96-104
  • Non-parametric estimates of high energy gamma-ray source distributions 4th Compton Symposium Dixon, D. D., Kolaczyk, E. D., Samimi, J., Saunders, M. A. AIP PRESS. 1997: 1601–5
  • 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 Saunders, M. A. SIAM. 1996: 92–100
  • SQP methods for large-scale optimization Gill, P., E., Murray, W., Saunders, M., A. 1996
  • On the stability of Cholesky factorization for symmetric quasidefinite systems SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS Gill, P. E., Saunders, M. A., Shinnerl, J. R. 1996; 17 (1): 35-46
  • Solution of sparse rectangular systems using LSQR and Craig BIT NUMERICAL MATHEMATICS Saunders, M. A. 1995; 35 (4): 588-604
  • Primal-dual methods for linear programming MATHEMATICAL PROGRAMMING Gill, P. E., Murray, W., PONCELEON, D. B., Saunders, M. A. 1995; 70 (3): 251-277
  • A PRACTICAL INTERIOR-POINT METHOD FOR CONVEX-PROGRAMMING SIAM JOURNAL ON OPTIMIZATION Jarre, F., Saunders, M. A. 1995; 5 (1): 149-171
  • MINOS(IIS) version 4.2: Analyzing infeasibilities inlinear programming Eur. J. Oper. Res. Chinneck, J., W., Saunders, M., A. 1995; 81: 217-218
  • Fortran software for optimization Gill, P., E., Murray, W., Saunders, M., A. 1995
  • THE SIMPLEX ALGORITHM WITH A NEW PRIMAL AND DUAL PIVOT RULE OPERATIONS RESEARCH LETTERS Chen, H. D., Pardalos, P. M., Saunders, M. A. 1994; 16 (3): 121-127
  • SOLVING REDUCED KKT SYSTEMS IN BARRIER METHODS FOR LINEAR-PROGRAMMING 15th Dundee Conference on Numerical Analysis Gill, P. E., Murray, W., PONCELEON, D. B., Saunders, M. A. LONGMAN SCIENTIFIC & TECHNICAL. 1994: 89–104
  • Fortran software for optimization 1995 NSF Design and Manufacturing Grantees Conference Gill, P. E., Murray, W., Saunders, M. A. SOC MANUFACTURING ENGINEERS. 1994: 31–32
  • Large-scale SQP methods and their applicationin trajectory optimization Control Applications of Opti-mization Gill, P., E., Murray, W., Saunders, M., A. edited by Bulirsch, R., Kraft, D. Birkhauser Verlag, Basel,Boston, Stuttgart. 1994: 29–42
  • Solving reduced KKT systems in barrier methods for linear programming Numerical Analysis 1993 Gill, P., E., Murray, W., Ponceleon, D., B., Saunders, M., A. edited by Watson, G., A., Grffiths, D. Pitman Research Notes in Mathematics 303, Longmans Press. 1994: 89–104
  • Major Cholesky would feel proud ORSA J. Comput. Saunders, M., A. 1994; 6: 23-27
  • PRECONDITIONERS FOR INDEFINITE SYSTEMS ARISING IN OPTIMIZATION SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS Gill, P. E., Murray, W., PONCELEON, D. B., Saunders, M. A. 1992; 13 (1): 292-311
  • Some theoretical properties of an augmented Lagrangian merit function Advances in Optimization and Parallel Computing Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. edited by Pardalos, P., M. North-Holland, Amsterdam. 1992: 101–128
  • The applicationof nonlinear programming and collocation to optimal aeroassisted orbital transfers Shi, Y., Y., Nelson, R., Young, D., H., Gill, P., E., Murray, W., Saunders, M., A. 1992
  • A BLOCK-LU UPDATE FOR LARGE-SCALE LINEAR-PROGRAMMING SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS ELDERSVELD, S. K., Saunders, M. A. 1992; 13 (1): 191-201
  • INERTIA-CONTROLLING METHODS FOR GENERAL QUADRATIC-PROGRAMMING SIAM REVIEW Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1991; 33 (1): 1-36
  • An adaptive primal-dual method for linear programming Math.Prog. Soc., Committee on Algorithms Newsletter Jarre, F., Saunders, M., A. 1991; 19: 7-16
  • A Schur-complement method forsparse quadratic programming Reliable Numerical Computation Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. edited by Cox, M., G., Hammarling, S. Oxford University Press, Oxford and New York. 1990: 113–138
  • A PRACTICAL ANTICYCLING PROCEDURE FOR LINEARLY CONSTRAINED OPTIMIZATION MATHEMATICAL PROGRAMMING Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1989; 45 (3): 437-474
  • Constrained nonlinear programming Optimization Handbooks in Operations Research and Management Science Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. edited by Nemhauser, G., L., G., A., H., Kan, R. North-Holland, Amsterdam. 1989: 171–210
  • 2 CONJUGATE-GRADIENT-TYPE METHODS FOR UNSYMMETRIC LINEAR-EQUATIONS SIAM JOURNAL ON NUMERICAL ANALYSIS Saunders, M. A., Simon, H. D., YIP, E. L. 1988; 25 (4): 927-940
  • RECENT DEVELOPMENTS IN CONSTRAINED OPTIMIZATION JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1988; 22 (2-3): 257-270
  • Two conjugate-gradient-type methods forunsymmetric linear equations SIAM J. Numer. Anal. Saunders, M., A., Simon, H., D., Yip, E., L. 1988; 25: 927-940
  • GAMS/MINOS GAMS: A User's Guide Gill, P., E., Murray, W., Murtagh, B., A., Saunders, M., A., Wright, M., H. edited by Brooke, A., Kendrick, D., Meeraus, A. The Scientic Press. 1988: 201–224
  • MAINTAINING LU FACTORS OF A GENERAL SPARSE-MATRIX LINEAR ALGEBRA AND ITS APPLICATIONS Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1987; 88-9: 239-270
  • ON PROJECTED NEWTON BARRIER METHODS FOR LINEAR-PROGRAMMING AND AN EQUIVALENCE TO KARMARKAR PROJECTIVE METHOD MATHEMATICAL PROGRAMMING Gill, P. E., Murray, W., Saunders, M. A., Tomlin, J. A., Wright, M. H. 1986; 36 (2): 183-209
  • CONSIDERATIONS OF NUMERICAL-ANALYSIS IN A SEQUENTIAL QUADRATIC-PROGRAMMING METHOD LECTURE NOTES IN MATHEMATICS Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1986; 1230: 46-62
  • Considerations of numerical analysis in sequential quadratic programming methods Numerical Analysis Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. edited by Hennart, J., P. Springer-Verlag, New York and London. 1986: 46–62
  • PROPERTIES OF A REPRESENTATION OF A BASIS FOR THE NULL SPACE MATHEMATICAL PROGRAMMING Gill, P. E., Murray, W., Saunders, M. A., Stewart, G. W., Wright, M. H. 1985; 33 (2): 172-186
  • Software and its relationship tomethods Numerical Optimization 1984 Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. 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 Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. edited by Schittkowski, K. NATO ASI, Springer-Verlag,Berlin and New York. 1985: 209–247
  • PROCEDURES FOR OPTIMIZATION PROBLEMS WITH A MIXTURE OF BOUNDS AND GENERAL LINEAR CONSTRAINTS ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1984; 10 (3): 282-298
  • Sequential quadratic programming methods for nonlinear programming Computer Aided Analysis and Optimization of Mechanical System Dynamics Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. edited by Haug, E., J. NATO ASI. 1984: 679–697
  • TRENDS IN NONLINEAR-PROGRAMMING SOFTWARE EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1984; 17 (2): 141-149
  • SPARSE-MATRIX METHODS IN OPTIMIZATION SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1984; 5 (3): 562-589
  • AQUIFER RECLAMATION DESIGN - THE USE OF CONTAMINANT TRANSPORT SIMULATION COMBINED WITH NONLINEAR-PROGRAMMING WATER RESOURCES RESEARCH Gorelick, S. M., Voss, C. I., Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1984; 20 (4): 415-427
  • A WEIGHTED GRAM-SCHMIDT METHOD FOR CONVEX QUADRATIC-PROGRAMMING MATHEMATICAL PROGRAMMING Gill, P. E., Gould, N. I., Murray, W., Saunders, M. A., Wright, M. H. 1984; 30 (2): 176-195
  • COMPUTING FORWARD-DIFFERENCE INTERVALS FOR NUMERICAL OPTIMIZATION SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1983; 4 (2): 310-321
  • ALGORITHM-583 - LSQR - SPARSE LINEAR-EQUATIONS AND LEAST-SQUARES PROBLEMS ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE Paige, C. C., Saunders, M. A. 1982; 8 (2): 195-209
  • Software for constrained optimization Nonlinear Optimization 1981 Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. edited by Powell, M. J., D. Academic Press, London and New York. 1982: 381–393
  • Linearly constrained optimization Nonlinear Optimization 1981 Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. edited by Powell, M. J., D. Academic Press, London and NewYork. 1982: 123–139
  • LSQR - AN ALGORITHM FOR SPARSE LINEAR-EQUATIONS AND SPARSE LEAST-SQUARES ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE Paige, C. C., Saunders, M. A. 1982; 8 (1): 43-71
  • A NOTE ON A SUFFICIENT-DECREASE CRITERION FOR A NON-DERIVATIVE STEP-LENGTH PROCEDURE MATHEMATICAL PROGRAMMING Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1982; 23 (3): 349-352
  • A PROJECTED LAGRANGIAN ALGORITHM AND ITS IMPLEMENTATION FOR SPARSE NON-LINEAR CONSTRAINTS MATHEMATICAL PROGRAMMING STUDY Murtagh, B. A., Saunders, M. A. 1982; 16 (MAR): 84-117
  • ASPECTS OF MATHEMATICAL-MODELING RELATED TO OPTIMIZATION APPLIED MATHEMATICAL MODELLING Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H. 1981; 5 (2): 71-83
  • QP-based methods for large-scale nonlinearly constrained optimization Nonlinear Programming 4 Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. 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 Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. edited by Dantzig, G., B., Dempster, M., A.H., Kallio, M. axenburg, Austria. 1981: 487–509
  • TOWARDS A GENERALIZED SINGULAR VALUE DECOMPOSITION SIAM JOURNAL ON NUMERICAL ANALYSIS Paige, C. C., Saunders, M. A. 1981; 18 (3): 398-405
  • Methods for large-scale nonlinear optimization Electric PowerProblems: The Mathematical Challenge Gill, P., E., Murray, W., Saunders, M., A., Wright, M., H. 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 Saunders, M., A. 1979
  • LARGE-SCALE LINEARLY CONSTRAINED OPTIMIZATION MATHEMATICAL PROGRAMMING Murtagh, B. A., Saunders, M. A. 1978; 14 (1): 41-72
  • LEAST-SQUARES ESTIMATION OF DISCRETE LINEAR DYNAMIC-SYSTEMS USING ORTHOGONAL TRANSFORMATIONS SIAM JOURNAL ON NUMERICAL ANALYSIS Paige, C. C., Saunders, M. A. 1977; 14 (2): 180-193
  • NONLINEAR OPTIMIZATION SUBJECT TO LINEAR-PROGRAMMING CONSTRAINTS Saunders, M. SIAM PUBLICATIONS. 1976: 825–26
  • A fast, stable implementation of the simplex method using Bartels-Golub updating Sparse Matrix Computations Saunders, M., A. 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 Saunders, M., A. edited by Brent, R., P. University of Queensland Press. 1976: 214–230
  • SOLUTION OF SPARSE INDEFINITE SYSTEMS OF LINEAR EQUATIONS SIAM JOURNAL ON NUMERICAL ANALYSIS Paige, C. C., Saunders, M. A. 1975; 12 (4): 617-629
  • METHODS FOR COMPUTING AND MODIFYING LDV FACTORS OF A MATRIX MATHEMATICS OF COMPUTATION Gill, P. E., Murray, W., Saunders, M. A. 1975; 29 (132): 1051-1077
  • Methods for computing and modifying the LDV factors of a matrix Math. Comput. Gill, P. E., Saunders, M. A. 1975; 29: 1051-1077
  • METHODS FOR MODIFYING MATRIX FACTORIZATIONS MATHEMATICS OF COMPUTATION Gill, P. E., Golub, G. H., Murray, W., Saunders, M. A. 1974; 28 (126): 505-535
  • Numerical stability in large-scale linear programming Approximation and Accuracy Saunders, M., A. edited by deHoog, F., R., Jarvis, C., L. University of Queensland Press. 1973: 144–158
  • Descent methods for minimization Optimization Osborne, M., R., Saunders, M., A. edited by Ryan, D., M. University of Queensland Press. 1972: 221–237
  • Linear least squares and quadratic programming Integer and Nonlinear Programming Golub, G., H., Saunders, M., A. edited by Abadie, J. North-Holland, Amsterdam. 1970: 229–256
  • Numerical techniques in mathematical programming Nonlinear Programming Bartels, R., H., Golub, G., H., Saunders, M., A. edited by Rosen, J., B., Mangasarian, O., L., Ritter, K. Academic Press, London and New York. 1970: 123–176