Michael A. Saunders

Stanford University | SU · Department of Management Science and Engineering

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...

We recently showed, in a simulation study using two artificial signals, that our PDCO (Primal Dual interior method for Convex Objectives) reconstruction algorithm can be efficiently used for the reconstruction of low-field proton nuclear magnetic resonance (¹H LF-NMR) relaxation signals into T1 (spin–lattice) vs. T2 (spin–spin) time 2D graphs of a...

We build upon Estrin et al. (2019) to develop a general constrained nonlinear optimization algorithm based on a smooth penalty function proposed by Fletcher (1970, 1973b). Although Fletcher's approach has historically been considered impractical, we show that the computational kernels required are no more expensive than those in other widely accept...

We develop a general equality-constrained nonlinear optimization algorithm based on 6 a smooth penalty function proposed by Fletcher (1970). Although it was historically considered to be 7 computationally prohibitive in practice, we demonstrate that the computational kernels required are 8 no more expensive than other widely accepted methods for no...

We propose an iterative method named LSLQ for solving linear least-squares problems of any shape. The method is based on the Golub and Kahan (1965) process, where the dominant cost is products with the linear operator and its transpose. In the rank-deficient case, LSLQ identifies the minimum-length least- squares solution. LSLQ is formally equivale...

For positive definite and semidefinite consistent Ax* = b, we use the Gauss-Radau approach of Golub and Meurant (1997) to obtain an upper bound on the error ||x* - x_k^L||_2 for SYMMLQ iterates, assuming exact arithmetic. Such a bound, computable in constant time per iteration, was not previously available. We show that the CG error ||x* - x_k^C||_...

In the original publication, the acknowledgment section was not published and it is given below now.

Two-dimensional low-field hydrogen nuclear magnetic resonance (2D ¹H LF-NMR) analysis of chemical compounds measures T1 and T2 relaxation times observed as exponential decay curves. Once relaxation curves are measured and stored in the format of discrete digital signals, they must be transformed, mathematically, into spectra that can be read and in...

We describe LNLQ for solving the least-norm problem min |x| subject to Ax=b. Craig's method is known to be equivalent to applying the conjugate gradient method to the normal equations of the second kind AA'y = b, x = A'y. LNLQ is equivalent to applying SYMMLQ. If an underestimate to the smallest singular value is available, error upper bounds for b...

For optimization problems involving many nonlinear inequality constraints, we extend the bound-constrained (BCL) and linearly constrained (LCL) augmented Lagrangian approaches of LANCELOT and MINOS to an algorithm that solves a sequence of nonlinearly constrained augmented Lagrangian subproblems whose nonlinear constraints satisfy the LICQ everywhe...

For optimization problems involving many nonlinear inequality constraints, we extend the bound-constrained (BCL) and linearly constrained (LCL) augmented Lagrangian approaches of LANCELOT and MINOS to an algorithm that solves a sequence of nonlinearly constrained augmented Lagrangian subproblems whose nonlinear constraints satisfy the LICQ everywhe...

Genome-scale models of metabolism and macromolecular expression (ME models) enable systems-level computation of proteome allocation coupled to metabolic phenotype. We develop dynamicME, an algorithm enabling time-course simulation of cell metabolism and protein expression. Our dynamicME correctly predicted the substrate utilization hierarchy on mix...

COnstraint-Based Reconstruction and Analysis (COBRA) provides a molecular mechanistic framework for integrative analysis of experimental data and quantitative prediction of physicochemically and biochemically feasible phenotypic states. The COBRA Toolbox is a comprehensive software suite of interoperable COBRA methods. It has found widespread appli...

We propose an iterative method named LSLQ for solving linear least-squares problems Ax ≈ b of any shape. The method is based on the Golub and Kahan (1965) process, where the dominant cost is products with A and its transpose. In the rank-deficient case, LSLQ identifies the minimum-length least-squares solution. LSLQ is formally equivalent to SYMMLQ...

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 varyi...

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 formula...

Background 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 maximiz...

For positive definite linear systems (or semidefinite consistent systems), we use Gauss-Radau quadrature to obtain a cheaply computable upper bound on the 2-norm error of SYMMLQ iterates. The close relationship between SYMMLQ and CG iterates lets us construct an upper bound on the 2-norm error for CG. For indefinite systems, the upper bound becomes...

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 can compute steady-state flux solutions to ME models, but flux values are spread over many orders of magnitude. Standard double-precision solver...

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 metho...

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 con...

Constraint‐based analysis of genome‐scale models (GEMs) arose shortly after the first genome sequences became available. As numerous reviews of the field show, this approach and methodology has proven to be successful in studying a wide range of biological phenomena (McCloskey et al, 2013; Bordbar et al, 2014). However, efforts to expand the user b...

Table EV1

Dataset EV1

Significance Defining a core functional proteome supporting the living process has importance for both developing fundamental understanding of cell functions and for synthetic biology applications. Comparative genomics has been the primary approach to achieve such a definition. Here, we use genome-scale models to define a core proteome that computa...

(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 a...

This paper concerns some practical issues associated with the formulation of sequential quadratic programming (SQP) methods for large-scale nonlinear optimization. SQP methods find approximate solutions of a sequence of quadratic programming (QP) subproblems in which a quadratic model of the Lagrangian is minimized subject to the linearized constra...

Systems biologists are developing increasingly large models of metabolism and integrated models of metabolism and macromolecular expression. These Metabolic Expression (ME) models lead to sequences of multiscale linear programs for which small solution values of order 10−6 to 10−10 are meaningful. Standard LP solvers do not give sufficiently accura...

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 instabilit...

A distributed optimal control problem with the constraint of a linear elliptic partial differential equation is considered. A necessary optimality condition for this problem forms a saddle point system, the efficient and accurate solution of which is crucial. A new factorization of the Schur complement for such a system is proposed and its characte...

Biological processes such as metabolism, signaling, and macromolecular synthesis can be modeled aslarge networks of biochemical reactions. Large and comprehensive networks, like integrated networksthat represent metabolism and macromolecular synthesis, are inherently multiscale because reactionrates can vary over many orders of magnitude. They requ...

We consider constrained bi-objective optimization problems. One of the extant issues in this area is that of uniform sampling of the Pareto front. We utilize equispacing constraints on the vector of objective values, as discussed in a previous paper dealing with the unconstrained problem.We present a formulation and a dual formulation based on arc-...

Low-resolution nuclear magnetic resonance (LR-NMR) relaxometry is a powerful tool that can be harnessed for characterizing constituents in complex materials. Conversion of the relaxation signal into a continuous distribution of relaxation components is an ill-posed inverse Laplace transform problem. The most common numerical method implemented toda...

Background Biodiesel production has increased dramatically over the last decade, raising the need for new rapid and non-destructive analytical tools and technologies. 1H Low Field Nuclear Magnetic Resonance (LF-NMR) applications, which offer great potential to the field of biodiesel, have been developed by the Phyto Lipid Biotechnology Lab research...

Template metaprogramming is a popular C ++ technique for implementing compile-time mech-anisms for numerical computing. We demonstrate how expression templates can be used for compile-time symbolic differentiation of algebraic expressions in C ++ computer programs. Given a positive integer N and an algebraic function of multiple variables, the comp...

We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type methods for minimizing smooth functions, even when sea...

For iterative solution of symmetric systems the conjugate gradient method (CG) is commonly used when A is positive definite, while the minimum residual method (MINRES) is typically reserved for indefinite systems. We investigate the sequence of approximate solutions generated by each method and suggest that even if A is positive definite, MINRES ma...

We seek to solve convex optimization problems in composite form: minimize xεℝn f(x) := g(x) + h(x); where g is convex and continuously differentiable and h : ℝn → ℝ is a convex but not necessarily differentiable function whose proximal mapping can be evaluated efficiently. We derive a generalization of Newton-type methods to handle such convex but...

A PDF format file, readable by Adobe Acrobat Reader. (PDF)

Higher-order generalized singular value decomposition (HO GSVD) of global mRNA expression datasets from three different organisms. A Mathematica 5.2 code file, executable by Mathematica 5.2 and readable by Mathematica Player, freely available at http://www.wolfram.com/products/player/. (NB)

HO GSVD of global mRNA expression datasets from three different organisms. A PDF format file, readable by Adobe Acrobat Reader. (PDF)

S. pombe global mRNA expression. A tab-delimited text format file, readable by both Mathematica and Microsoft Excel, reproducing the relative mRNA expression levels of = 3167 S. pombe gene clones at = 17 time points during about two cell-cycle periods from Rustici et al. [15] with the cell-cycle classifications of Rustici et al. or Oliva et al. [16...

Human global mRNA expression. A tab-delimited text format file, readable by both Mathematica and Microsoft Excel, reproducing the relative mRNA expression levels of = 13,068 human genes at = 17 time points during about two cell-cycle periods, including cell-cycle classifications, from Whitfield et al. [18]. (TXT)

S. cerevisiae global mRNA expression. A tab-delimited text format file, readable by both Mathematica and Microsoft Excel, reproducing the relative mRNA expression levels of = 4772 S. cerevisiae open reading frames (ORFs), or genes, at = 17 time points during about two cell-cycle periods, including cell-cycle classifications, from Spellman et al. [1...

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 lim...

We describe a parallel iterative least squares solver named \texttt{LSRN} that is based on random normal projection. \texttt{LSRN} computes the min-length solution to $\min_{x \in \mathbb{R}^n} \|A x - b\|_2$, where $A \in \mathbb{R}^{m \times n}$ with $m \gg n$ or $m \ll n$, and where $A$ may be rank-deficient. Tikhonov regularization may also be...

A procedure for global optimization of PID type controller parameters for SISO plants with model uncertainty is presented. Robustness to the uncertainties is guaranteed by the use of Horowitz bounds,which are used as constraints when low frequency performance is optimized. The basic idea of both the optimization and the parameter tuning is to formu...

We establish that mass conserving single terminal-linkage networks of chemical reactions admit positive steady states regardless of network deficiency and the choice of reaction rate constants. This result holds for closed systems without material exchange across the boundary, as well as for open systems with material exchange at rates that satisfy...

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...

CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimu...

An iterative method LSMR is presented for solving linear systems $Ax=b$ and least-squares problem $\min \norm{Ax-b}_2$, with $A$ being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation $A\T Ax = A\T b$, so that the quantitie...

An automated procedure for optimization of proportional-integral-derivative (PID)-type controller parameters for single-input, single-output (SISO) plants with explicit model uncertainty is presented. Robustness to the uncertainties is guaranteed by the use of Horowitz-Sidi bounds, which are used as constraints when low-frequency performance is opt...

An algorithm is described for solving large-scale nonlinear programs whose objective and constraint functions are smooth and continuously differentiable. The algorithm is of the projected Lagrangian type, involving a sequence of sparse, linearly constrained subproblems whose objective functions include a modified Lagrangian term and a modified quad...

Today's focus on sustainability within industry presents a modeling challenge that maybedealtwithusingdynamicprogrammingoveraninflnitetimehorizon. However,thecurseof dimensionality often results in a large number of states in these models. These large-scale models requirenumerically stable solution methods. The best method for inflnite-horizon dyna...

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...

SNOPT is a general-purpose system for constrained optimization. It minimizes a linear or nonlinear function subject to bounds on the variables and sparse linear or nonlinear constraints. It is suitable for large-scale linear and quadratic programming and for linearly constrained optimization, as well as for general nonlinear programs. SNOPT finds s...

We pay homage to George B. Dantzig by describing a less well-known part of his legacy–his early and dedicated championship of the importance of systems optimization in solving complex real-world problems.

Discussion of ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' [math/0506081]

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...

Numerical tests are used to validate a practical estimatefor the optimal backward errors of linear least squares problems. Thissolves a thirty-year-old problem suggested by Stewart andWilkinson.

This paper describes some of the important issues of numerical analysis in implementing a sequential quadratic programming method for nonlinearly constrained optimization. We consider the separate treatment of linear constraints, design of a specialized quadratic programming algorithm, and control of ill-conditioning. The results of applying the me...

An adaptive rule-based algorithm, SpaseLoc, is described to solve localization prob- lems for ad hoc wireless sensor networks. A large problem is solved as a sequence of very small subproblems, each of which is solved by semidefinite programming relaxation of a geometric opti- mization model. The subproblems are generated according to a set of sens...

SQOPT is a set of Fortran subroutines for minimizing a convex quadratic function subject to both equality and inequality constraints. (SQOPT may also be used for linear programming and for finding a feasible point for a set of linear equalities and inequalities.) The method of SQOPT is of the two-phase, active-set type, and is related to the method...

The lasso penalizes a least squares regression by the sum of the absolute values ("L"₁-norm) of the coefficients. The form of this penalty encourages sparse solutions (with many coefficients equal to 0). We propose the 'fused lasso', a generalization that is designed for problems with features that can be ordered in some meaningful way....

For optimization problems with nonlinear constraints, linearly constrained Lagran- gian (LCL) methods solve a sequence of subproblems of the form "minimize an augmented Lagran- gian function subject to linearized constraints." Such methods converge rapidly near a solution but may not be reliable from arbitrary starting points. Nevertheless, the wel...

An adaptive rule-based algorithm is proposed to solve localization problems for ad hoc wireless sensor networks. For scalability reasons, a large problem is decomposed into a sequence of smaller subproblems. Each subproblem is solved by semidefinite programming relaxation of a geometric optimization model, and then the next subproblem is generated...

Apro cedure foglo79 o797,1j(9N o PID type cotro79; parametersfo SISO plants withmo del uncertainty is presented.Ro,jUQ)U, to the uncertainties is guaranteed by the use oHoU witz bo unds,which are used ascoj;)((, ts whenlo w frequency perfo;U;,1 isoUUN9K,1j The basic ideao boj theo ptimizatio and the parameter tuning isto foU ulate separate criteria...

A new approach to computing the mixed /spl mu/ upper bound (v) is presented. The method exploits the fact that a positive definite matrix V(/spl alpha/) becomes singular when the scalar parameter a decreases to a critical value for a given frequency. A two-level optimization strategy is used with a bisection algorithm branching on the definiteness...

We present a novel implementation of a two-level iterative method for the solution of discrete linear ill-posed problems. The algorithm is algebraically equivalent to the two-level Schur complement conjugate gradient algorithm of M. Hanke and C. R. Vogel [Numer. Math. 83, No. 3, 385–402 (1999; Zbl 0941.65056)], but involves less work per iteration....

We study the use of black-box LDL factorizations for solving the augmented systems (KKT systems) associated with least-squares problems and barrier methods for linear programming (LP). With judicious regularization parameters, stability can be achieved for arbitrary data and arbitrary permutations of the KKT matrix.

For optimization problems with nonlinear constraints, linearly constrained Lagrangian (LCL) methods sequentially minimize a Lagrangian function subject to linearized constraints. These methods converge rapidly near a solution but may not be reliable from arbitrary starting points. The well known example \MINOS\ has proven effective on many large pr...

This document describes the GAMS interface to MINOS which is a general purpose nonlinear programming solver