M. Saunders

Stanford University, Management Science & Engineering, Faculty Member

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Papers by M. Saunders

A Globally Convergent LCL Method for Nonlinear Optimization

arXiv (Cornell University), Jan 10, 2003

For optimization problems with nonlinear constraints, linearly constrained Lagrangian (LCL) metho... more 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 problems. Its success motivates us to propose a globally convergent variant. Our stabilized LCL method possesses two important properties: the subproblems are always feasible, and they may be solved inexactly. These features are present in MINOS only as heuristics. The new algorithm has been implemented in Matlab, with the option to use either the MINOS or SNOPT Fortran codes to solve the linearly constrained subproblems. Only first derivatives are required. We present numerical results on a nonlinear subset of the COPS, CUTE, and HS testproblem sets, which include many large examples. The results demonstrate the robustness and efficiency of the stabilized LCL procedure.

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Some issues in implementing a sequential quadratic programming algorithm

ACM Signum Newsletter, Apr 1, 1985

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Sequential Quadratic Programming Methods for Nonlinear Programming

Springer eBooks, 1984

Sequential quadratic programming (SQP) methods are known to be efficient for solving a series of ... more Sequential quadratic programming (SQP) methods are known to be efficient for solving a series of related nonlinear optimization problems because of desirable hot and warm start properties-a solution for one problem is a good estimate of the solution of the next. However, standard SQP solvers contain elements to enforce global convergence that can interfere with the potential to take advantage of these theoretical local properties in full. We present two new predictor-corrector procedures for solving a nonlinear program given a sufficiently accurate estimate of the solution of a similar problem. The procedures attempt to trace a homotopy path between solutions of the two problems, staying within the local domain of convergence for the series of problems generated. We provide theoretical convergence and tracking results, as well as some numerical results demonstrating the robustness and performance of the methods.

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George B. Dantzig and systems optimization

Discrete Optimization, May 1, 2008

We pay homage to George B. Dantzig by describing a less well known part of his legacy-his early a... more

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The application of nonlinear programming and collocation to optimal aeroassisted orbital transfers

30th Aerospace Sciences Meeting and Exhibit, Jan 6, 1992

ABSTRACT Sequential quadratic programming (SQP) and collocation of the differential equations of ... more ABSTRACT Sequential quadratic programming (SQP) and collocation of the differential equations of motion were applied to optimal aeroassisted orbital transfers. The Optimal Trajectory by Implicit Simulation (OTIS) computer program codes with updated nonlinear programming code (NZSOL) were used as a testbed for the SQP nonlinear programming (NLP) algorithms. The state-of-the-art sparse SQP method is considered to be effective for solving large problems with a sparse matrix. Sparse optimizers are characterized in terms of memory requirements and computational efficiency. For the OTIS problems, less than 10 percent of the Jacobian matrix elements are nonzero. The SQP method encompasses two phases: finding an initial feasible point by minimizing the sum of infeasibilities and minimizing the quadratic objective function within the feasible region. The orbital transfer problem under consideration involves the transfer from a high energy orbit to a low energy orbit.

Methods for computing and modifying the $LDV$ factors of a matrix

Mathematics of Computation, 1975

Methods are given for computing the LDV factorization of a matrix B and

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User's Guide for QPSOL (Version 3.2): A Fortran Package for Quadratic Programming. Revision

ABSTRACT

A Numerical Investigation of Ellipsoid Algorithms for Large-Scale Linear Programming

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A practical anti-cycling procedure for linearly constrained optimization

Mathematical Programming, Aug 1, 1989

A procedure is described for preventing cycling in active-set methods for linearly constrained op... more A procedure is described for preventing cycling in active-set methods for linearly constrained optimization, including the simplex method. The key ideas are a limited acceptance ofinfeasibilities in all variables, and maintenance of a "working" feasibility tolerance that increases over a long sequence of iterations. The additional work per iteration is nominal, and "stalling" cannot occur with exact arithmetic. The method appears to be reliable, based on computational results for the first 53 linear programming problems in the Netlib set.

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Methods for Large-Scale Nonlinear Optimization

Abstract : The application of optimization to electrical power technology often requires the nume... more Abstract : The application of optimization to electrical power technology often requires the numerical solution of systems that are very large and possibly non-differentiable. A brief survey of the state of the art of numerical optimization is presented in which those methods that are directly applicable to power system problems will be highlighted. The areas of current research that are most likely to yield direct benefit to practical computation are identified. The paper concludes with a survey of available software. (Author)

User’s Guide for SNOPT Version 7, A Fortran Package for Large-Scale Nonlinear Programming∗

SNOPT is a general-purpose system for constrained optimization. It minimizes a linear or nonlinea... more 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 solutions that are locally optimal, and ideally any nonlinear functions should be smooth and users should provide gradients. It is often more widely useful. For example, local optima are often global solutions, and discontinuities in the function gradients can often be tolerated if they are not too close to an optimum. Unknown gradients are estimated by finite differences. SNOPT uses a sequential quadratic programming (SQP) algorithm. Search directions are obtained from QP subproblems that minimize a quadratic model of the Lagrangian function subject to linearized constraints. An augmented Lagrangian merit function is reduced along each search direction to ensure convergence from any starting point. On large problems, SNOPT is most efficient if only some of the variables enter nonlinearly, or if many constraints (including simple bounds) are active. SNOPT requires relatively few evaluations of the problem functions. Hence it is especially effective if the objective or constraint functions (and their gradients) are expensive to evaluate. The source code for SNOPT is suitable for any machine with a Fortran compiler or the f2c translator. SNOPT may be called from a driver program (typically in Fortran, C or MATLAB). It can also be used as a stand-alone package, reading data in the MPS format used by commercial mathematical programming systems.

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Gams/Snopt: An SQP Algorithm for Large-Scale Constrained Optimization

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Primal—dual methods for linear programming

Mathematical Programming, Oct 1, 1995

Some theoretical properties of an augmented lagrangian merit function

Sequential quadratic programming (SQP) methods for nonlinearly constrained optimization typically... more Sequential quadratic programming (SQP) methods for nonlinearly constrained optimization typically use a merit function to enforce convergence from an arbitrary starting point. We define a smooth augmented Lagrangian merit function in which the Lagrange multiplier estimate is treated as a separate variable, and inequality constraints are handled by means of non-negative slack variables that are included in the linesearch. Global convergence is proved for an SQP algorithm that uses this merit function. We also prove that steps of unity are accepted in a neighborhood of the solution when this merit function is used in a suitable superlinearly convergent algorithm. Finally, some numerical results are presented to illustrate the performance of the associated SQP method.

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User’s Guide for Snopt 5.3: A Fortran Package for Large-Scale Nonlinear Programming

SNOPT is a set of Fortran subroutines for minimizing a smooth function subject to constraints, wh... more SNOPT is a set of Fortran subroutines for minimizing a smooth function subject to constraints, which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints. SNOPT is a general-purpose optimizer, designed to find locally optimal solutions for models involving smooth nonlinear functions. They are often more widely useful. (For example, local optima are often global solutions, and discontinuities in the function gradients can often be tolerated if they are not too close to an optimum.) Ideally, users should provide gradients. Unknown components are estimated by finite differences. SNOPT incorporates a sequential quadratic programming (SQP) method that obtains search directions from a sequence of quadratic programming subproblems. Each QP subproblem minimizes a quadratic model of a certain Lagrangian function subject to a linearization of the constraints. An augmented Lagrangian merit function is reduced along each search direction to ensure convergence from any starting point. SNOPT is most efficient if only some of the variables enter nonlinearly, or if the number of active constraints (including simple bounds) is nearly as large as the number of variables. SNOPT requires relatively few evaluations of the problem functions. Hence it is especially effective if the objective or constraint functions are expensive to evaluate. The source code for SNOPT is suitable for any machine with a reasonable amount of memory and a Fortran compiler. SNOPT may be called from a driver program (typically in Fortran, C or MATLAB).

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Considerations of numerical analysis in a sequential quadratic programming method

Lecture Notes in Mathematics, 1986

ABSTRACT This paper describes some of the important issues of numerical analysis in implementing ... more ABSTRACT 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 method to two specific examples are analyzed in detail.

QP-Based Methods for Large-Scale Nonlinearly Constrained Optimization

Abstract: Several methods for nonlinearly constrained optimization have been suggested in recent ... more

User's Guide for SNOPT Version 7.5: Software for Large-Scale Nonlinear Programming

SNOPT is a general-purpose system for constrained optimization. It minimizes a linear or nonlinea... more 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 solutions that are locally optimal, and ideally any nonlinear functions should be smooth and users should provide gradients. It is often more widely useful. For example, local optima are often global solutions, and discontinuities in the function gradients can often be tolerated if they are not too close to an optimum. Unknown gradients are estimated by finite differences. SNOPT uses a sequential quadratic programming (SQP) algorithm. Search directions are obtained from QP subproblems that minimize a quadratic model of the Lagrangian function subject to linearized constraints. An augmented Lagrangian merit function is reduced along each search direction to ensure convergence from any starting point. On large problems, SNOPT is most efficient if only some of the variables enter nonlinearly, or there are relatively few degrees of freedom at a solution (i.e., many constraints are active). SNOPT requires relatively few evaluations of the problem functions. Hence it is especially effective if the objective or constraint functions (and their gradients) are expensive to evaluate. The source code is re-entrant and suitable for any machine with a Fortran compiler. SNOPT may be called from a driver program in Fortran, Matlab, or C/C++ with the new interface based on the Fortran 2003 standard on Fortran-C interoperability. A f2c translation of SNOPT to the C language is still provided, although this feature will be discontinued in the future (users should migrate to the new C/C++ interface).

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A variational principle for computing nonequilibrium fluxes and potentials in genome-scale biochemical networks

Journal of Theoretical Biology, 2012

We derive a convex optimization problem on a steady-state nonequilibrium network of biochemical r... more 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.

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Subspace Preconditioned LSQR for Discrete Ill-Posed Problems

BIT Numerical Mathematics, 2003

We present a novel implementation of a two-level iterative method for the solution of discrete li... more 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 CG algorithm of Hanke and Vogel, but involves less work per iteration. We review the algorithm, discuss our implementation, and show promising results from numerical experiments that give insight into the proper use of the algorithm.

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M. Saunders

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