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Snopt: An SQP Algorithm For LargeScale Constrained Optimization
, 1997
"... Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first deriv ..."
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Cited by 328 (18 self)
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Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available, and that the constraint gradients are sparse.
User's Guide For SNOPT 5.3: A Fortran Package For LargeScale Nonlinear Programming
, 1999
"... SNOPT is a generalpurpose system for solving optimization problems involving many variables and constraints. It minimizes a linear or nonlinear function subject to bounds on the variables and sparse linear or nonlinear constraints. It is suitable for largescale linear and quadratic programming ..."
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Cited by 76 (1 self)
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SNOPT is a generalpurpose system for solving optimization problems involving many variables and constraints. It minimizes a linear or nonlinear function subject to bounds on the variables and sparse linear or nonlinear constraints. It is suitable for largescale 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 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 (and their gradients) are expensive to evaluate. The source code for SNOPT is suitable for any machine with a Fortran compiler. SNOPT may be called from a driver program (typically in Fortran, C or MATLAB). SNOPT can also be used as a standalone package, reading data in the MPS ...
Solving reduced KKT systems in barrier methods for linear and quadratic programming
, 1991
"... In barrier methods for constrained optimization, the main work lies in solving large linear systems Kp = r, where K is symmetric and indefinite. For linear programs, these KKT systems are usually reduced to smaller positivedefinite systems AH −1 A T q = s, where H is a large principal submatrix of ..."
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Cited by 22 (7 self)
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In barrier methods for constrained optimization, the main work lies in solving large linear systems Kp = r, where K is symmetric and indefinite. For linear programs, these KKT systems are usually reduced to smaller positivedefinite systems AH −1 A T q = s, where H is a large principal submatrix of K. These systems can be solved more efficiently, but AH −1 A T is typically more illconditioned than K. In order to improve the numerical properties of barrier implementations, we discuss the use of “reduced KKT systems”, whose dimension and condition lie somewhere in between those of K and AH −1 A T. The approach applies to linear programs and to positive semidefinite quadratic programs whose Hessian H is at least partially diagonal. We have implemented reduced KKT systems in a primaldual algorithm for linear programming, based on the sparse indefinite solver MA27 from the Harwell Subroutine Library. Some features of the algorithm are presented, along with results on the netlib LP test set.
User’s Guide for SNOPT Version 7: Software for LargeScale Nonlinear Programming
"... SNOPT is a generalpurpose 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 largescale linear and quadratic programming and for linearly constrained optimization, as wel ..."
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Cited by 13 (0 self)
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SNOPT is a generalpurpose 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 largescale 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.
GAMS/SNOPT: An SQP Algorithm For LargeScale Constrained Optimization
, 2000
"... Introduction This section describes the GAMS interface to the generalpurpose NLP solver SNOPT, (Sparse Nonlinear Optimizer) which implements a sequential quadratic programming (SQP) method for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints ..."
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Cited by 3 (0 self)
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Introduction This section describes the GAMS interface to the generalpurpose NLP solver SNOPT, (Sparse Nonlinear Optimizer) which implements a sequential quadratic programming (SQP) method for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. The optimization problem is assumed to be stated in the form NP minimize x or maximize x f(x) subject to F (x) # b 1 Gx # b 2 l # x # u, (1) where x # # n , f(x) is a linear or nonlinear smooth objective function, l and u are constant lower and upper bounds, F (x) is a set of nonlinear constraint functions, G is a sparse ma
Gams/minos: A Solver For LargeScale Nonlinear Optimization Problems
, 2002
"... This document describes the GAMS interface to MINOS which is a general purpose nonlinear programming solver ..."
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Cited by 2 (0 self)
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This document describes the GAMS interface to MINOS which is a general purpose nonlinear programming solver
User’s Guide for SQOPT Version 7: Software for LargeScale Linear and Quadratic Programming ∗
, 2008
"... SQOPT is a software package 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. SQOPT uses a twophase, activeset, reduceHessi ..."
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Cited by 1 (0 self)
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SQOPT is a software package 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. SQOPT uses a twophase, activeset, reduceHessian method. It is most efficient on problems with relatively few degrees of freedom (for example, if only some of the variables appear in the quadratic term, or the number of active constraints and bounds is nearly as large as the number of variables). However, unlike previous versions of SQOPT, there is no limit on the number of degrees of freedom. SQOPT is primarily intended for large linear and quadratic problems with sparse constraint matrices. A quadratic term 1 2 xTHx in the objective function is represented by a user subroutine that returns the product Hx for a given vector x. SQOPT uses stable numerical methods throughout and includes a reliable basis package (for maintaining sparse LU factors of the basis matrix), a practical antidegeneracy procedure, scaling, and elastic bounds on any number of constraints and variables. SQOPT is part of the SNOPT package for largescale nonlinearly constrained optimization. The source code is reentrant and suitable for any machine with a Fortran compiler (or the f2c translator and a C compiler). SQOPT may be called from a driver program in Fortran, C, or Matlab. It can also be used as a standalone package, reading data in the MPS format used by commercial mathematical programming systems.
A General Pricing Scheme for the Simplex Method
, 2001
"... Pricing is a term in the simplex method for linear programming used to refer to the step of checking the reduced costs of nonbasic variables. If they are all of the `right sign' the current basis (and solution) is optimal, if not, this procedure selects a candidate vector that looks profitable for i ..."
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Cited by 1 (0 self)
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Pricing is a term in the simplex method for linear programming used to refer to the step of checking the reduced costs of nonbasic variables. If they are all of the `right sign' the current basis (and solution) is optimal, if not, this procedure selects a candidate vector that looks profitable for inclusion in the basis. While theoretically the choice of any profitable vector will lead to a finite termination (provided degeneracy is handled properly) but the number of iterations until termination depends very heavily on the actual choice (which is defined by the selection rule applied). Pricing has long been an area of heuristics to help make better selection. As a result, many different and sophisticated pricing strategies have been developed, implemented and tested. So far none of them is known to be dominating all others in all cases. Therefore, advanced simplex solvers need to be equipped with many strategies so that the most suitable one can be activated for each individual problem instance. In this paper we present a general pricing scheme. It creates a large flexibility in pricing. It is controlled by three parameters. With different settings of the parameters many of the known strategies can be reproduced as special cases. At the same time, the framework makes it possible to define new strategies or variants of them. The scheme is equally applicable to general and network simplex algorithms.
Gams/minos
"... Contents: ..................................................................................................................................... 1. INTRODUCTION.............................................................................................. 2 2. HOW TO RUN A MODEL WITH GAMS/MINOS..... ..."
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Contents: ..................................................................................................................................... 1. INTRODUCTION.............................................................................................. 2 2. HOW TO RUN A MODEL WITH GAMS/MINOS.......................................... 2 3. OVERVIEW OF GAMS/MINOS ...................................................................... 2 3.1. Linear programming .................................................................................... 3 3.2. Problems with a Nonlinear Objective .......................................................... 4 3.3. Problems with Nonlinear Constraints .......................................................... 6 4. GAMS OPTIONS .............................................................................................. 7 4.1. Options specified through the option statement........................................... 7 4.2. Options specified throu