Stanford University Systems Optimization Laboratory
Overview
Works:  97 works in 105 publications in 1 language and 222 library holdings 

Roles:  Researcher 
Classifications:  TD353, 333.82 
Publication Timeline
.
Most widely held works by
Stanford University
Modeling water supply for the energy sector : final report by
Nathan Buras(
Book
)
1 edition published in 1982 in English and held by 11 WorldCat member libraries worldwide
1 edition published in 1982 in English and held by 11 WorldCat member libraries worldwide
Determining the feasibility of incorporating water resource constraints into energy models : final report by
Nathan Buras(
Book
)
2 editions published in 1979 in English and held by 10 WorldCat member libraries worldwide
2 editions published in 1979 in English and held by 10 WorldCat member libraries worldwide
PILOT1980 energyeconomic model by
George B Dantzig(
Book
)
1 edition published in 1982 in English and held by 10 WorldCat member libraries worldwide
1 edition published in 1982 in English and held by 10 WorldCat member libraries worldwide
MINOS : a largescale nonlinear programming system (for problems with linear constraints) user's guide by
Bruce A Murtagh(
Book
)
4 editions published between 1977 and 1981 in English and held by 5 WorldCat member libraries worldwide
4 editions published between 1977 and 1981 in English and held by 5 WorldCat member libraries worldwide
Stanford PILOT energy/economic model : interim report by
George B Dantzig(
Book
)
1 edition published in 1978 in English and held by 5 WorldCat member libraries worldwide
1 edition published in 1978 in English and held by 5 WorldCat member libraries worldwide
Mathematical decomposition techniques for power system expansion planning : final report(
Book
)
1 edition published in 1988 in English and held by 4 WorldCat member libraries worldwide
1 edition published in 1988 in English and held by 4 WorldCat member libraries worldwide
Stanford Pilot energy/economic model : interim report, May, 1978 by
George B Dantzig(
Book
)
1 edition published in 1978 in English and held by 4 WorldCat member libraries worldwide
1 edition published in 1978 in English and held by 4 WorldCat member libraries worldwide
Planning under uncertainty using parallel computing by
George B Dantzig(
Book
)
1 edition published in 1987 in English and held by 4 WorldCat member libraries worldwide
Abstract: "Industry and government routinely solve deterministic mathematical programs for planning and scheduling purposes, some involving thousands of variables with a linear or nonlinear objective and inequality constraints. The solutions obtained are often ignored because they don't properly hedge against future contingencies. It is relatively easy to reformulate models to include uncertainty. The bottleneck has been (and is) our capability to solve them. The time is now ripe for finding a way to do so. To this end, we describe in this paper how largescale system methods for solving multistaged systems, such as Bender's Decomposition, highspeed sampling or Monte Carlo simulation, and parallel processors can be combined to solve some important planning problems involving uncertainty
1 edition published in 1987 in English and held by 4 WorldCat member libraries worldwide
Abstract: "Industry and government routinely solve deterministic mathematical programs for planning and scheduling purposes, some involving thousands of variables with a linear or nonlinear objective and inequality constraints. The solutions obtained are often ignored because they don't properly hedge against future contingencies. It is relatively easy to reformulate models to include uncertainty. The bottleneck has been (and is) our capability to solve them. The time is now ripe for finding a way to do so. To this end, we describe in this paper how largescale system methods for solving multistaged systems, such as Bender's Decomposition, highspeed sampling or Monte Carlo simulation, and parallel processors can be combined to solve some important planning problems involving uncertainty
Primal barrier methods for linear programming by
Aeneas Marxen(
Book
)
1 edition published in 1989 in English and held by 3 WorldCat member libraries worldwide
Abstract: "The linear program min c[superscript T]x subject to Ax=b, x [greater than or equal to] 0, is solved by the projected Newton barrier method. The method consists of solving a sequence of subproblems of the form [formula] subject to Ax=b. Extensions for upper bounds, free and fixed variables are given. A linear modification is made to the logarithmic barrier function, which results in the solution being bounded in all cases. It also facilitates the provision of a good starting point. The solution of each subproblem involves repeatedly computing a search direction and taking a step along this direction. Ways to find an initial feasible solution, step sizes and convergence criteria are discussed
1 edition published in 1989 in English and held by 3 WorldCat member libraries worldwide
Abstract: "The linear program min c[superscript T]x subject to Ax=b, x [greater than or equal to] 0, is solved by the projected Newton barrier method. The method consists of solving a sequence of subproblems of the form [formula] subject to Ax=b. Extensions for upper bounds, free and fixed variables are given. A linear modification is made to the logarithmic barrier function, which results in the solution being bounded in all cases. It also facilitates the provision of a good starting point. The solution of each subproblem involves repeatedly computing a search direction and taking a step along this direction. Ways to find an initial feasible solution, step sizes and convergence criteria are discussed
Optimal design of pitched tapered laminated wood beams by
M Avriel(
Book
)
2 editions published in 1976 in English and held by 3 WorldCat member libraries worldwide
The optimal design of a pitched tapered laminated woood beam is considered. An engineering formulation is given in which the volume of the beam is minimized. The problem is then reformulated and solved as a generalized geometric (signomial) program. Sample designs are presented. (Author)
2 editions published in 1976 in English and held by 3 WorldCat member libraries worldwide
The optimal design of a pitched tapered laminated woood beam is considered. An engineering formulation is given in which the volume of the beam is minimized. The problem is then reformulated and solved as a generalized geometric (signomial) program. Sample designs are presented. (Author)
Origins of the simplex method by
George B Dantzig(
Book
)
1 edition published in 1987 in English and held by 3 WorldCat member libraries worldwide
Today we know that before 1947 that four isolated papers had been published on special cases of the linear programming problem by Fourier (1824) [5], de la Vallʹee Poussin (1911) [6], Kantorovich (1939) [7] and Hitchcock (1941) [8]. All except Kantorovich's paper proposed as a solution method descent along the outside edges of the polyhedral set which is the way we describe the simplex method today. There is no evidence that these papers had any influence on each other. Evidently the sparked zero interest on the part of other mathematicians and were unknown to me when I first proposed the simplex method. As we shall see the simplex algorithm evolved from a very different geometry, one in which it appeared to be very efficient."
1 edition published in 1987 in English and held by 3 WorldCat member libraries worldwide
Today we know that before 1947 that four isolated papers had been published on special cases of the linear programming problem by Fourier (1824) [5], de la Vallʹee Poussin (1911) [6], Kantorovich (1939) [7] and Hitchcock (1941) [8]. All except Kantorovich's paper proposed as a solution method descent along the outside edges of the polyhedral set which is the way we describe the simplex method today. There is no evidence that these papers had any influence on each other. Evidently the sparked zero interest on the part of other mathematicians and were unknown to me when I first proposed the simplex method. As we shall see the simplex algorithm evolved from a very different geometry, one in which it appeared to be very efficient."
An analysis of an available set of linear programming test problems by Irvin J Lustig(
Book
)
1 edition published in 1987 in English and held by 3 WorldCat member libraries worldwide
Abstract: "A set of linear programming test problems is analyzed with MINOS, Version 5.1. The problems have been run with different options for scaling and partial pricing to illustrate the effects of these options on the performance of the simplex method. The results indicate that the different options can significantly improve or degrade the performance of the simplex method, and that these options must be chosen wisely. For each problem, a picture of the nonzero structure of the matrix A is also presented so that the problems can be classified according to structure."
1 edition published in 1987 in English and held by 3 WorldCat member libraries worldwide
Abstract: "A set of linear programming test problems is analyzed with MINOS, Version 5.1. The problems have been run with different options for scaling and partial pricing to illustrate the effects of these options on the performance of the simplex method. The results indicate that the different options can significantly improve or degrade the performance of the simplex method, and that these options must be chosen wisely. For each problem, a picture of the nonzero structure of the matrix A is also presented so that the problems can be classified according to structure."
On the numerical stability of quasidefinite systems(
)
1 edition published in 1993 in English and held by 0 WorldCat member libraries worldwide
The authors discuss the solution of sparse linear equations Kd = r, where K is a symmetric and specially structured indefinite matrix that often arises in numerical optimization. For such K, the indefinite factorization K = LDL{sup T} is known to exist in exact arithmetic with 1 x 1 pivots and no row or column interchanges. It is shown that the stability of the LDL{sup T} factorization of this matrix is naturally connected with the stability of the LDM{sup T} factorization of a closely related unsymmetric positivedefinite matrix. Conditions are given that allow the stable numerical solution of this system by Gaussian elimination without row and column interchanges
1 edition published in 1993 in English and held by 0 WorldCat member libraries worldwide
The authors discuss the solution of sparse linear equations Kd = r, where K is a symmetric and specially structured indefinite matrix that often arises in numerical optimization. For such K, the indefinite factorization K = LDL{sup T} is known to exist in exact arithmetic with 1 x 1 pivots and no row or column interchanges. It is shown that the stability of the LDL{sup T} factorization of this matrix is naturally connected with the stability of the LDM{sup T} factorization of a closely related unsymmetric positivedefinite matrix. Conditions are given that allow the stable numerical solution of this system by Gaussian elimination without row and column interchanges
A discountedcost continuoustime flexible manufacturing and operator scheduling model solved by deconvexification over time(
)
1 edition published in 1990 in English and held by 0 WorldCat member libraries worldwide
A discountedcost, continuoustime, infinitehorizon version of a flexible manufacturing and operator scheduling model is solved. The solution procedure is to convexify the discrete operatorassignment constraints to obtain a linear program, and then to regain the discreteness and obtain an approximate manufacturing schedule by deconvexification of the solution of the linear program over time. The strong features of the model are the accommodation of linear inequality relations among the manufacturing activities and the discrete manufacturing scheduling, whereas the weak features are intraperiod relaxation of inventory availability constraints, and the absence of inventory costs, setup times, and setup charges
1 edition published in 1990 in English and held by 0 WorldCat member libraries worldwide
A discountedcost, continuoustime, infinitehorizon version of a flexible manufacturing and operator scheduling model is solved. The solution procedure is to convexify the discrete operatorassignment constraints to obtain a linear program, and then to regain the discreteness and obtain an approximate manufacturing schedule by deconvexification of the solution of the linear program over time. The strong features of the model are the accommodation of linear inequality relations among the manufacturing activities and the discrete manufacturing scheduling, whereas the weak features are intraperiod relaxation of inventory availability constraints, and the absence of inventory costs, setup times, and setup charges
Algorithmic advances in stochastic programming(
)
1 edition published in 1993 in English and held by 0 WorldCat member libraries worldwide
Practical planning problems with deterministic forecasts of inherently uncertain parameters often yield unsatisfactory solutions. Stochastic programming formulations allow uncertain parameters to be modeled as random variables with known distributions, but the size of the resulting mathematical programs can be formidable. Decompositionbased algorithms take advantage of special structure and provide an attractive approach to such problems. We consider two classes of decompositionbased stochastic programming algorithms. The first type of algorithm addresses problems with a ''manageable'' number of scenarios. The second class incorporates Monte Carlo sampling within a decomposition algorithm. We develop and empirically study an enhanced Benders decomposition algorithm for solving multistage stochastic linear programs within a prespecified tolerance. The enhancements include warm start basis selection, preliminary cut generation, the multicut procedure, and decision tree traversing strategies. Computational results are presented for a collection of ''realworld'' multistage stochastic hydroelectric scheduling problems. Recently, there has been an increased focus on decompositionbased algorithms that use sampling within the optimization framework. These approaches hold much promise for solving stochastic programs with many scenarios. A critical component of such algorithms is a stopping criterion to ensure the quality of the solution. With this as motivation, we develop a stopping rule theory for algorithms in which bounds on the optimal objective function value are estimated by sampling. Rules are provided for selecting sample sizes and terminating the algorithm under which asymptotic validity of confidence interval statements for the quality of the proposed solution can be verified. Issues associated with the application of this theory to two samplingbased algorithms are considered, and preliminary empirical coverage results are presented
1 edition published in 1993 in English and held by 0 WorldCat member libraries worldwide
Practical planning problems with deterministic forecasts of inherently uncertain parameters often yield unsatisfactory solutions. Stochastic programming formulations allow uncertain parameters to be modeled as random variables with known distributions, but the size of the resulting mathematical programs can be formidable. Decompositionbased algorithms take advantage of special structure and provide an attractive approach to such problems. We consider two classes of decompositionbased stochastic programming algorithms. The first type of algorithm addresses problems with a ''manageable'' number of scenarios. The second class incorporates Monte Carlo sampling within a decomposition algorithm. We develop and empirically study an enhanced Benders decomposition algorithm for solving multistage stochastic linear programs within a prespecified tolerance. The enhancements include warm start basis selection, preliminary cut generation, the multicut procedure, and decision tree traversing strategies. Computational results are presented for a collection of ''realworld'' multistage stochastic hydroelectric scheduling problems. Recently, there has been an increased focus on decompositionbased algorithms that use sampling within the optimization framework. These approaches hold much promise for solving stochastic programs with many scenarios. A critical component of such algorithms is a stopping criterion to ensure the quality of the solution. With this as motivation, we develop a stopping rule theory for algorithms in which bounds on the optimal objective function value are estimated by sampling. Rules are provided for selecting sample sizes and terminating the algorithm under which asymptotic validity of confidence interval statements for the quality of the proposed solution can be verified. Issues associated with the application of this theory to two samplingbased algorithms are considered, and preliminary empirical coverage results are presented
The simplex algorithm with a new primal and dual pivot rule(
)
1 edition published in 1993 in English and held by 0 WorldCat member libraries worldwide
We present a simplextype algorithm for linear programming that works with primalfeasible and dualfeasible points associated with bases that differ by only one column
1 edition published in 1993 in English and held by 0 WorldCat member libraries worldwide
We present a simplextype algorithm for linear programming that works with primalfeasible and dualfeasible points associated with bases that differ by only one column
A strictly improving linear programming alorithm based on a series of Phase 1 problems(
)
1 edition published in 1992 in English and held by 0 WorldCat member libraries worldwide
When used on degenerate problems, the simplex method often takes a number of degenerate steps at a particular vertex before moving to the next. In theory (although rarely in practice), the simplex method can actually cycle at such a degenerate point. Instead of trying to modify the simplex method to avoid degenerate steps, we have developed a new linear programming algorithm that is completely impervious to degeneracy. This new method solves the Phase II problem of finding an optimal solution by solving a series of Phase I feasibility problems. Strict improvement is attained at each iteration in the Phase I algorithm, and the Phase II sequence of feasibility problems has linear convergence in the number of Phase I problems. When tested on the 30 smallest NETLIB linear programming test problems, the computational results for the new Phase II algorithm were over 15% faster than the simplex method; on some problems, it was almost two times faster, and on one problem it was four times faster
1 edition published in 1992 in English and held by 0 WorldCat member libraries worldwide
When used on degenerate problems, the simplex method often takes a number of degenerate steps at a particular vertex before moving to the next. In theory (although rarely in practice), the simplex method can actually cycle at such a degenerate point. Instead of trying to modify the simplex method to avoid degenerate steps, we have developed a new linear programming algorithm that is completely impervious to degeneracy. This new method solves the Phase II problem of finding an optimal solution by solving a series of Phase I feasibility problems. Strict improvement is attained at each iteration in the Phase I algorithm, and the Phase II sequence of feasibility problems has linear convergence in the number of Phase I problems. When tested on the 30 smallest NETLIB linear programming test problems, the computational results for the new Phase II algorithm were over 15% faster than the simplex method; on some problems, it was almost two times faster, and on one problem it was four times faster
Decomposition and (importance) sampling techniques for multistage stochastic linear programs by Stanford University(
)
1 edition published in 1993 in English and held by 0 WorldCat member libraries worldwide
The difficulty of solving largescale multistage stochastic linear programs arises from the sheer number of scenarios associated with numerous stochastic parameters. The number of scenarios grows exponentially with the number of stages and problems get easily out of hand even for very moderate numbers of stochastic parameters per stage. Our method combines dual (Benders) decomposition with Monte Carlo sampling techniques. We employ importance sampling to efficiently obtain accurate estimates of both expected future costs and gradients and righthand sides of cuts. The method enables us to solve practical largescale problems with many stages and numerous stochastic parameters per stage. We discuss the theory of sharing and adjusting cuts between different scenarios in a stage. We derive probabilistic lower and upper bounds, where we use importance path sampling for the upper bound estimation. Initial numerical results turned out to be promising
1 edition published in 1993 in English and held by 0 WorldCat member libraries worldwide
The difficulty of solving largescale multistage stochastic linear programs arises from the sheer number of scenarios associated with numerous stochastic parameters. The number of scenarios grows exponentially with the number of stages and problems get easily out of hand even for very moderate numbers of stochastic parameters per stage. Our method combines dual (Benders) decomposition with Monte Carlo sampling techniques. We employ importance sampling to efficiently obtain accurate estimates of both expected future costs and gradients and righthand sides of cuts. The method enables us to solve practical largescale problems with many stages and numerous stochastic parameters per stage. We discuss the theory of sharing and adjusting cuts between different scenarios in a stage. We derive probabilistic lower and upper bounds, where we use importance path sampling for the upper bound estimation. Initial numerical results turned out to be promising
Two characterizations of sufficient matrices(
)
1 edition published in 1990 in English and held by 0 WorldCat member libraries worldwide
Two characterizations are given for the class of sufficient matrices defined by Cottle, Pang, and Venkateswaran. The first is a direct translation of the definition into linear programming terms. The second can be thought of as a generalization of a theorem of T.D. Parsons on Pmatrices. 19 refs
1 edition published in 1990 in English and held by 0 WorldCat member libraries worldwide
Two characterizations are given for the class of sufficient matrices defined by Cottle, Pang, and Venkateswaran. The first is a direct translation of the definition into linear programming terms. The second can be thought of as a generalization of a theorem of T.D. Parsons on Pmatrices. 19 refs
Planning under uncertainty solving largescale stochastic linear programs(
)
2 editions published in 1992 in English and held by 0 WorldCat member libraries worldwide
For many practical problems, solutions obtained from deterministic models are unsatisfactory because they fail to hedge against certain contingencies that may occur in the future. Stochastic models address this shortcoming, but up to recently seemed to be intractable due to their size. Recent advances both in solution algorithms and in computer technology now allow us to solve important and general classes of practical stochastic problems. We show how largescale stochastic linear programs can be efficiently solved by combining classical decomposition and Monte Carlo (importance) sampling techniques. We discuss the methodology for solving twostage stochastic linear programs with recourse, present numerical results of large problems with numerous stochastic parameters, show how to efficiently implement the methodology on a parallel multicomputer and derive the theory for solving a general class of multistage problems with dependency of the stochastic parameters within a stage and between different stages
2 editions published in 1992 in English and held by 0 WorldCat member libraries worldwide
For many practical problems, solutions obtained from deterministic models are unsatisfactory because they fail to hedge against certain contingencies that may occur in the future. Stochastic models address this shortcoming, but up to recently seemed to be intractable due to their size. Recent advances both in solution algorithms and in computer technology now allow us to solve important and general classes of practical stochastic problems. We show how largescale stochastic linear programs can be efficiently solved by combining classical decomposition and Monte Carlo (importance) sampling techniques. We discuss the methodology for solving twostage stochastic linear programs with recourse, present numerical results of large problems with numerous stochastic parameters, show how to efficiently implement the methodology on a parallel multicomputer and derive the theory for solving a general class of multistage problems with dependency of the stochastic parameters within a stage and between different stages
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Related Identities
 Electric Power Research Institute Sponsor
 United States Department of Energy Oakland Operations Office Researcher
 United States Department of Energy Office of Scientific and Technical Information Distributor
 Dantzig, George B. (George Bernard) 19142005 Author Editor
 National Science Foundation (U.S.) Sponsor
 Buras, Nathan Author
 Gill, Philip E. Author
 Connolly, Thomas J. 1923
 United States Department of Defense Sponsor
 Saunders, Michael A. Author
Associated Subjects
Decomposition (Mathematics) Energy policyEconomic aspects Industrial water supplyMathematical models Laminated wood Linear programming Linear programmingComputer programs Nonlinear programming Nonlinear programmingData processing Parallel processing (Electronic computers) Power resourcesMathematical models Production scheduling Programming (Mathematics) Simplexes (Mathematics) United States Water resources developmentMathematical models WatersupplyMathematical models
Alternative Names
S.O.L.
SOL
Stanford University Department of Operations Research Systems Optimization Laboratory
Stanford University. Dept. of Operations Research. Systems Optimization Laboratory
Stanford University Systems Optimization Laboratory
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