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Title: A strictly improving Phase 1 algorithm using least-squares subproblems

Abstract

Although the simplex method`s performance in solving linear programming problems is usually quite good, it does not guarantee strict improvement at each iteration on degenerate problems. Instead of trying to recognize and avoid degenerate steps in the simplex method, we have developed a new Phase I algorithm that is completely impervious to degeneracy, with strict improvement attained at each iteration. It is also noted that the new Phase I algorithm is closely related to a number of existing algorithms. When tested on the 30 smallest NETLIB linear programming test problems, the computational results for the new Phase I algorithm were almost 3.5 times faster than the simplex method; on some problems, it was over 10 times faster.

Authors:
; ;
Publication Date:
Research Org.:
Stanford Univ., CA (United States). Systems Optimization Lab.
Sponsoring Org.:
USDOE, Washington, DC (United States)
OSTI Identifier:
10153254
Report Number(s):
SOL-92-1
ON: DE92015904; CNN: Grant ECS-8906260; Grant DMS-8913089; N00014-89-J-1659
DOE Contract Number:
FG03-92ER25116
Resource Type:
Technical Report
Resource Relation:
Other Information: PBD: Apr 1992
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; LINEAR PROGRAMMING; ALGORITHMS; ITERATIVE METHODS; LEAST SQUARE FIT; CONVERGENCE; PERFORMANCE; 990200; MATHEMATICS AND COMPUTERS

Citation Formats

Leichner, S.A., Dantzig, G.B., and Davis, J.W. A strictly improving Phase 1 algorithm using least-squares subproblems. United States: N. p., 1992. Web. doi:10.2172/10153254.
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Leichner, S.A., Dantzig, G.B., & Davis, J.W. A strictly improving Phase 1 algorithm using least-squares subproblems. United States. doi:10.2172/10153254.
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Leichner, S.A., Dantzig, G.B., and Davis, J.W. Wed . "A strictly improving Phase 1 algorithm using least-squares subproblems". United States. doi:10.2172/10153254. https://www.osti.gov/servlets/purl/10153254.
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@article{osti_10153254,
title = {A strictly improving Phase 1 algorithm using least-squares subproblems},
author = {Leichner, S.A. and Dantzig, G.B. and Davis, J.W.},
abstractNote = {Although the simplex method`s performance in solving linear programming problems is usually quite good, it does not guarantee strict improvement at each iteration on degenerate problems. Instead of trying to recognize and avoid degenerate steps in the simplex method, we have developed a new Phase I algorithm that is completely impervious to degeneracy, with strict improvement attained at each iteration. It is also noted that the new Phase I algorithm is closely related to a number of existing algorithms. When tested on the 30 smallest NETLIB linear programming test problems, the computational results for the new Phase I algorithm were almost 3.5 times faster than the simplex method; on some problems, it was over 10 times faster.},
doi = {10.2172/10153254},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Wed Apr 01 00:00:00 EST 1992},
month = {Wed Apr 01 00:00:00 EST 1992}
}
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Technical Report:
View Technical Report (2.21 MB)
DOI:  10.2172/10153254
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  • ; ;
    Although the simplex method's performance in solving linear programming problems is usually quite good, it does not guarantee strict improvement at each iteration on degenerate problems. Instead of trying to recognize and avoid degenerate steps in the simplex method, we have developed a new Phase I algorithm that is completely impervious to degeneracy, with strict improvement attained at each iteration. It is also noted that the new Phase I algorithm is closely related to a number of existing algorithms. When tested on the 30 smallest NETLIB linear programming test problems, the computational results for the new Phase I algorithm weremore » almost 3.5 times faster than the simplex method; on some problems, it was over 10 times faster.« less

  • ; ;
    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 Imore » 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.« less

  • ; ;
    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 Imore » 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.« less

  • No abstract prepared.

  • ;
    The least squares method of rock noise source location presented is shown to be more accurate and reliable than standard direct solution methods. This method is particularly effective in improving solutions from rock noise locations outside of geophone array boundaries. A least squares solution for a medium in which the seismic velocity is the same in all directions is also presented. In this special case, one can solve for both the source coordinates and the velocity.