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Title: A build-up interior method for linear programming: Affine scaling form

Abstract

We propose a build-up interior method for solving an m equation n variable linear program which has the same convergence properties as their well known analogues in dual affine and projective forms but requires less computational effort. The algorithm has three forms, an affine scaling form, a projective scaling form, and an exact form. In this paper, we present the first of these. It differs from Dikin's algorithm of dual affine form in that the ellipsoid chosen to generate the improving direction {bar {Delta}} in dual space is constructed from only a subset of the dual constraints. At the start of each major iteration t, we are given an interior iterate y{sup t}. A selection of m dual constraints is being made using an order-columns'' rule as to which constraints show the most promise'' of being tight in the optimal dual solution. An ellipsoid centered at y{sup t} is then inscribed in convex region defined by these promising constraints and an improving direction {bar {Delta}} computed that points to the optical point y{sup t} + {bar {Delta}} on the ellipsoid boundary. Minor cycling within a major iteration is then started. During a minor cycle, the constraints selected to define themore » ellipsoid centered at y{sup t} is built up to include the constraint that first blocks feasible movement from y{sup t} to y{sup t} + {bar {Delta}}. If one blocks, it is used to augment the set of promising constraints and the ellipsoid is revised; the improving direction {bar {Delta}} is recomputed by means of a rank-one update, and the minor cycle repeated until none blocks movement from y{sup t} to y{sup t} + {bar {Delta}}. When none blocks, the minor cycling ends. y{sup t+1} = y{sup t} + {bar {Delta}} initiates the next major iteration. Major iterations stop when an optimum solution is reached. We prove this will occur in a finite number of iterations.« less

Authors:
;
Publication Date:
Research Org.:
Stanford Univ., CA (USA). Systems Optimization Lab.
Sponsoring Org.:
DOE/ER; National Science Foundation (NSF)
OSTI Identifier:
6937814
Report Number(s):
SOL-90-4
ON: DE90010878; CNN: DDM-8814253; DMS-8913089; N00014-89-J-1659
DOE Contract Number:  
FG03-87ER25028
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; LINEAR PROGRAMMING; ALGORITHMS; CONVERGENCE; ITERATIVE METHODS; MATHEMATICAL LOGIC; PROGRAMMING; 990200* - Mathematics & Computers

Citation Formats

Dantzig, G B, and Yinyu, Ye. A build-up interior method for linear programming: Affine scaling form. United States: N. p., 1990. Web. doi:10.2172/6937814.
Dantzig, G B, & Yinyu, Ye. A build-up interior method for linear programming: Affine scaling form. United States. https://doi.org/10.2172/6937814
Dantzig, G B, and Yinyu, Ye. 1990. "A build-up interior method for linear programming: Affine scaling form". United States. https://doi.org/10.2172/6937814. https://www.osti.gov/servlets/purl/6937814.
@article{osti_6937814,
title = {A build-up interior method for linear programming: Affine scaling form},
author = {Dantzig, G B and Yinyu, Ye},
abstractNote = {We propose a build-up interior method for solving an m equation n variable linear program which has the same convergence properties as their well known analogues in dual affine and projective forms but requires less computational effort. The algorithm has three forms, an affine scaling form, a projective scaling form, and an exact form. In this paper, we present the first of these. It differs from Dikin's algorithm of dual affine form in that the ellipsoid chosen to generate the improving direction {bar {Delta}} in dual space is constructed from only a subset of the dual constraints. At the start of each major iteration t, we are given an interior iterate y{sup t}. A selection of m dual constraints is being made using an order-columns'' rule as to which constraints show the most promise'' of being tight in the optimal dual solution. An ellipsoid centered at y{sup t} is then inscribed in convex region defined by these promising constraints and an improving direction {bar {Delta}} computed that points to the optical point y{sup t} + {bar {Delta}} on the ellipsoid boundary. Minor cycling within a major iteration is then started. During a minor cycle, the constraints selected to define the ellipsoid centered at y{sup t} is built up to include the constraint that first blocks feasible movement from y{sup t} to y{sup t} + {bar {Delta}}. If one blocks, it is used to augment the set of promising constraints and the ellipsoid is revised; the improving direction {bar {Delta}} is recomputed by means of a rank-one update, and the minor cycle repeated until none blocks movement from y{sup t} to y{sup t} + {bar {Delta}}. When none blocks, the minor cycling ends. y{sup t+1} = y{sup t} + {bar {Delta}} initiates the next major iteration. Major iterations stop when an optimum solution is reached. We prove this will occur in a finite number of iterations.},
doi = {10.2172/6937814},
url = {https://www.osti.gov/biblio/6937814}, journal = {},
number = ,
volume = ,
place = {United States},
year = {1990},
month = {2}
}