Geodesic
Matrix correction of a dual pair of improper linear programming problems with a block structure
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 1, pp. 80-89

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The following problem is considered: how to modify the coefficient matrix of a dual pair of improper linear programs with a block structure so as to make these problems proper and minimize the sum of the squares of the Euclidean norms of the blocks in the correction matrix? Two variants of this problem are examined: (1) all the blocks in the coefficient matrix are modified, and (2) the upper block, which constraints all the primal variables, is left unchanged. Methods are presented for reducing these problems to minimizing quadratic fractional functions subject to linear equality and inequality constraints. The latter problem allows the use of conventional methods for constrained minimization. A numerical example is given. 
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     author = {V. I. Erokhin and A. S. Krasnikov},
     title = {Matrix correction of a~dual pair of improper linear programming problems with a~block structure},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
     volume = {48},
     number = {1},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_1_a5/}
}
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V. I. Erokhin; A. S. Krasnikov. Matrix correction of a dual pair of improper linear programming problems with a block structure. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 1, pp. 80-89. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_1_a5/