On Stability In Linear Programming

다국어 초록 (Multilingual Abstract)

We shall be concerned with the pair of dual linear programming problem given by
Min<c,x>
Ax-bεP, xεP, (P)
And max<u,b>
c-uAεP^*, uεQ, (D)
where P and D are nonempty convex cones in R^n and R^m respectively, and where the asterisk denotes the dual cone P^*={zεR^n｜ <z, x>≥0 for each xεP}. It is clear that any pair of dual linear programming problems can be conditions must be placed on (P) and (D). The purpose of the paper is to establish what conditions must be placed on (P) and on (D) in order that for all small but otherwise arbitrary perturbations in the data A, b, c, the problem (P) and (D) will remain solvable,
We now state our principal result.
The following are equivalent:
(a) The constraints of (P) and of (D) are regular.
(b) The sets of optimal solutions of (P) and of (D) are nonempty and bounded.