A Characterization of Stability in Linear Programming

We prove that a necessary and sufficient condition for the primal and dual solution sets of a solvable, finite-dimensional linear programming problem to be stable under small but arbitrary perturbations in the data of the problem is that both of these sets be bounded. The distance from any pair of solutions of the perturbed problem to the solution sets of the original problem is then bounded by a constant multiple of the norm of the perturbations. These results extend earlier work of Williams.

[1]  J.J.M. Evers.  On the existence of balanced solutions in optimal economic growth and investment problems , 1974 .

[2]  S. M. Robinson.  Bounds for error in the solution set of a perturbed linear program , 1973 .

[3]  A. Hoffman.  On approximate solutions of systems of linear inequalities , 1952 .

[4]  Floyd J. Gould,et al.  Stability in Nonlinear Programming , 1970, Oper. Res..

[5]  S. M. Robinson.  Stability Theory for Systems of Inequalities. Part I: Linear Systems , 1975 .

[6]  Volker Bã–hm.  On the continuity of the optimal policy set for linear programs , 1975 .

[7]  Stephen M. Robinson,et al.  Regularity and Stability for Convex Multivalued Functions , 1976, Math. Oper. Res..

[8]  Adi Ben-Israel.  Linear equations and inequalities on finite dimensional, real or complex, vector spaces: A unified theory☆ , 1969 .

[9]  Harvey J. Greenberg,et al.  Extensions of the Evans-Gould Stability Theorems for Mathematical Programs , 1972, Oper. Res..

[10]  G. Dantzig,et al.  On the continuity of the minimum set of a continuous function , 1967 .

[11]  D. H. Martin.  On the continuity of the maximum in parametric linear programming , 1975 .

[12]  H. Mills.  8. Marginal Values of Matrix Games and Linear Programs , 1957 .

[13]  A. J. Goldman.  3 . Resolution and Separation Theorems for Polyhedral Convex Sets , 1957 .

[14]  Donald M. Topkis,et al.  Rates of Stability in Nonlinear Programming , 1976, Oper. Res..

[15]  A. Williams.  Marginal Values in Linear Programming , 1963 .

[16]  Abraham Charnes,et al.  Some Properties of Redundant Constraints and Extraneous Variables in Direct and Dual Linear Programming Problems , 1962 .

[17]  David Gale.  The theory of linear economic models , 1960 .

[18]  R. Bellman,et al.  Linear Programming and Economic Analysis. , 1958 .