Abstract
Sensitivity analysis is applied to the robust linear programming problem in this paper. The coefficients of the linear program are assumed to be perturbed in three perturbation manners within ellipsoidal sets. Our robust sensitivity analysis is to calculate the maximal radii of the perturbation sets to keep some properties of the robust feasible set. Mathematical models are formulated for the robust sensitivity analysis problems and all models are either reformulated into linear programs or convex quadratic programs except for the bi-convex programs where more than one row of the constraint matrix is perturbed. For the bi-convex programs, we develop a binary search algorithm.
Highlights
In this study, the perturbed linear programming problem is as follows: min cT0 x s.t
Where x ∈ Rn is a variable, c0 ∈ Rn is fixed and (A, b) ∈ Rm×n × Rm is in an ellipsoidal uncertainty set U (l) determined by a variable l of perturbation radius that is nonnegative
We will not consider the simultaneous perturbation of the constraint matrix and the right-hand-side vector since we can reform it into min cT0 x x s.t
Summary
The perturbed linear programming problem is as follows: min cT0 x s.t. In classical sensitivity analysis for the linear programming problem, the ranges of the objective coefficients and the ranges of the right-hand-side vector are analyzed to keep the optimal solution unchanged or the optimal basis unchanged, respectively, which can be found in books about linear programming In both cases, the ranges of the parameters are given in a "box" form. [15] presented copositive programs to state the best-case and worst-case optimal values when the coefficients in the objective function and the right-hand side are perturbed They developed tight, tractable conic relaxations to provide the lower and upper bounds. Their idea is quite different from ours Their uncertainty set is fixed rather than variable, their feasible solution set is not suitable for all possible realizations of the parameters and perturbation only in objective and right-hand side is allowed, while the constraint matrix is not considered.
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