Some perturbation theory for linear programming

1. Introducfion This paper examines a few relations between solution characteristics of an LP and the amount by which the LP must be perturbed to obtain either a primal infeasible LP or a dual infeasible LP. We consider such solution characteristics as the size of the optimal solution and the sensitivity of the optimal value to data perturbations. We show, for example, that an LP has a large optimal solution, or has a sensitive optimal value, only if the instance is nearly primal infeasible or dual infeasible. The results are not particularly surprising but they do formalize an interesting viewpoint which apparently has not been made explicit in the linear programming literature. The results are rather general. Several of the results are valid for linear programs defined in arbitrary real normed spaces. A Hahn-Banach Theorem is the main tool employed in the analysis; given a closed convex set in a normed vector space and a point in the space but not in the set, there exists a continuous linear functional strictly separating the set from the point. We introduce notation, then the results. LetX, Ydenote real vector spaces, each with a norm. We use the same notation (i.e. II II) for all norms, it being clear from context which norm is referred to. Let X* denote the dual space for X; this is the space of all continuous linear functionals c * :X ~ ~ (continuous with respect to the norm topology). Endow X* with the operator norm; ifc* eX* then IIc*ll .'= sup{c*x; Ilxll = 1}.