We aim to quantify the stability of systems of (possibly infinitely many) linear inequalities under arbitrary perturbations of the data. Our focus is on the Aubin property (also called pseudo-Lipschitz) of the solution set mapping, or, equivalently, on the metric regularity of its inverse mapping. The main goal is to determine the regularity modulus of the latter mapping exclusively in terms of the system's data. In our context, both, the right-and the left-hand side of the system are subject to possible perturbations. This fact entails notable differences with respect to previous developments in the framework of linear systems with perturbations of the right-hand side. In these previous studies, the feasible set mapping is sublinear (which is not our current case) and the wellknown Radius Theorem constitutes a useful tool for determining the modulus. In our current setting we do not have an explicit expression for the radius of metric regularity, and we have to tackle the modulus directly. As an application we approach, under appropriate assumptions, the regularity modulus for a semi-infinite system associated with the Lagrangian dual of an ordinary nonlinear programming problem.

This paper is devoted to quantify the Lipschitzian behavior of the optimal solutions set in linear optimization under perturbations of the objective function and the right hand side of the constraints (inequalities). In our model, the set indexing the constraints is assumed to be a compact metric space and all coefficients depend continuously on the index. The paper provides a lower bound on the Lipschitz modulus of the optimal set mapping (also called argmin mapping), which, under our assumptions, is single-valued and Lipschitz continuous near the nominal parameter. This lower bound turns out to be the exact modulus in ordinary linear programming, as well as in the semi-infinite case under some additional hypothesis which always holds for dimensions n 3. The expression for the lower bound (or exact modulus) only depends on the nominal problem's coefficients, providing an operative formula from the practical side, specially in the particular framework of ordinary linear programming, where it constitutes the sharp Lipschitz constant. In the semi-infinite case, the problem of whether or not the lower bound equals the exact modulus for n > 3 under weaker hypotheses (or none) remains as an open problem.

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