Point-based neighborhoods for sharp calmness constants in linear programming

Juan Parra López, M.J. Cánovas, J.-J- Rückmann, F.J. Toledo

This talk is focused on recent advances on the calmness property of ordinary (finite) linear programs under canonical perturbations (i.e., perturbations of the objective function coefficient vector and the right-hand side of the constraint system). We show that the expression for the calmness modulus of the argmin mapping given in [1] is indeed a calmness constant in a certain neighborhood, which is also provided, of the nominal minimizer. We emphasize the fact that the expressions for both the (sharp) calmness constant and the neighborhood can be easily computed as far as they only depend on the nominal data (in this sense we call them point-based). As an intermediate step, we show that an analogous fact occurs for the feasible set mapping associated with linear inequality systems under right-hand side perturbations: the point-based expression for the corresponding calmness modulus given in [2] turns out to be a calmness constant in a certain neighborhood, for which we also give a point-based expression. We also show that this result cannot be extended to general convex systems.

Key words. Calmness, feasible set mapping, optimal set mapping, linear programming, variational analysis.

Mathematics Subject Classification: 90C31, 49J53, 90C05, 49K40, 65F22.

[1] M. J. CÁNOVAS, R. HENRION, M.A. LÓPEZ, J. PARRA, Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming, J. Optim. Theory Appl. 169 (2016), pp. 925-952.

[2] M. J. CÁNOVAS, M. A. LÓPEZ, J. PARRA, F.J. TOLEDO, Calmness of the feasible set mapping for linear inequality systems, Set-Valued Var. Anal. 22 (2014), pp. 375-389.