Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming

Abstract

With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions, which are the maximum of a finite amount of continuously differentiable functions of n real variables, paying special attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a Karush-Kuhn-Tucker index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems, without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given. B M. A. López marco.antonio@ua.es M. J. Cánovas canovas@umh.es R. Henrion henrion@wias-berlin.de

Perturbation of error bounds

“…Thanks to (25) and (26), it follows from Theorem 2.1 that Er g(x) ≤ 2ξ . As ξ > 0 can be chosen arbitrarily small, we conclude that Er {Ptb l ( f ,x, ε)}(x) = 0, which proves (15). ⊓ ⊔ (14) for any |∂ f | bd (x) as well as the first equality in (12) in the case |∂ f | bd (x) = 0.…”

“…Many authors have recently studied error bounds in connection with the metric regularity and subregularity (cf. [21]) as well as Aubin property and calmness of setvalued mappings: [3,15,16,26,33,34,36,[39][40][41]54,56,58,59,68,69]. The connections between the error bounds and weak sharp minima were studied in [13].…”

Outer limits of subdifferentials for min–max type functions

“…We are now ready to generalise Theorem 3.1 from [4]. We first prove that for positively homogeneous functions the inclusion (23) can be replaced by an equality.…”

“…where J is a finite index set. As in [4] define the collection D(x) of index subsets D ⊂ J(x) such that the following system is consistent with respect to d Proof. We begin by showing the following identity:…”

Marco A. López, a Pioneer of Continuous Optimization in Spain

“…Cánovas, R. Henrion, A. Kruger, J. Parra and M. Théra) are as follows: an expression for the calmness modulus of the mapping in linear programming under canonical perturbations (objective function and right-hand side of the constraints), involving limits of subdifferentials [10]; left-hand-side perturbations of the constraints system added into the analysis in [18]; and the outer limits of subdifferentials of max-functions and calmness moduli for feasible and optimal set mappings dealt with in [8]. -In [13,14,16], the distance to ill-posedness (in terms of the distance to infeasibility and to unsolvability) is studied.…”