On Calmness of the Argmin Mapping in Parametric Optimization Problems

Abstract

Recently, Cánovas et al. presented an interesting result: the argmin mapping of a linear semi-infinite program under canonical perturbations is calm if and only if some associated linear semiinfinite inequality system is calm. Using classical tools from parametric optimization, we show that the if-direction of this condition holds in a much more general framework of optimization models, while the opposite direction may fail in the general case. In applications to special classes of problems, we apply a more recent result on the intersection of calm multifunctions. Abstract Recently, Cánovas et al. presented an interesting result: the argmin mapping of a linear semi-infinite program under canonical perturbations is calm if and only if some associated linear semi-infinite inequality system is calm. Using classical tools from parametric optimization, we show that the if-direction of this condition holds in a much more general framework of optimization models, while the opposite direction may fail in the general case. In applications to special classes of problems, we apply a more recent result on the intersection of calm multifunctions.

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

“…observe that x is a regular point of the surface S defined by (25). Moreover, if we denote by ∇h (x) the matrix whose columns are ∇h 1 (x) , .…”

“…In relation to the second problem (calmness modulus of the argmin mapping for nonlinear problems), we would like to mention the gap which occurs when passing from linear to nonlinear problems pointed out in [25,Section 3], where a characterization of the calmness property of the argmin mapping becomes just a sufficient condition when replacing a linear function (either in the objective function or in the constraints) with a convex quadratic one.…”

Optimality Conditions for Set-Valued Optimisation Problems Using a Modified Demyanov Difference

“…In order to discuss this issue, we need the definition of calmness of a multifunction (cf. [24]). Let a set-valued map F : R n ⇒ R p be given.…”

“…If V (ȳ) = R m can be chosen, then the map F is calm at the pointx. Assume that for a given l ∈ T F(x) ∩(K * ∪(−K * )), the map F is calm and Lipschitz lower semicontinuous at (x, ψ l (x)) [24], i.e. there are U (x) and σ > 0 such that…”

“…Then ψ l as the map x → {ψ l (x)} is calm at (x, ψ l (x)) (cf. [24,Theorem3.1]) and therefore, we obtain: Remark 3.2 Let us consider a set-valued map F : R n ⇒ R p , the pointx ∈ R n and assume that a set T F(x) exists which is constant for all x ∈ R n . Additionally, suppose that there is a neighbourhood U (x) such that for all l ∈ T F(x) ∩ (K * ∪ (−K * )) ψ l (cf.…”