Uniqueness and stability of Lagrange multipliers and associated qualification conditions

This paper is concerned with uniqueness and stability of Lagrange multipliers for constrained optimization problems in abstract spaces. It is well known that validity of the strict Robinson-Zowe-Kurcyusz condition implies the so-called isolated calmness, a one-sided Lipschitz property tailored for set-valued mappings, of some Lagrange multiplier mapping associated with a perturbed version of the original optimization problem, and the latter indeed is enough to guarantee uniqueness of the Lagrange multiplier. The paper studies the isolated calmness of the Lagrange multiplier mapping in detail. Exemplary, it is shown that this condition is sufficient for the Robinson-Zowe-Kurcyusz constraint qualification and, in the presence of additional assumptions, even equivalent to the strict Robinson-Zowe-Kurcyusz condition. Illustrative examples are presented to underline the necessity of postulated assumptions.