## Regularity conditions in vector optimization

Giancarlo Bigi and Massimo Pappalardo
### Summary

In the field of scalar Optimization
regularity is conceived as a condition which
guarantees the existence of generalized Karush-Kuhn-Tucker (in short KKT)
multipliers such that the one corresponding to the objective
function is nonzero. In the field of Vector Optimization, the
analysis is more difficult; in fact, since there are many objective
functions, different concepts of regularity can be considered.
In a recent paper, new concepts of regularity and total
regularity for linear separation between a set and a convex cone have
been studied. Here, we aim to apply the results of that paper to the study
of regularity in Vector Optimization.
Most of the known optimality conditions can be expressed in terms of the
impossibility of generalized systems; following the image space approach,
the impossibility of such systems can be turned into the disjunction
between a set and a convex cone and studied by means of linear
separation tools. Besides, the gradient of a separating hyperplane is
a vector of KKT multipliers.
We prove that regularity conditions for linear separation provide
regularity conditions for vector optimization problems. Moreover, we
show that total regularity for linear separation is equivalent to very
general Mangasarian-Fromovitz type conditions, which can be employed
to achieve additional properties of KKT multipliers. Besides, we also
show that many well-known regularity conditions and constraint
qualifications are less general than the ones developed in
our scheme and they can be recovered as particular cases.

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