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We study a parameter (σ)
dependent relaxation of the Travelling Salesman Problem on .
The relaxed problem is reduced to the Travelling Salesman Problem
as 0. For increasing σ it is also an
ordered clustering algorithm for a set of points in .
A dual formulation is introduced, which reduces the problem to a
convex optimization, provided the minimizer is in the domain of
convexity of the relaxed functional. It is shown that this last
condition is generically satisfied, provided σ is large
enough.
...
The paper concerns the study of tilt stability of local minimizers in standard problems of nonlinear programming. This notion plays an important role in both theoretical and numerical aspects of optimization and has drawn a lot of attention in optimization theory and its applications, especially in recent years. Under the classical Mangasarian-Fromovitz Constraint Qualification, we establish relationships between tilt stability and some other stability notions in constrained optimization. Involving...
Two conditions are given each of which is both necessary and sufficient for a point to be a global Pareto minimum. The first one is obtained by studying programs where each criterion appears as a single objective function, while the second one is given in terms of a "restricted Lagrangian". The conditions are compared with the familiar characterizations of properly efficient and weakly efficient points of Karlin and Geoffrion.
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