Computing optimal distances to Pareto sets of multi-objective optimization problems in asymmetric normed lattices. (English) Zbl 1482.47150
Summary: Given a finite dimensional asymmetric normed lattice, we provide explicit formulae for the optimization of the associated (non-Hausdorff) asymmetric “distance” among a subset and a point. Our analysis has its roots and finds its applications in the current development of effective algorithms for multi-objective optimization programs. We are interested in providing the fundamental theoretical results for the associated convex analysis, fixing in this way the framework for this new optimization tool. The fact that the associated topology is not Hausdorff forces us to define a new setting and to use a new point of view for this analysis. Existence and uniqueness theorems for this optimization are shown. Our main result is the translation of the original abstract optimal distance problem to a clear optimization scheme. Actually, this justifies the algorithms and shows new aspects of the numerical and computational methods that have been already used in visualization of multi-objective optimization problems.
MSC:
| 47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |
| 46B85 | Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science |
| 46L85 | Noncommutative topology |
| 90C29 | Multi-objective and goal programming |
| 65K05 | Numerical mathematical programming methods |
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