Abstract
Let f be a smooth nondegenerate real valued function on a finite dimensional, compact and connected Riemannian manifold. The bipartite min-max graph Γ is defined as follows. Its nodes are formed by the set of local minima and the set of local maxima. Two nodes (a local minimum and a local maximum) are connected in Γ by means of an edge if some trajectory of the corresponding gradient flow connects them. Given a natural number k, we construct a function f such that the length of the shortest path in Γ between two specific local minima exceeds k. The latter construction is independent of the underlying Riemannian metric.
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GÜNZEL, H., Jongen, H.T. Global Optimization: On Pathlengths in Min-Max Graphs. Journal of Global Optimization 17, 161–165 (2000). https://doi.org/10.1023/A:1026579317417
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DOI: https://doi.org/10.1023/A:1026579317417