Global Optimization: Local Minima and Transition Points |
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Authors: | Floudas C A Jongen H Th |
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Institution: | (1) Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA;(2) Department of Mathematics, RWTH Aachen University, D-52056 Aachen, Germany |
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Abstract: | We consider the minimization of smooth functions of the Euclidean space with a finite number of stationary points having moderate
asymptotic behavior at infinity. The crucial role of transition points of first order (i.e., saddle points of index 1) is
emphasized. It is shown that (generically) any two local minima can be connected via an alternating sequence of local minima
and transition points of first order. In particular, the graph with local minima as its nodes and first order transition points
representing the edges turns out to be connected (Theorem A). On the other hand, any connected (finite) graph can be realized
in the above sense by means of a smooth function of three variables having a minimal number of stationary points (Theorem
B). |
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Keywords: | Global optimization Local minima Saddle points Transition points of first order |
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