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On a class of nonconvex problems where all local minima are global. (English) Zbl 1220.90097
The author characterizes a class of optimization problems having a convex objective function and a nonconvex feasible region with the property that all local minima are global. The property of globality of all local minima applies to many problem instances, apart from the class of convex problems. This suggests the use of nonconvex relaxations (having the same local-to-global minimality property) which may be much tighter than an ordinary convex relaxations and, thus, considerably speed up the global optimization software acting on the nonlinear problem. Here, a topological approach is followed. The author proves that certain convex functions defined on nonconvex sets satisfying a special set of conditions have the desired property.
MSC:
90C26 Nonconvex programming, global optimization
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
26B25 Convexity of real functions of several variables, generalizations
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