Summary
Many important classes of decision models give rise to the problem of finding a global minimum of a concave function over a convex set. Since many local minima can occur, concave minimization belongs to the “hard” global optimization problems, where standard nonlinear programming procedures fail.
After a brief survey on important classes of decision models that can be formulated as concave minimization problems, the two main solution approaches are discussed: branch-and-bound combined with convex underestimation and outer approximation by cutting planes.
Zusammenfassung
Eine Reihe wichtiger Klassen von Entscheidungsmodellen führt auf die globale Minimierung konkaver Funktionen über konvexen Mengen. Da viele lokale Minima auftreten können, ist die konkave Minimierung zu den “harten” globalen Optimierungsaufgaben zu rechnen, bei denen Standardverfahren der nichtlinearen Optimierung nicht zum Ziel führen.
Nach einer kurzen Übersicht über wichtige Klassen von Entscheidungsmodellen, die sich als konkave Minimierungsprobleme formulieren lassen, werden die beiden wichtigsten Lösungsansätze diskutiert: Eine Kombination von Branch und Bound mit konvexer Approximation und äußere Approximationsverfahren mit Hilfe von Schnittebenen.
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Horst, R. On the global minimization of concave functions. OR Spektrum 6, 195–205 (1984). https://doi.org/10.1007/BF01720068
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DOI: https://doi.org/10.1007/BF01720068