Abstract
The simplex algorithm for linear programming is based on the well-known equivalence between the problem of maximizing a linear functionf on a polyhedronP and the problem of maximizingf over the setV P of all vertices ofP. The equivalence between these two problems is also exploited by some methods for maximizing a convex or quasi-convex function on a polyhedron.
In this paper we determine some very general conditions under which the problem of maximizingf overP is equivalent, in some sense, to the problem of maximizingf overV P . In particular, we show that these two problems are equivalent whenf is convex or quasi-convex on all the line segments contained inP and parallel to some edge ofP.
In the case whereP is a box our results extend a well-known result of Rosenberg for 0–1 problems. Furthermore, whenP is a box or a simplex, we determine some classes of functions that can be maximized in polynomial time overP.
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This paper has been partially written while the author was visiting the Rutgers Center for Operations Research (RUTCOR). The support of the Air Force grants AFORS-89-0512 and AFORS-90-0008 is gratefully acknowledged.
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Tardella, F. On the equivalence between some discrete and continuous optimization problems. Ann Oper Res 25, 291–300 (1990). https://doi.org/10.1007/BF02283701
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DOI: https://doi.org/10.1007/BF02283701