Computational complexity of norm-maximization


This paper discusses the problem of maximizing a quasiconvex functionφ over a convex polytopeP inn-space that is presented as the intersection of a finite number of halfspaces. The problem is known to beNP-hard (for variablen) whenφ is thep th power of the classicalp-norm. The present reexamination of the problem establishesNP-hardness for a wider class of functions, and for thep-norm it proves theNP-hardness of maximization overn-dimensionalparallelotopes that are centered at the origin or have a vertex there. This in turn implies theNP-hardness of {−1, 1}-maximization and {0, 1}-maximization of a positive definite quadratic form. On the “good” side, there is an efficient algorithm for maximizing the Euclidean norm over an arbitraryrectangular parallelotope.

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The authors are indebted to J. O'Rourke, P, Pardalos and R. Freund for useful references. The second and third authors are indebted to the Institute for Mathematics and its Applications in Minneapolis, where much of this paper was written: they acknowledge additional support from the Alexander von Humboldt Stiftung and the National Science Foundation, respectively.

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Bodlaender, H.L., Gritzmann, P., Klee, V. et al. Computational complexity of norm-maximization. Combinatorica 10, 203–225 (1990).

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