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 thepth 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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