# how to find vertex of parallelotope closest to given point P in R^n ? (Or minimize quadratic form over {+-1}) Is it NP ?

Consider a parallelotope in R^n and some point "P" in R^n. What algorithms (except of brute force) can be suggested to find the closest vertex of paralleloptope to "P" ?

Is it NP ?

Parallelotope has 2^n vertex, not arbitrary 2^n point in R^n are vertex of paralleloptope, so clever algorithm should somehow use this additional information, while brute force search over 2^n points does not use.

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Reformulation:

after choosing origin in the center of parallelotope we can come to the following algebraic version of the problem: minimize over x_i = {-1,+1} the quadratic form: \sum a_ij x_i x_j - \sum x_i v_i

This looks almost like the Max-CUT problem to me (you have minimize instead of maximize, but you can just flip the signs of the matrix $A$)