Efficient computation of the binary vector that maximizes a rank-deficient quadratic form

  • Authors:

  • George N. Karystinos;Athanasios P. Liavas

  • Affiliations:

  • Department of Electronic and Computer Engineering, Technical University of Crete, Chania, Greece;Department of Electronic and Computer Engineering, Technical University of Crete, Chania, Greece

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2010

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Abstract

The maximization of a full-rank quadratic form over the binary alphabet can be performed through exponential-complexity exhaustive search. However, if the rank of the form is not a function of the problem size, then it can be maximized in polynomial time. By introducing auxiliary spherical coordinates, we show that the rank-deficient quadratic-form maximization problem is converted into a double maximization of a linear form over a multidimensional continuous set, the multidimensional set is partitioned into a polynomial-size set of regions which are associated with distinct candidate binary vectors, and the optimal binary vector belongs to the polynomial-size set of candidate vectors. Thus, the size of the candidate set is reduced from exponential to polynomial. We also develop an algorithm that constructs the polynomial-size candidate set in polynomial time and show that it is fully parallelizable and rank-scalable. Finally, we demonstrate the efficiency of the proposed algorithm in the context of adaptive spreading code design.