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Sparse solutions to random standard quadratic optimization problems

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Abstract

The standard quadratic optimization problem (StQP) refers to the problem of minimizing a quadratic form over the standard simplex. Such a problem arises from numerous applications and is known to be NP-hard. In this paper we focus on a special scenario of the StQP where all the elements of the data matrix Q are independently identically distributed and follow a certain distribution such as uniform or exponential distribution. We show that the probability that such a random StQP has a global optimal solution with k nonzero elements decays exponentially in k. Numerical evaluation of our theoretical finding is discussed as well.

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Authors and Affiliations

  1. Department of Industrial and Enterprise System Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Xin Chen & Jiming Peng

  2. Industrial and Systems Engineering Program, University of Minnesota, Minneapolis, MN, 55455, USA

    Shuzhong Zhang

Authors
  1. Xin Chen
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  2. Jiming Peng
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  3. Shuzhong Zhang
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Correspondence to Jiming Peng.

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Chen, X., Peng, J. & Zhang, S. Sparse solutions to random standard quadratic optimization problems. Math. Program. 141, 273–293 (2013). https://doi.org/10.1007/s10107-012-0519-x

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  • DOI: https://doi.org/10.1007/s10107-012-0519-x

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