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Semidefinite Relaxations, Multivariate Normal Distributions, and Order Statistics

In: Handbook of Combinatorial Optimization

Author

Listed:
  • Dimitris Bertsimas

    (Massachusetts Institute of Technology, Sloan School of Management)

  • Yinyu Ye

    (The University of Iowa, Department of Management Science)

Abstract

Given the symmetric matrix $$Q = \left\{ {qij} \right\} \in {R^{n \times n}}$$ and the constraint matrix $$A = \left\{ {{a_{ij}}} \right\} \in {R^{m \times n}}$$ , we consider the quadratic programming (QP) problem with linear and boolean constraints QP $$\begin{array}{l} Maximize q\left( x \right): = x'Qx\\ subject to \left| {\sum\limits_{j = 1}^n {{a_{ij}}{x_j}} } \right| = {b_i} = 1,...,m,\\ x_j^2 = 1,j = 1,...,n. \end{array}$$ Note that the constraint x j 2 =1 will force x j = 1 or x j = −1, making it a boolean variable.

Suggested Citation

  • Dimitris Bertsimas & Yinyu Ye, 1998. "Semidefinite Relaxations, Multivariate Normal Distributions, and Order Statistics," Springer Books, in: Ding-Zhu Du & Panos M. Pardalos (ed.), Handbook of Combinatorial Optimization, pages 1473-1491, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-0303-9_24
    DOI: 10.1007/978-1-4613-0303-9_24
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