Handbook of Combinatorial Optimization pp 1473-1491 | Cite as

# Semidefinite Relaxations, Multivariate Normal Distributions, and Order Statistics

Chapter

## 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 Note that the constraint x

$$\begin{array}{l}
Maximize q\left( x \right): = x'Qx\\
subject to \left| {\sum\limits_{j = 1}^n {{a_{ij}}{x_j}} } \right| = {b_i} = 1,...,m,\\
x_j^2 = 1,j = 1,...,n.
\end{array}$$

(QP)

_{ j }^{2}=1 will force x_{ j }= 1 or x_{ j }= −1, making it a boolean variable.## Keywords

Approximation Algorithm Quadratic Programming Positive Semidefinite Multivariate Normal Distribution Semidefinite Programming
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]F. Alizadeh,
*Combinatorial optimization with interior point methods and semi-definite matrices*, PhD thesis, University of Minnesota, Minneapolis, MN, 1991.Google Scholar - [2]M. Bellare and P. Rogaway, “The complexity of approximating a nonlinear program,”
*Mathematical Programming*69 (1995) 429–442.zbMATHMathSciNetGoogle Scholar - [3]S. Benson Y. Ye and X. Zhang, “Solving large-scale sparse semidefinite programs for combinatorial optimization,” Working Paper, Computational and Applied Mathematics, The University of Iowa, Iowa City, IA 52242, 1997.Google Scholar
- [4]D. Bertsimas, C. Teo and R. Vohra, “On dependent randomized rounding algorithms,”
*Proc. 5th IPCO Conference*(1996) 330–344.Google Scholar - [5]D. Bertsimas, C. Teo and R. Vohra, “Nonlinear relaxations and improved randomized approximation algorithms for multicut problems,”
*Proc., nth IPCO Conference*(1995) 29–39.Google Scholar - [6]A. Frieze and M. Jerrum, “Improved approximation algorithms for max k-cut and max bisection,”
*Proc. 4th IPCO Conference*(1995) 1–13.Google Scholar - [7]L. E. Gibbons, D. W. Hearn and P. M. Pardalos, “A continuous based heuristic for the maximum clique problem,”
*DIMACS Series in Discrete Mathematics and Theoretical Computer Science*26 (1996) 103–124.MathSciNetGoogle Scholar - [8]M. X. Goemans and D. P. Williamson, “ Improved approximation algorithms for Maximum Cut and Satisfiability problems using semidefinite programming,”
*Journal of ACM*42 (1995) 1115–1145.CrossRefzbMATHMathSciNetGoogle Scholar - [9]C. Helmberg and F. Rendl, “A spectral bundle method for semidefinite programming,” ZIB Preprint SC 97–37, Konrad-Zuse-Zentrum fuer Informationstechnik Berlin, Takustrasse 7, D-14195 Berlin, Germany, August 1997.Google Scholar
- [10]N. Johnson and S. Kotz,
*Distributions in Statistics: Continuous Multivariate Distributions*, John Wiley & Sons, 1972.Google Scholar - [11]L. Lovâsz and A. Shrijver, “Cones of matrices and setfunctions, and 0 — 1 optimization,”
*SIAM Journal on Optimization**1*(1990) 166–190.CrossRefGoogle Scholar - [12]Yu. E. Nesterov, “Quality of semidefinite relaxation for nonconvex quadratic optimization,” CORE Discussion Paper, #9719, Belgium, March 1997.Google Scholar
- [13]Yu. E. Nesterov and A. S. Nemirovskii,
*Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms*, SIAM Publications, SIAM, Philadelphia, 1993.Google Scholar - [14]Yu. E. Nesterov, M. J. Todd, and Y. Ye, “Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems,” Technical Report No. 1156, School of Operations Research and and Industrial Engineering, Cornell University, Ithaca, NY 14853–3801, 1996, to appear in
*Mathematical Programming.*Google Scholar - [15]W. F. Sheppard, “On the calculation of the double integral expressing normal correlation,”
*Transactions of the Cambridge Philosophical Society*19 (1900) 23–66.Google Scholar - [16]Y. L. Tong,
*The Multivariate Normal Distribution*, Springer-Verlag, New York, 1990.CrossRefzbMATHGoogle Scholar - [17]Y. Ye, “Approximating quadratic programming with quadratic con-straints,” Working Paper, Department of Management Science, The University of Iowa, Iowa City, IA 52242, 1997, to appear in
*Mathematical Programming.*Google Scholar

## Copyright information

© Kluwer Academic Publishers 1998