Semidefinite Relaxations, Multivariate Normal Distributions, and Order Statistics

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

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

(QP)

Note that the constraint x ² _j =1 will force x _j = 1 or x _j = −1, making it a boolean variable.