Constrained Quadratic Models

The constrained quadratic model (CQM) are problems of the form:

\[\begin{split}\begin{align} \text{Minimize an objective:} & \\ & \sum_{i} a_i x_i + \sum_{i<j} b_{ij} x_i x_j + c, \\ \text{Subject to constraints:} & \\ & \sum_i a_i^{(c)} x_i + \sum_{i<j} b_{ij}^{(c)} x_i x_j+ c^{(c)} \le 0, \quad c=1, \dots, C_{\rm ineq.}, \\ & \sum_i a_i^{(d)} x_i + \sum_{i<j} b_{ij}^{(d)} x_i x_j + c^{(d)} = 0, \quad d=1, \dots, C_{\rm eq.}, \end{align}\end{split}\]

where \(\{ x_i\}_{i=1, \dots, N}\) can be binary, integer, and real1 variables, \(a_{i}, b_{ij}, c\) are real values and \(C_{\rm ineq.}, C_{\rm eq,}\) are the number of inequality and equality constraints respectively.

The dimod.ConstrainedQuadraticModel class can contain this model and its methods provide convenient utilities for working with representations of a problem.

1

Real-valued variables are currently not supported in quadratic interactions.