.. _cqm_sdk:

============================
Constrained Quadratic Models
============================

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

.. math::

    \begin{align}
        \text{Minimize an objective:} & \\
        & \sum_{i} a_i x_i + \sum_{i \le j} b_{ij} x_i x_j + c, \\
        \text{Subject to constraints:} & \\
        & \sum_i a_i^{(m)} x_i + \sum_{i \le j} b_{ij}^{(m)} x_i x_j+ c^{(m)} \circ 0,
        \quad m=1, \dots, M,
    \end{align}

where :math:`\{ x_i\}_{i=1, \dots, N}` can be binary\ [#]_, integer, or 
continuous\ [#]_ variables, :math:`a_{i}, b_{ij}, c` are real values,
:math:`\circ \in \{ \ge, \le, = \}` and  :math:`M` is the total number of constraints.

.. [#]
    For binary variables, the range of the quadratic-term summation is
    :math:`i < j` because :math:`x^2 = x` for binary values :math:`\{0, 1\}`
    and :math:`s^2 = 1` for spin values :math:`\{-1, 1\}`.

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

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