Quadratization

A Quadratization is a mapping $\mathcal{Q}\{f\}: \mathscr{F} \to \mathscr{F}^{2}$ that preserves the optimization over auxiliary variables.

For minimization, the default sign = 1 gives

\[\min_{\mathbf{y}} \mathcal{Q}\{f\}(\mathbf{x}, \mathbf{y}) = f(\mathbf{x}) ~\forall \mathbf{x} \in \mathbb{B}^{n}.~\forall f \in \mathscr{F}.\]

For maximization, use sign = -1 so that

\[\max_{\mathbf{y}} \mathcal{Q}\{f\}(\mathbf{x}, \mathbf{y}) = f(\mathbf{x}) ~\forall \mathbf{x} \in \mathbb{B}^{n}.~\forall f \in \mathscr{F}.\]

The sign keyword controls the objective sense used to select and validate term reductions. The quadratized function keeps coefficients in the original objective scale.