Abstract
A linearization technique for binary quadratic programs (BQPs) that comprise linear constraints is presented. The technique, called “inductive linearization”, extends concepts for BQPs with particular equation constraints, that have been referred to as “compact linearization” before, to the general case. Quadratic terms may occur in the objective function, in the set of constraints, or in both. For several relevant applications, the linear programming relaxations obtained from applying the technique are proven to be at least as strong as the one obtained with a well-known classical linearization. It is also shown how to obtain an inductive linearization automatically. This might be used, e.g., by general-purpose mixed-integer programming solvers.
Highlights
We present a linearization technique for binary quadratic programs (BQPs) that comprise linear and possibly quadratic constraints
The inductive linearization technique generalizes on a method by Liberti (2007) that exploits the special case of equations with right hand side and left hand side coefficients equal to one, and on its later revision
The inductive linearization technique has been extended to binary quadratic problems with arbitrary linear constraints
Summary
We present a linearization technique for binary quadratic programs (BQPs) that comprise linear and possibly quadratic constraints. This can be assumed (or established) without of loss of generality (see the appendix) If this requirement is neither fulfilled in the original problem nor established for some factors, this does not affect a successful inductive linearization of all the products whose factors do fulfill the requirement. The inductive linearization technique generalizes on a method by Liberti (2007) that exploits the special case of equations with right hand side and left hand side coefficients equal to one, and on its later revision (cf Mallach 2018) In his original article, Liberti coined the name “compact linearization” because it typically adds fewer constraints to such problems than the “standard linearization”
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