Speeding up IP-based algorithms for constrained quadratic 0–1 optimization

Abstract

Abstract. In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP). Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming. Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvátal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastical speedup in the solution of constrained quadratic 0-1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions.

A computational study and survey of methods for the single-row facility layout problem

“…However the hyperplane rounding idea suggested in [18] cannot be applied directly to (SRFLP) to get a good layout because it yields a {−1, 1} vectorỹ, which need not be feasible with respect to the three cycle equations (11). That is why Anjos et al [5] propose a different procedure to obtain a good feasible layout from the optimal solution of the SDP relaxation whereas Hungerländer and Rendl [26] suggest to apply a repair strategy to the infeasibleỹ.…”

“…The main contributions of this paper are the following: First we describe and compare the most successful modelling approaches to (SRFLP), pointing out their common connections to the maximum cut [10,21,38] and the quadratic ordering problem [11,12]. For further details on this subject see also the recent survey of (SRFLP) by Anjos and Liers [7].…”

An Exact Algorithm for Robust Network Design

“…Finally, the inequality is lifted to become valid (not necessarily facet-defining) for P . In [8], target cuts were successfully used for solving several constrained binary quadratic optimization problems.…”

Global Approaches for Facility Layout and VLSI Floorplanning

“…Therefore, approaches exploiting the cut-polytope structure can lead to effective solution algorithms. This connection has been exploited in the work by Buchheim et al [15] where several different applications were studied for which the linear constraints always induce faces of some cut polytope. General separation methods for constrained quadratic problems of form (QP) were designed that can complement or replace detailed polyhedral studies of the polytope of the linearized problem and that can be used as a black box.…”

“…We will introduce these applications in the next section. The main contribution in [15] was to show that these general approaches lead to a dramatic decrease of both the number of nodes in the enumeration tree and the running time when compared to an algorithm that only uses the standard separation routines for the well-studied polytope P from (QP).…”