Reformulation and solution approach for non-separable integer quadratic programs

Dominique Quadri and Eric Soutil
Additional contact information
Dominique Quadri: Laboratoire de Recherche en Informatique (LRI), Université Paris Sud—XI, Orsay, France
Eric Soutil: Conservatoire National des Arts et Métiers, Paris, France

Journal of the Operational Research Society, 2015, vol. 66, issue 8, 1270-1280

Abstract: We consider quadratic programs with pure general integer variables. The objective function is quadratic and convex and the constraints are linear. An exact solution approach is proposed. It is decomposed into two phases. In the first phase, the initial problem is reformulated into an equivalent problem with a separable objective function. This is done by use of a Gauss decomposition of the Hessian matrix of the initial problem and requires the addition of some continuous variables and constraints. In the second phase, the reformulated problem is linearized by an approximation of each squared term by a set of K linear functions that correspond to the tangents of a hyperbola in K points. We give a proof of the intuitive property that when K is large enough, the optimal value of the obtained linear program is very close to optimal value of the two previous problems, the initial problem and the reformulated separable problem. The reminder is dedicated to the implementation of a branch-and-bound algorithm for the solution of linearized problem, and its application to a set of instances. Several points are considered among which choice of the right value for parameter K and the implementation of a sophisticated heuristic solution algorithm. The numerical comparison is done with CPLEX 12.2 since, in this case, the initial problem as well as the problem reformulated by the first step can be solved by CPLEX. We show that with our approach, the total CPU time is divided by a factor ranging from 1.2 to 131.6 for instances with 40–60 variables.

Date: 2015
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.palgrave-journals.com/jors/journal/v66/n8/pdf/jors201476a.pdf Link to full text PDF (application/pdf)
http://www.palgrave-journals.com/jors/journal/v66/n8/full/jors201476a.html Link to full text HTML (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:pal:jorsoc:v:66:y:2015:i:8:p:1270-1280

Ordering information: This journal article can be ordered from
http://www.springer. ... search/journal/41274

Access Statistics for this article

Journal of the Operational Research Society is currently edited by Tom Archibald and Jonathan Crook

More articles in Journal of the Operational Research Society from Palgrave Macmillan, The OR Society
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().