Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound

Burer, Samuel; Vandenbussche, Dieter

doi:10.1007/s10589-007-9137-6

Cited by 66 publications

(65 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A survey of research on QPB up to 1997 was given by De Angelis, Pardalos, and Toraldo [7]. More recent relevant papers include Yajima and Fujie [29], Vandenbussche and Nemhauser [27,28], Burer and Vandenbussche [6], Anstreicher [1], and Anstreicher and Burer [3].…”

Section: Introduction Nonconvex Quadratic Programming With Box Constmentioning

confidence: 99%

On Nonconvex Quadratic Programming with Box Constraints

Burer¹,

Letchford²

2009

SIAM J. Optim.

Self Cite

View full text Add to dashboard Cite

Abstract. Nonconvex quadratic programming with box constraints is a fundamental N P-hard global optimization problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterize their extreme points and vertices, show their invariance under certain affine transformations, and show that various linear inequalities induce facets. We also show that the sets are closely related to the Boolean quadric polytope, a fundamental polytope in the field of polyhedral combinatorics. Finally, we give a classification of valid inequalities and show that this yields a finite recursive procedure to check the validity of any proposed inequality.

show abstract

Section: Introduction Nonconvex Quadratic Programming With Box Constmentioning

confidence: 99%

On Nonconvex Quadratic Programming with Box Constraints

Burer¹,

Letchford²

2009

SIAM J. Optim.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We need just one additional lemma, whose proof is a straightforward adaptation of the proof of proposition 3.2 in Burer and Vandenbussche (2006):…”

Section: Additional Properties Of (Sdp 0 )mentioning

confidence: 99%

“…There are numerous methods for solving (1) and more general nonconvex quadratic programs, including local methods (Gould and Toint, 2002) and global methods (Pardalos, 1991). For a survey of methods to globally solve (1), see De Angelis et al (1997) as well as Vandenbussche and Nemhauser (2005a,b) and Burer and Vandenbussche (2006).…”

Section: Introductionmentioning

confidence: 99%

“…Vandenbussche and Nemhauser (2005a,b) and Burer and Vandenbussche (2006) have previously considered linear and semidefinite relaxations, respectively, involving only the first-order conditions. The contributions of the current paper are to demonstrate how also to incorporate the secondorder conditions and to illustrate the positive effects of doing so.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Relaxing the optimality conditions of box QP

Burer

Chen

2009

Comput Optim Appl

Self Cite

View full text Add to dashboard Cite

We present semidefinite relaxations of nonconvex, box-constrained quadratic programming, which incorporate the first-and second-order necessary optimality conditions. We compare these relaxations with a basic semidefinite relaxation due to Shor, particularly in the context of branch-and-bound to determine a global optimal solution, where it is shown empirically that the new relaxations are significantly stronger. We also establish theoretical relationships between the new relaxations and Shor's relaxation.

show abstract

“…A variety of methods are available for solving the quadratic programming problem [1][2][3][4][5][6]. These include interior point, extensions of the simplex, gradient projecttion, conjugate gradient, augmented Lagrangian and active set methods.…”

Section: Introductionmentioning

confidence: 99%

Minimizing Complementary Pivots in a Simplex-Based Solution Method for a Quadratic Programming Problem

Munapo¹

2012

AJOR

View full text Add to dashboard Cite

The paper presents an approach for avoiding and minimizing the complementary pivots in a simplex based solution method for a quadratic programming problem. The linearization of the problem is slightly changed so that the simplex or interior point methods can solve with full speed. This is a big advantage as a complementary pivot algorithm will take roughly eight times as longer time to solve a quadratic program than the full speed simplex-method solving a linear problem of the same size. The strategy of the approach is in the assumption that the solution of the quadratic programming problem is near the feasible point closest to the stationary point assuming no constraints.

show abstract

Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound

Abstract: Nonconcave quadratic maximization, Nonconvex quadratic programming, Branch-and-bound, Lift-and-project relaxations, Semidefinite programming,

Cited by 66 publications

References 17 publications

On Nonconvex Quadratic Programming with Box Constraints

On Nonconvex Quadratic Programming with Box Constraints

Relaxing the optimality conditions of box QP

Minimizing Complementary Pivots in a Simplex-Based Solution Method for a Quadratic Programming Problem

Contact Info

Product

Resources

About