New global algorithms for quadratic programming with a few negative eigenvalues based on alternative direction method and convex relaxation

We consider a quadratic program with a few negative eigenvalues (QP- r -NE) subject to linear and convex quadratic constraints that covers many applications and is known to be NP-hard even with one negative eigenvalue (QP1NE). In this paper, we first introduce a new global algorithm (ADMBB), which i...

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Bibliographic Details
Published inMathematical programming computation Vol. 11; no. 1; pp. 119 - 171
Main Authors Luo, Hezhi, Bai, Xiaodi, Lim, Gino, Peng, Jiming
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 14.03.2019
Springer Nature B.V
Subjects
Algorithms
Complexity
Digital Object Identifier
Eigenvalues
Full Length Paper
Mathematics
Mathematics and Statistics
Mathematics of Computing
Operations Research/Decision Theory
Optimization
Optimization techniques
Quadratic programming
Search algorithms
Theory of Computation
Line search
90C20
Branch-and-bound
90C22
Convex relaxation
90C26
Quadratic programming
Computational complexity
Alternative direction method
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ISSN1867-2949
1867-2957
DOI10.1007/s12532-018-0142-9

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Summary:We consider a quadratic program with a few negative eigenvalues (QP- r -NE) subject to linear and convex quadratic constraints that covers many applications and is known to be NP-hard even with one negative eigenvalue (QP1NE). In this paper, we first introduce a new global algorithm (ADMBB), which integrates several simple optimization techniques such as alternative direction method, and branch-and-bound, to find a globally optimal solution to the underlying QP within a pre-specified ϵ -tolerance. We establish the convergence of the ADMBB algorithm and estimate its complexity. Second, we develop a global search algorithm (GSA) for QP1NE that can locate an optimal solution to QP1NE within ϵ -tolerance and estimate the worst-case complexity bound of the GSA. Preliminary numerical results demonstrate that the ADMBB algorithm can effectively find a global optimal solution to large-scale QP- r -NE instances when r ≤ 10 , and the GSA outperforms the ADMBB for most of the tested QP1NE instances. The software reviewed as part of this submission was given the DOI (digital object identifier) https://doi.org/10.5281/zenodo.1344739 .
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ISSN:1867-2949
1867-2957
DOI:10.1007/s12532-018-0142-9