Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

A nonconvex quadratically constrained quadratic programming (QCQP) with one constraint is usually solved via a dual SDP problem, or Moré’s algorithm based on iteratively solving linear systems. In this work we introduce an algorithm for QCQP that requires finding just one eigenpair of a generalized...

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Bibliographic Details
Published in:Mathematical programming Vol. 173; no. 1-2; pp. 79 - 116
Main Authors: Adachi, Satoru, Nakatsukasa, Yuji
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2019
Springer Nature B.V
Subjects:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Eigenvalues
Full Length Paper
Linear systems
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Quadratic programming
Theoretical
Generalized eigenvalue problem
90C20
65K05
90C30
QCQP
Canonical form for symmetric matrix pair
49M37
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:A nonconvex quadratically constrained quadratic programming (QCQP) with one constraint is usually solved via a dual SDP problem, or Moré’s algorithm based on iteratively solving linear systems. In this work we introduce an algorithm for QCQP that requires finding just one eigenpair of a generalized eigenvalue problem, and involves no outer iterations other than the (usually black-box) iterations for computing the eigenpair. Numerical experiments illustrate the efficiency and accuracy of our algorithm. We also analyze the QCQP solution extensively, including difficult cases, and show that the canonical form of a matrix pair gives a complete classification of the QCQP in terms of boundedness and attainability, and explain how to obtain a global solution whenever it exists.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-017-1206-8