The trust region subproblem with non-intersecting linear constraints

Burer, Samuel; Yang, Boshi

doi:10.1007/s10107-014-0749-1

Cited by 62 publications

(46 citation statements)

References 9 publications

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“…The convex hull question has also received attention for certain strengthened relaxations of simple QCQPs [12,13,15,35]. In this line of work, the standard SDP relaxation is strengthened by additional inequalities derived using the Reformulation-Linearization Technique (RLT).…”

Section: Introductionmentioning

confidence: 99%

On Convex Hulls of Epigraphs of QCQPs

Wang

Kılınç-Karzan

2020

Integer Programming and Combinatorial Optimization

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Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the convex hull result holds whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

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Section: Introductionmentioning

confidence: 99%

On Convex Hulls of Epigraphs of QCQPs

Wang

Kılınç-Karzan

2020

Integer Programming and Combinatorial Optimization

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“…The TRS first arose in trust region methods for nonlinear optimization [6] and also finds applications in the least square problems [31] and robust optimization [2]. Various approaches have been derived to solve the TRS and its variant with additional linear constraints, see [17,19,29,22,25,30,4,5,28]. Recently, Hazan and Koren [9] proposed the first linear-time algorithm (with respect to the nonzero entries in the input) for the TRS, via a linear-time eigenvalue oracle and a linear-time semidefinite programming (SDP) solver based on approximate eigenvector computations [16].…”

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confidence: 99%

A Linear-Time Algorithm for Generalized Trust Region Subproblems

Jiang¹,

Li²

2020

SIAM J. Optim.

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In this paper, we provide the first provable linear-time (in term of the number of non-zero entries of the input) algorithm for approximately solving the generalized trust region subproblem (GTRS) of minimizing a quadratic function over a quadratic constraint under some regularity condition. Our algorithm is motivated by and extends a recent linear-time algorithm for the trust region subproblem by Hazan and Koren [Math. Program., 2016, 158(1-2): 363-381]. However, due to the non-convexity and non-compactness of the feasible region, such an extension is nontrivial. Our main contribution is to demonstrate that under some regularity condition, the optimal solution is in a compact and convex set and lower and upper bounds of the optimal value can be computed in linear time. Using these properties, we develop a linear-time algorithm for the GTRS.

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“…Generally, when all the linear cuts (45) are non-intersecting in the ball (44), (T m ) has the following SOC-SDP reformulation [21]:…”

Section: Trust-region Subproblem and Extensionsmentioning

confidence: 99%

A Survey of Hidden Convex Optimization

Xia

2020

J. Oper. Res. Soc. China

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Motivated by the fact that not all nonconvex optimization problems are difficult to solve, we survey in this paper three widely-used ways to reveal the hidden convex structure for different classes of nonconvex optimization problems. Finally, ten open problems are raised.Keywords convex programming · quadratic programming · quadratic matrix programming · fractional programming · Lagrangian dual · semidefinite programming Mathematics Subject Classification (2010) 90C20, 90C25, 90C26, 90C32

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The trust region subproblem with non-intersecting linear constraints

Cited by 62 publications

References 9 publications

On Convex Hulls of Epigraphs of QCQPs

On Convex Hulls of Epigraphs of QCQPs

A Linear-Time Algorithm for Generalized Trust Region Subproblems

A Survey of Hidden Convex Optimization

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