TY - JOUR

T1 - A linear-time algorithm for trust region problems

AU - Hazan, Elad

AU - Koren, Tomer

N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - We consider the fundamental problem of minimizing a general quadratic function over an ellipsoidal domain, also known as the trust region (sub)problem. We give the first provable linear-time (in the number of non-zero entries of the input) algorithm for approximately solving this problem. Specifically, our algorithm returns an ϵ-approximate solution in runtime of order N/ϵ, where N is the number of non-zero entries in the input. This matches the runtime of Nesterov’s accelerated gradient descent, suitable for the special case in which the quadratic objective is convex, and the runtime of the Lanczos method which is applicable when the problem is purely quadratic.

AB - We consider the fundamental problem of minimizing a general quadratic function over an ellipsoidal domain, also known as the trust region (sub)problem. We give the first provable linear-time (in the number of non-zero entries of the input) algorithm for approximately solving this problem. Specifically, our algorithm returns an ϵ-approximate solution in runtime of order N/ϵ, where N is the number of non-zero entries in the input. This matches the runtime of Nesterov’s accelerated gradient descent, suitable for the special case in which the quadratic objective is convex, and the runtime of the Lanczos method which is applicable when the problem is purely quadratic.

KW - Approximation algorithms

KW - Linear time complexity

KW - Semidefinite programming

KW - Trust region methods

KW - Trust region subproblem

UR - http://www.scopus.com/inward/record.url?scp=84937053404&partnerID=8YFLogxK

U2 - 10.1007/s10107-015-0933-y

DO - 10.1007/s10107-015-0933-y

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AN - SCOPUS:84937053404

SN - 0025-5610

VL - 158

SP - 363

EP - 381

JO - Mathematical Programming

JF - Mathematical Programming

IS - 1-2

ER -