TY - JOUR

T1 - A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid

AU - Beck, Amir

AU - Teboulle, Marc

PY - 2009/4

Y1 - 2009/4

N2 - We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the so-called regularized total least squares problem (RTLS), which is a special case of the problem's class we study. We prove that under a certain mild assumption on the problem's data, problem (RQ) admits an exact semidefinite programming relaxation. We then study a simple iterative procedure which is proven to converge superlinearly to a global solution of (RQ) and show that the dependency of the number of iterations on the optimality tolerance ε grows as O(√ In ε-1).

AB - We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the so-called regularized total least squares problem (RTLS), which is a special case of the problem's class we study. We prove that under a certain mild assumption on the problem's data, problem (RQ) admits an exact semidefinite programming relaxation. We then study a simple iterative procedure which is proven to converge superlinearly to a global solution of (RQ) and show that the dependency of the number of iterations on the optimality tolerance ε grows as O(√ In ε-1).

KW - Convergence analysis

KW - Fixed point algorithms

KW - Nonconvex quadratic minimization

KW - Ratio of quadratic minimization

KW - Regularized total least squares

KW - Semidefinite programming

KW - Strong duality

UR - http://www.scopus.com/inward/record.url?scp=58149508826&partnerID=8YFLogxK

U2 - 10.1007/s10107-007-0181-x

DO - 10.1007/s10107-007-0181-x

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AN - SCOPUS:58149508826

SN - 0025-5610

VL - 118

SP - 13

EP - 35

JO - Mathematical Programming

JF - Mathematical Programming

IS - 1

ER -