SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices

We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric mat...

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Vydáno v:Mathematical programming Ročník 169; číslo 2; s. 531 - 563
Hlavní autoři: Jiang, Rujun, Li, Duan, Wu, Baiyi
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2018
Springer Nature B.V
Témata:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Economic models
Full Length Paper
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Matrix methods
Numerical Analysis
Theoretical
90C20
90C26
Trust region subproblem
Quadratically constrained quadratic programming
Canonical form
ISSN:0025-5610, 1436-4646
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Shrnutí:We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions. By exploiting the block separability of the canonical form, we show that all GTRSs with an optimal value bounded from below are second order cone programming (SOCP) representable. Our result generalizes the recent work of Ben-Tal and den Hertog (Math. Program. 143(1–2):1–29, 2014 ), which establishes the SOCP representability of the GTRS under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions. We then derive a closed-form solution for the GTRS when the two matrices are not simultaneously diagonalizable. We further extend our method to two variants of the GTRS in which the inequality constraint is replaced by either an equality constraint or an interval constraint.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-017-1145-4