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October  2019, 15(4): 1677-1699. doi: 10.3934/jimo.2018117

## Recovering optimal solutions via SOC-SDP relaxation of trust region subproblem with nonintersecting linear constraints

 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695, USA

* Corresponding author: Wenxun Xing < wxing@tsinghua.edu.cn >

Received  April 2017 Revised  April 2018 Published  August 2018

Fund Project: Xing's research has been supported by the NNSF of China Grants #11571029 and #11771243, Fang's research has been supported by the US ARO Grant #W911NF-15-1-0223.

In this paper, we study an extended trust region subproblem (eTRS) in which the unit ball intersects with $m$ linear inequality constraints. In the literature, Burer et al. proved that an SOC-SDP relaxation (SOCSDPr) of eTRS is exact, under the condition that the nonredundant constraints do not intersect each other in the unit ball. Furthermore, Yuan et al. gave a necessary and sufficient condition for the corresponding SOCSDPr to be a tight relaxation when $m = 2$. However, there lacks a recovering algorithm to generate an optimal solution of eTRS from an optimal solution $X^*$ of SOCSDPr when rank $(X^*)≥ 2$ and $m≥ 3$. This paper provides such a recovering algorithm to complement those known works.

Citation: Jinyu Dai, Shu-Cherng Fang, Wenxun Xing. Recovering optimal solutions via SOC-SDP relaxation of trust region subproblem with nonintersecting linear constraints. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1677-1699. doi: 10.3934/jimo.2018117
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