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Hsia Yong(dearyxiagmail.com) Abstract: The trust region subproblem with a fixed number m additional linear inequality constraints, denoted by (T_m), have drawn much attention recently. The question as to whether Problem ( T_m) is in Class P or Class NP remains open. So far, the only affirmative general result is that (T_1) has an exact SOCP/SDP reformulation and thus is polynomially solvable. By adopting an early result of Martinez on local nonglobal minimum of the trust region subproblem, we can inductively reduce any instance in (T_m) to a sequence of trust region subproblems (T_0). Although the total number of (T_0) to be solved takes an exponential order of m, the reduction scheme still provides an argument that the class (T_m) has polynomial complexity for each fixed m. In contrast, we show by a simple example that, solving the class of extended trust region subproblems which contains more linear inequality constraints than the problem dimension; or the class of instances consisting of an arbitrarily number of linear constraints is NPhard. When m is small such as m=1,2, our inductive algorithm should be more efficient than the SOCP/SDP reformulation since at most 2 or 5 subproblems of (T_0), respectively, are to be handled. In the end of the paper, we improve a very recent dimension condition by Jeyakumar and Li under which (T_m) admits an exact SDP relaxation. Examples show that such an improvement can be strict indeed. Keywords: Trust region subproblem; Computational complexity; Nonconvex quadratic programming; Local nonglobal minimizer; Semidefinite relaxation; Hidden convexity Category 1: Global Optimization (Theory ) Category 2: Nonlinear Optimization (Quadratic Programming ) Citation: Download: [PDF] Entry Submitted: 12/01/2013 Modify/Update this entry  
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