2024/09/03 by Ai, Wenbao, Zhang, Mengxiao, Yuan, Jianhua
#90C20 #90C26 #90C32 #90C46 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2409.01697
In this paper, we study the local-nonglobal minimizers of the Generalized Trust-Region subproblem (GTR) and its Equality-constrained version (GTRE). Firstly, the equivalence is established between the local-nonglobal minimizers of both (GTR) and (GTRE) under assumption of the joint definiteness. By the way, a counterexample is presented to disprove a conjecture of Song-Wang-Liu-Xia [SIAM J. Optim., 33(2023), pp.267-293]. Secondly, if the Hessian matrix pair is jointly positive definite, it is proved that each of (GTR) and (GTRE) has at most two local-nonglobal minimizers. This result first confirms the correctness of another conjecture of Song-Wang-Liu-Xia [SIAM J. Optim., 33(2023), pp.267-293]. Thirdly, if the Hessian matrix pair is jointly negative definite, it is verified that each of (GTR) and (GTRE) has at most one local-nonglobal minimizer. In special, if the constraint is reduced to be a ball or a sphere, the above result is just the classical Martínez's. Finally, an algorithm is proposed to find all the local-nonglobal minimizers of (GTR) and (GTRE) or confirm their nonexistence in a tolerance. Preliminary numerical results demonstrate the effectiveness of the algorithm.