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Local nonglobal minima for solving large-scale extended trust-region subproblems. (English) Zbl 1391.90496
Summary: We study large-scale extended trust-region subproblems (eTRS) i.e., the minimization of a general quadratic function subject to a norm constraint, known as the trust-region subproblem (TRS) but with an additional linear inequality constraint. It is well known that strong duality holds for the TRS and that there are efficient algorithms for solving large-scale TRS problems. It is also known that there can exist at most one local non-global minimizer (LNGM) for TRS. We combine this with known characterizations for strong duality for eTRS and, in particular, connect this with the so-called hard case for TRS. We begin with a recent characterization of the minimum for the TRS via a generalized eigenvalue problem and extend this result to the LNGM. We then use this to derive an efficient algorithm that finds the global minimum for eTRS by solving at most three generalized eigenvalue problems.

90C26 Nonconvex programming, global optimization
90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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