Strong duality for generalized trust region subproblem: S-lemma with interval bounds

The generalized trust region subproblem

The interval bounded generalized trust region subproblem (GTRS) consists in minimizing a general quadratic objective, q0(x) → min, subject to an upper and lower bounded general quadratic constraint, l ≤ q1(x) ≤ u. This means that there are no definiteness assumptions on either quadratic function. We first study characterizations of optimality for this implicitly convex problem under a constrain...

Solving the Trust-Region Subproblem By a Generalized Eigenvalue Problem

The state-of-the-art algorithms for solving the trust-region subproblem are based on an iterative process, involving solutions of many linear systems, eigenvalue problems, subspace optimization, or line search steps. A relatively underappreciated fact, due to Gander, Golub and von Matt in 1989, is that trust-region subproblems can be solved by one generalized eigenvalue problem, with no outer i...

Strong Duality for a Trust-Region Type Relaxation of QAP

Regularization using a parameterized trust region subproblem

We present a new method for regularization of ill-conditioned problems, such as those that arise in image restoration or mathematical processing of medical data. The method extends the traditional trust-region subproblem, TRS, approach that makes use of the L-curve maximum curvature criterion, a strategy recently proposed to find a good regularization parameter. We apply a parameterized trust r...

On SOCP/SDP Formulation of the Extended Trust Region Subproblem

We consider the extended trust region subproblem (eTRS) as the minimization of an indefinite quadratic function subject to the intersection of unit ball with a single linear inequality constraint. Using a variation of the S-Lemma, we derive the necessary and sufficient optimality conditions for eTRS. Then, an OCP/SDP formulation is introduced for the problem. Finally, several illustrative examp...