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Inertia Conditions for the Minimization of Quadratic Forms

We study the relation between the solutions of two minimization problems with indefinite quadratic forms. We show that a complete link between both solutions can be established by invoking a fundamental set of inertia conditions. While these inertia conditions are automatically satisfied in a standard Hilbert space setting, which is the case of classical least-squares problems in both the deterministic and stochastic frameworks, they nevertheless turn out to mark the differences between the two optimization problems in indefinite metric spaces. Applications to H∞-filtering, robust adaptive filtering, and approximate total-least-squares methods are included.

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