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Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method. (English) Zbl 1236.65070

The author considers a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the quadratic Hermitian matrix function ϕ(X)=Q-XPX * with respect to the variable matrix X by using a linearization method and some known formulas for extremum ranks and inertias of linear Hermitian matrix functions, where both P and Q are complex Hermitian matrices and X * is the conjugate transpose of X.

Examples are presented to illustrative applications of the equality-constrained quadratic optimization in some matrix completion problems.

65K05Mathematical programming (numerical methods)
15A09Matrix inversion, generalized inverses
15A24Matrix equations and identities
15A63Quadratic and bilinear forms, inner products
15B10Orthogonal matrices
15B57Hermitian, skew-Hermitian, and related matrices
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