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Robust Solutions To LeastSquares Problems With Uncertain Data
, 1997
"... . We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpret ..."
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Cited by 149 (13 self)
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. We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomialtime using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknownbutbounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worstcase residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Leastsquares, uncertainty, robustness, secondorder cone...
Full SignInvertibility and Symplectic Matrices
, 1993
"... An n \Theta n sign pattern H is said to be signinvertible if there exists a sign pattern H \Gamma1 (called the signinverse of H) such that, for all matrices A 2 Q(H), A \Gamma1 exists and A \Gamma1 2 Q(H \Gamma1 ). If, in addition, H \Gamma1 is signinvertible (implying (H \Gamma1 ) ..."
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An n \Theta n sign pattern H is said to be signinvertible if there exists a sign pattern H \Gamma1 (called the signinverse of H) such that, for all matrices A 2 Q(H), A \Gamma1 exists and A \Gamma1 2 Q(H \Gamma1 ). If, in addition, H \Gamma1 is signinvertible (implying (H \Gamma1 ) \Gamma1 = H), H is said to be fully signinvertible and (H; H \Gamma1 ) is called a signinvertible pair. Given an n \Theta n sign pattern H, a Symplectic Pair in Q(H) is a pair of matrices (A; D) such that A 2 Q(H);D 2 Q(H), and A T D = I. (Symplectic Pairs are a patterngeneralization of orthogonal matrices which arise from a special symplectic matrix found in nbody problems in celestial mechanics [1].) We discuss the digraphical relationship between a signinvertible pattern H and its signinverse H \Gamma1 , and use this to cast a necessary condition for full signinvertibility of H. We proceed to develop sufficient conditions for H's full signinvertibility in terms of allow...
Powers of Ray Pattern Matrices
"... We examine the similarities and differences between results for powers of sign pattern matrices and for powers of ray pattern matrices. In particular, we investigate what is known about kpotent patterns and powerful patterns in both the irreducible and reducible cases. 1 Signs, Rays, Sign & Ray Pat ..."
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We examine the similarities and differences between results for powers of sign pattern matrices and for powers of ray pattern matrices. In particular, we investigate what is known about kpotent patterns and powerful patterns in both the irreducible and reducible cases. 1 Signs, Rays, Sign & Ray Patterns Since at least the 1960’s, the qualitative theory of real matrices has been an extremely fruitful area of research. When posing qualitative questions, we ask what aspects of a matrix — such as stability, invertibility, controllability — are determined entirely by the sign pattern — the arrangement of positive, negative and zero entries — without reference to the magnitudes of the entries. During the last decade, there has been a vigorous effort to develop a corresponding qualitative theory of complex matrices. One of the main approaches has been to recognize that positivity and negativity naturally generalize to rays of fixed argument in the complex plane. That is, extending the view that all positive numbers as equivalent to the number +1, and all negative numbers as equivalent to the number −1, we will treat the complex ray consisting of all complex numbers of the form reiθ with r>0 and θ ∈ R as equivalent to eiθ. Of course, we need to introduce an appropriate arithmetic on such rays. Clearly, r1eiθ + r2eiθ will be equivalent to eiθ provided r1> 0 and r2> 0. In general, however, the choice of θ for r1eiθ1 + r2eiθ2 to be equivalent to eiθ will depend not only on θ1 and θ2, but also on r1 and r2. Consequently, we will require an additional symbol, #, to represent the result of adding nonzero complex numbers that lie on distinct rays. A ray operation that results in # will be called ambiguous. The complete arithmetic properties of # can be found