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Additional info for Algorithm for approximating complex polynomial zeros (1998)
Such approach not only simpliﬁes the analysis, but sometimes enables us to obtain stronger results. Since the results of this section are useful also in the analysis of more general optimization problems that are locally approximated by quadratic problems, this chapter may also serve as a simple introduction to nonlinear optimization. Systematic exposition of the optimization theory in the framework of nonlinear optimization may be found in the books by Bertsekas , Nocedal and Wright , Conn, Gould, and Toint , or Bazaraa, Sherali, and Shetty .
26) implies dT Ad ≥ 0. Thus A|KerB must be positive semideﬁnite. 26) is determined by the sign of α(Ax − b)T d. 24) holds for any d ∈ KerB. 24) holds for a vector x ∈ ΩE and A|KerB is positive semideﬁnite. 18). 24) for any d ∈ KerB. 25) holds. Let A|KerB be positive semideﬁnite. 18). 8(i) may be easily modiﬁed to characterize the solutions of the minimization problem whose feasible set is a manifold deﬁned by a vector and a subspace; this modiﬁcation is often useful in what follows. 9. Let f be a convex quadratic function on Rn , let S be a subspace of Rn , and let x0 ∈ Rn .
15. 50) be deﬁned by a symmetric matrix A ∈ Rn×n , the constraint matrix B ∈ Rm×n whose column rank is less than n, and the vectors b, c. Let C denote the cone of directions of the feasible set ΩI . 52) for any feasible direction d of ΩI at x. 50), then there is λ ∈ Rm such that λ ≥ o, Ax − b + BT λ = o, and λT (Bx − c) = 0. 53) Proof. 54) 2 is nonnegative for all suﬃciently small α > 0. 54) implies that α(Ax − b)T d ≥ 0.