By L. R. A. Casse, W. D. Wallis
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2y0 Since the above quadratic form is nonnegative for any h ∈ Rn+1 , it follows that the matrix Toep(2y0 , y1 , . . , yn ) is positive semidefinite. Knowing the form of the dual cone, we can build easier the duals of optimization problems with nonnegative trigonometric polynomials. 22), where, for simplicity we take Γ = I. 33) is g(y) = inf f (r) − y T r , r where the Lagrangean multiplier y belongs to the dual cone. The minimum is obtained trivially for y = 2(r − rˆ ) and so 1 g(y) = − y T y − y T rˆ .
2 .. .. i=k k=0 yn yn−1 . . 2y0 Since the above quadratic form is nonnegative for any h ∈ Rn+1 , it follows that the matrix Toep(2y0 , y1 , . . , yn ) is positive semidefinite. Knowing the form of the dual cone, we can build easier the duals of optimization problems with nonnegative trigonometric polynomials. 22), where, for simplicity we take Γ = I. 33) is g(y) = inf f (r) − y T r , r where the Lagrangean multiplier y belongs to the dual cone. The minimum is obtained trivially for y = 2(r − rˆ ) and so 1 g(y) = − y T y − y T rˆ .
Q 0, y ≤ α This is a standard SQLP problem in equality form. 13 (Complexity issues) As discussed in Appendix A, the complexity of an SDP problem in equality form is O(n2 m2 ), where n × n is the size of the variable positive semidefinite matrix and m is the number of equality constraints. 23), respectively) do not change significantly the complexity. Since the size of the Gram matrix Q is (n+1)×(n+1) and the number of equality constraints is n+1, we can appreciate that the complexity of the three problems—Most positive Gram matrix, Min poly value and Nearest autocorrelation—formulated in SDP form in this section is O(n4 ).