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A class of optimal design algorithms is based on the multiplicative updating of the weights of the current design measure ξ (k) with some function of ϕ(x, ξ), see Chapter 1 in this volume. 7). Assume that our measure is discrete and concentrated on [m, M ]. Assume also that ∂Φ(M )/∂μ2K−2 > 0; that is, the (K, K)-element of the matrix ∂Φ(M )/∂M is positive. Then ϕ(x, ξ) has a well-deﬁned minimum c(ξ) = min ϕ(x, ξ) > −∞ . 11) where b(ξ) is a normalizing constant M M ϕ(x, ξ) − c(ξ) ξ(dx) = b(ξ) = m ϕ(x, ξ)ξ(dx) − c(ξ) m ◦ = tr M (ξ) Φ (ξ) − c(ξ) .
Theorem 3. For almost all starting points x0 , the gradient algorithm corresponding to the A-optimality criterion for d = 2 has a superlinear convergence in the sense that the sequence of rates vk tends to 0 as k → ∞. If the dimension d is larger than 2, then the convergence of the optimization algorithm generated by the A-criterion is no longer superlinear. In fact, the algorithm tries to attract to the two-dimensional plane with the basis e1 , ed by reducing the weights of the designs ξ (k) at the other eigenvalues.
Q. 13) since T (t) has degree q − 1 ≤ s − 1. 46 L. Pronzato et al. Corollary 1. If x0 is such that ν0 is supported on n0 = s + 1 points at least, then (gk , gk ) > 0 for all k ≥ 1. 9), are strictly positive for all k ≥ 1. Theorem 3. Assume that ν0 is supported on n0 = s+1 points. Then ν2k = ν0 for all k. Proof. It is enough to prove that g2k is parallel to g0 , and thus that g2 is parallel to g0 . Since the updating rule only concerns nonzero components, we may assume that d = s + 1. We have g1 = Q0s (A)g0 , g2 = Q1s (A)g1 , g1 is orthogonal to g0 , Ag0 , .