On Φ<sub>p</sub>-optimality of Incomplete Block Designs: An Algorithm

Faghihi, Mohammad Reza; Tat, Samaneh

doi:10.1080/03610918.2013.873130

Cited by 2 publications

(1 citation statement)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Concerning optimality, this distinguished graph shows another interesting and unique character. In [18], an algorithm is developed which searches for optimal designs within D v,b,k . Implementing that algorithm, we looked for the optimal designs in the family of graphs with 10 vertices and 15 edges.…”

Section: Now We Letmentioning

confidence: 99%

On Optimality of Designs with Three Distinct Eigenvalues

Faghihi¹,

Ghorbani

Khosrovshahi³

et al. 2013

Electron. J. Combin.

Self Cite

View full text Add to dashboard Cite

Let D v,b,k denote the family of all connected block designs with v treatments and b blocks of size k. Let d ∈ D v,b,k . The replication of a treatment is the number of times it appears in the blocks of d. The matrix C(d) = R(d) − 1 k N (d)N (d) ⊤ is called the information matrix of d where N (d) is the incidence matrix of d and R(d) is a diagonal matrix of the replications. Since d is connected, C(d) has v − 1 nonzero eigenvalues µ1(d), . . . , µv−1(d). Let D be the class of all binary designs of D v,b,k . We prove that if there is a design d * ∈ D such that (i) C(d * ) has three distinct eigenvalues, (ii) d * minimizes trace of C(d) 2 over d ∈ D, (iii) d * maximizes the smallest nonzero eigenvalue and the product of the nonzero eigenvalues of C(d) over d ∈ D, then for all p > 0, d * minimizes v−1 i=1 µi(d) −p 1/p over d ∈ D. In the context of optimal design theory, this means that if there is a design d * ∈ D such that its information matrix has three distinct eigenvalues satisfying the condition (ii) above and that d * is E-and D-optimal in D, then d * is Φp-optimal in D for all p > 0. As an application, we demonstrate the Φp-optimality of certain group divisible designs. Our proof is based on the method of KKT conditions in nonlinear programming.

show abstract

Section: Now We Letmentioning

confidence: 99%