COMPUTING EXACT D-OPTIMAL DESIGNS BY MIXED INTEGER SECOND-ORDER CONE PROGRAMMING

• 成果类型: Article
• 署名作者: Sagnol, Guillaume; Harman, Radoslav
• 署名单位: Zuse Institute Berlin; Comenius University Bratislava
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1339
• 发表日期: 2015
• 页码: 2198-2224
• 关键词: nonlinear models; CONSTRUCTION; algorithm; optimization

摘要

Let the design of an experiment be represented by an s-dimensional vector w of weights with nonnegative components. Let the quality of w for the estimation of the parameters of the statistical model be measured by the criterion of D-optimality, defined as the mth root of the determinant of the information matrix M(w) = Sigma(s)(i=1) w(i)A(i)A(i)(T), where A(i), i = 1, ... , s are known matrices with in rows. In this paper, we show that the criterion of D-optimality is second-order cone representable. As a result, the method of second-order cone programming can be used to compute an approximate D-optimal design with any system of linear constraints on the vector of weights. More importantly, the proposed characterization allows us to compute an exact D-optimal design, which is possible thanks to high-quality branch-and-cut solvers specialized to solve mixed integer second-order cone programming problems. Our results extend to the case of the criterion of D-K-optimality, which measures the quality of w for the estimation of a linear parameter subsystem defined by a full-rank coefficient matrix K. We prove that some other widely used criteria are also second-order cone representable, for instance, the criteria of A-, A(K)-, G- and I-optimality. We present several numerical examples demonstrating the efficiency and general applicability of the proposed method. We show that in many cases the mixed integer second-order cone programming approach allows us to find a provably optimal exact design, while the standard heuristics systematically miss the optimum.