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...cost experimenting and collecting data can lead one to wish to cast the problem as a multi-response model. It has been shown that the optimal designs to estimate the parameters of single response models are different from those of the corresponding multi-response model except in special circumstances (Shah et al., 2004). Thus, one has to construct multi-response optimal designs.
There exists an extensive literature on single response optimal design methods including analytical methods, optimization methods and recursive search algorithms (Fedorov, 1972; Cook and Nachtsheim, 1980; Atkinson and Donev, 1992; Pukelsheim, 1993; Gaffke and Heiligers, 1995). In the last 15 years evolutionary algorithms have also been applied to find single response optimal designs (Jung and Yum, 1996; Angelis et al., 2001; Borkowski, 2003; Drain et al., 2004; Heredia-Langner et al., 2004).
The development of techniques for multi-response experiments began with the original work of Draper and Hunter (1966) on the design of experiments for parameter estimation of multi-response models. Roy et al. (1971) extended classical designs to multi-response experimental designs. Later, Fedorov (1972) established a theoretical foundation for multi-response experiments and developed a recursive algorithm able to generate multi-response D-optimal designs. Chang (1994) studied the properties of D-optimal designs for multi-response models and proved that the optimal design of a multi-response model whose response functions have the same form coincide with that of a single response model of the same form. Khuri and Cornell (1996) devoted a chapter of their book to multi-response experiments and described Wijesinha's algorithm (Wijesinha, 1984) for generating D-optimal designs. Chang (1997) proposed an algorithm which generates near D-optimal designs for some special multi-response linear models. Other related papers such as those of Krafft and Schaefer (1992), Bischoff (1993), Imhof (2000), and Chang et al. (2001) report analytical solutions for specific problems. They do not, however, provide a general algorithm for solving this class of problems.
There exist three general algorithms able to generate multi-response D-optimal designs:
1. Fedorov's recursive algorithm constructs multi-response D-optimal designs when the variance-covariance matrix of the responses ([SIGMA]) is known.
2. The second algorithm is a sequential algorithm and is due to Wijesinha (1984) who extended the work of Fedorov (1972) to the case of an unknown [SIGMA].
3. The third one is Chang's algorithm that constructs near D-optimal designs for special cases of a multi-response model.
The main drawback of existing algorithms is that they involve the solution of many optimization problems in the process of generating an optimal design. The optimization problem has to be solved at each step of the algorithms for every design point added to a previously generated design as shown in Section 3.
The motivation of this research is to develop an alternative method which avoids this drawback. In this paper, we formulate a multi-response D-optimal design problem as a Semi-Definite Programming (SDP) model and solve a relaxed form of it to generate D-optimal designs. The proposed formulation is a one-shot optimization model whose solution selects the optimal design points among all possible points in the design space. It is an extension of the single response model developed by Boyd and Vandenberghe (2004) and can be solved efficiently using an SDP solver.
2. Definition and notation
2.1. Model definition
A linear multi-response model can be presented as
[Y.sub.i] = [f.sub.i.sup.T] (x)[[beta].sub.i] + [[epsilon].sub.i], i = 1,..., r, (1)
where [f.sub.i](x) is a vector representing the model form, x = ([x.sub.l],..., [x.sub.q]) is a design point for q input factors, [[beta].sub.i] is a vector of [p.sub.i] unknown parameters, and [[epsilon].sub.i] is a random variable associated with the ith response [Y.sub.i] and is correlated with [[epsilon].sub.j]. For example, a second-order model with one input variable at any design point x in the design space x [member of] R, may be written as [f.sub.i.sup.T](x) = (1, x, [x.sup.2]).
It is generally assumed that the error terms are normally distributed with the following parameters:
E([[epsilon].sub.i]) = 0,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)
where [SIGMA] is the variance-covariance matrix...
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